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Calculus 1 / AB
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September 4, 2025
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September 4 of 2025
a) For all $n > 1, 0 \le \frac{\sin^2(n)}{n^2} \le \frac{1}{n^2}$, and the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so by the Comparison Test, the series $\sum_{n=1}^{\infty} \frac{\sin^2(n)}{n^2}$ converges. Correct b) For all $n > 1, 0 \le \frac{\arctan(n)}{n^3} <…
Do the following with the given information. $$ \int_{0}^{1} 39 \cos \left(x^{2}\right) d x $$ (a) Find the approximations $$ T_{8} $$ and $$ M_{8} $$ for the given integral. (Round your answer to six decimal places.) $$ T_{8}= $$ $$ M_{8}= $$ (b) Estimate the errors in the approximations $$…
1. 2.26 / 5.26 Points DETAILS MY NOTES PREVIOUS ANSWERS ASK YOUR TEACHER SCalcET9 9.1.016. (a) For what values of k does the function $y = \cos(kt)$ satisfy the differential equation $25y'' = -81y$? (Enter your answers as a comma-separated list.) $k = \frac{9}{5}, -\frac{9}{5}$ (b) For those…
2. (3 pts total) We know that tangent lines approximations typically get worse the further you go from the point $(a, f(a))$. However, this is not always the case. In fact, sometimes, the tangent line can even touch the function at more than one point. Recall that the error of a tangent line…
(a) Find the approximations $T_{10}$, $M_{10}$ and $S_{10}$ for $\int_{0}^{\pi} 32 \sin(x) dx$. (Round your answers to six decimal places.) $T_{10} = 63.472753$ $M_{10} = 64.263949$ $S_{10} = 64.003506$ Find the corresponding errors $E_T$, $E_M$ and $E_S$. (Round your answers to six decimal…
Question 7 If g is differentiable at $(l, m, n)$, then the normal line to the surface given by $g(x, y, z) = 0$ at $(l, m, n)$ can be expressed as $x = l + g_x(l, m, n)t$, $y = m + g_y(l, m, n)t$, $z = n + g_z(l, m, n)t$, $t \in \mathbb{R}$ $\frac{x - l}{g_x(l, m, n)} = \frac{y - m}{g_y(l, m,…
The graph below illustrates approximating rectangles with left endpoints for $f(x) = (16/x)$ on the interval $[2, 6]$. The estimated area based on these rectangles is and this sum is an overestimate of the area of the region enclosed by $y = f(x)$, the x-axis, and the vertical lines $x = 2$ and…
Find the general solution in powers of z of the differential equation $$(z^2 - 1)y'' + 4zy' + 2y = 0$$ Assume the form $$y(z) = \sum_{n=0}^{\infty} c_n z^n$$ Then $$y'(z) = \sum_{n=1}^{\infty} n c_n z^{n-1}$$ $$y''(z) = \sum_{n=2}^{\infty} n(n-1) c_n z^{n-2}$$ $$z^2 y''(z) = \sum_{n=2}^{\infty}…
Tools Window Help signment 3.pdf Q Search ing out this form. AMAT 112 Calculus 1 Written Assignment 3 Question 1) You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16…
Which of the following equations will determine the centroidal x-axis for the beam cross sections shown in the Figure below? 0.441" S 200 x 34 A = 6.77 in2 12.5" y W 310 x 52 A = 10.3 in2 A. $$ \bar{y} = \frac{6.77 \left( 12.5 + \frac{0.441}{2} \right) + 10.3 \left( \frac{12.5}{2} \right)}{6.77…
7. Use Stokes' Theorem to evaluate: (a) 4 marks $$ \iint_S \text{curl} \vec{F} \cdot d\vec{S} $$ where $$ \vec{F}(x, y, z) = ze^y \vec{i} + x \cos y \vec{j} + xz \sin y \vec{k}, $$ S is the hemisphere $$ x^2 + y^2 + z^2 = 16, y \geq 0, $$ oriented in the direction of the positive y-axis. (b) 4…
An Energy Drink company wants to design a new can for their new product. The material to be used for the can is aluminum; and is cylindrical in shape. Design a can that requires the least amount of aluminum to be used; and can contain exactly 16 fluid ounces (28.875 cubic inches). Find the…
Find a parametrization for the curve. The upper half of the parabola x - 2 = y² Choose the correct answer below. A. x=t, y=t² + 2, t≤2 B. x=t² + 2, y=t, t≤0 C. x=t² + 2, y=t, t≥0 D. x=t, y=t² - 2, t≥0 E. x=t² - 2, y=t, t≥2 F. x=t, y=t² - 2, t≥2
2. (Chapter 17, Section 19.1) Consider the intersection R between the two circles $$x^2 + y^2 = 2$$ and $$(x - 2)^2 + y^2 = 2$$. y R x (a) Find a 2-dimensional vector field $$F = (M(x, y), N(x, y))$$ such that $$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 1$$. (3) (b) Using…
Find the approximations $T_n$, $M_n$, and $S_n$ for $n = 6$ and 12. Then compute the corresponding errors $E_T$, $E_M$, and $E_S$. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) $\int_1^4 \frac{27}{\sqrt{x}} dx$ $n$ $T_n$ $M_n$…
Previous Problem Problem List Next Problem Homework 8: Problem 1 (1 point) Use Green's Theorem to find the counterclockwise circulation of the vector field $\mathbf{F} = \langle 3xy, 5x + 2y \rangle$ along the curve $C$, where $C$ is the triangle with vertices $(0,0)$, $(2,0)$, and $(0,2)$. To…
We are asked to use polar coordinates to find the volume under the paraboloid $z = x^2 + y^2$ and above the disk $x^2 + y^2 \leq 49$. Recall that as the paraboloid is given in the form $z = f(x, y)$, the volume below $f(x, y)$ and above the given disc is given by the following. $V = \iint_D…
Evaluate the integral. $$ \int \frac{\sqrt{y^2 - 64}}{y} dy, y > 8 $$ Which substitution transforms the given integral into one that can be evaluated directly in terms of $$ \theta $$? O A. $$ y = 8 \sec \theta $$ O B. $$ y = 8 \sin \theta $$ O C. $$ y = 8 \tan \theta $$ Given the expression…
For each function below find the requested partial derivatives. Note that $f_x$ is equivalent to $\partial f/\partial x$. If $f(x,y) = x^6 y + x^3 + 3$ then $f_x =$ $f_y =$ If $f(x,t) = \ln(t) \sqrt{x}$ then $f_x =$ $f_t =$ Remember that square roots can be entered using "sqrt" (e.g. $\sqrt{2}$…
Express D as a region of type II. $$D = \{(x, y) | 0 \le y \le x, y \le x \le 3\}$$ $$D = \{(x, y) | 0 \le y \le x, 0 \le x \le y\}$$ $$D = \{(x, y) | 0 \le y \le 3, 0 \le x \le y\}$$ $$D = \{(x, y) | 0 \le y \le 3, y \le x \le 3\}$$ $$D = \{(x, y) | 0 \le y \le 3, 0 \le x \le 3\}$$ Evaluate…
The given curve is rotated about the x-axis. Set up, but do not evaluate, an integral for the area of the resulting surface by integrating (a) with respect to x and (b) with respect to y. $$x = \ln(8y + 1), \quad 0 \le y \le 1$$ (a) Integrate with respect to x. $$\int_{0}^{\ln(9)} \left( \left(…
2. A cable company determines that the number of hours of service N needed to install a new line is $N = f(d)$, where d is the distance in meters from the nearest service hub. Write mathematical expressions for the following statements. (a) The company estimates the number of hours needed and…
Compute area of the region bounded by $y = 3x^2 - 4x$ and $y = 2x$. The figure below gion(shaded) and coordinates of the intersection points of the two curves. $y$ $(2, 4)$ $x$ $-1$ $-0.5 (0, 0)$ $0.5$ $1$ $1.5$ $2$ $2.5$ $3$ $-1$ $1$ $2$ $3$ $4$ $5$
Question 4 Mark this question The equation $y = -16t^2 + 40t + 9$ represents the height of a projectile, $y$, in feet at a particular time $t$, in seconds. For what interval (or intervals) of time will the projectile's height be above 25 feet? Between 0.5 and 2 seconds Between 1 and 3…
7. [0 / 1 Points] Use part one of the fundamental theorem of calculus to find the derivative of the function. $f(x) = \int_{x}^{0} \sqrt{6 + \sec(3t)} dt$ [Hint: $\int_{x}^{0} \sqrt{6 + \sec(3t)} dt = - \int_{0}^{x} \sqrt{6 + \sec(3t)} dt$] $f'(x) = |$ Check the plus and minus signs of all…
Question Comp Use a change of variables or the table of general integration formulas to evaluat 8 $$ \int_{7}^{8} \frac{x}{\sqrt[3]{x^{2}-8}} d x $$ Click to view the table of general integration formulas. 8 $$ \int_{7}^{8} \frac{x}{\sqrt[3]{x^{2}-8}} d x = $$ (Type an exact answer.) Get more…
SCalcET9 11.10.01 Find the Maclaurin series for $f(x)$ using the definition of a Maclaurin series. [Assume that $f$ has a power series expansion. Do not show that $R_n(x) \to 0$.] $f(x) = 3^x$ $f(x) = \sum_{n=0}^{\infty} (\quad)$ Find the associated radius of convergence $R$. $R = \quad$
Determine the moment of the strip area $A_2$ about the x-axis in the Figure below. y $A_2$ 4" $A_1$ 2" $A_3$ 4" x 2" 2" 3" A. $A_2d_2 = 7 \times 4 \times 8$ in.$^3$ B. $A_2d_2 = 7 \times 4 \times 2$ in.$^3$ C. $A_2d_2 = 7 \times 4 \times 10$ in.$^3$ D. $A_2d_2 = 7 \times 4 \times 3.5$…
ASSIGNMENT #1 Classify the order, degree, and linearity of the following DE. 1. $f'''(x) + 3t^3f''(x) = \cos 5t$ 2. $y''' + 3x^2y'' + \ln y' = 0$ 3. $f(x)f''(x) + f'(x) = 0$ 4. $y''' + yy'' + 2y^2 + \sin(y'') + e^ty = 0$ 5. $3x^2y'' + 2\ln(x)y' + e^xy = 3x \cos x$ 6. $4yy''' - x^3y' + \cos y =…
Student Enrollment The enrollment at a local college has been decreasing linearly. In 2000, there where 830 students enrolled. By 2005, there were only 575 students enrolled. Determine the average rate of change of the school's enrollment during this time period, and write a sentence explaining…
An object was launched from the ground. The height $h$ of the object, in meters, above the ground $t$ seconds after it was launched can be modeled by the function $h(t) = -4.9t^2 + 29.4t$, where $0 \le t \le 6$. According to the model, for which of the following values of $t$ was the height of…
Use Stokes' Theorem to compute the counterclockwise circulation of the vector field $\mathbf{F} = \langle 4y, 1z, 4x \rangle$ along the curve $C$, where $C$ is the rectangle with vertices $(0, 0, 3)$, $(2, 0, 3)$, $(2, 5, 3)$, and $(0, 5, 3)$. To apply Stokes' Theorem, you must first find the…
Survey Final Portal Forensic Scien Flower Deliv Order Food Basic quizzes/89664/take Page Mathematics 2450, Calculus 3, Final Exam Show all details. A correct answer with no work (or unreadable work) counts as zero. 1. Let the velocity vector be $v(t) = \sin t \mathbf{i} + e^{2t} \mathbf{j} -…
Laplace Transform 1. [Medium] (Laplace Transform (Higher order)) Use Laplace transform method to find the PS of $$ \begin{cases} y''' + y'' - y' - y = 1 + \cos x + \cos 2x + e^x \\ y(0) = y'(0) = y''(0) = 0 \end{cases} $$
ourses/2387204/quizzes/5372830/take/questions/97713564 Question 6 10 pts 6. A cube of ice is melting so that the edge, x, is decreasing at the rate of 2 inches per hour. Write the equation for the volume, V, first and then determine how fast the volume of the ice is decreasing per hour at the…
Use the given graph of the function $f$ to answer the following questions. 1. Find the open interval(s) on which $f$ is concave upward. Answer (in help (intervals) ): $(0,2)U(5,8)$ 2. Find the open interval(s) on which $f$ is concave downward. Answer (in help (intervals) ): $(2,4)U(4,5)$ 3.…
1. 6 marks Verify Stokes' theorem (i.e. show that the line integral of F over curve $C = $ Double integral of curl F.$\hat{n}$ over the surface) if $F = < y, z, x >$ and S is the portion of the plane $x + y + z = 0$ cut out by the cylinder $x^2 + y^2 = 1$, and C is its boundary (an ellipse).
Consider the vector field $\vec{F}$ and the curve C below. $\vec{F}(x, y) = (7 + 4xy^2)\vec{i} + 4x^2y\vec{j}$, C is the arc of the hyperbola $y = 1/x$ from $(1, 1)$ to $(3, \frac{1}{3})$ (a) Find a potential function $f$ such that $\vec{F} = \nabla f$. $f(x, y) = $ (b) Use part (a) to…
6:22 Test 3 MULTIPLE CHOICE 1/1 CORRECT Write the equation of a line that is perpendicular to the straight line $$y = \frac{13}{11}x - \frac{19}{23}$$ that goes through the point $$(\frac{4}{9}, -\frac{5}{11})$$ Correct: $$y = -\frac{11}{13}x - \frac{101}{1287}$$ Correct answer B $$y =…
View History Bookmarks Window Help buckeyelink3.osu.edu Question 20 of 25 Solve for x: $|x - 4| > 6$. A) $-10 < x < 2$ B) $x > 10$ C) $x > 10$ or $x < -2$ D) $-2 > x > 10$ E) None of the above Go to question: Go Prev Click the Go button. ('Enter' or 'Return' will take you to Question #1). Next
Company E-Denim sells jeans. Their weekly revenue is modeled by the function $R(x) = -x^2 + 80x$. Their weekly expenses are modeled by the cost function $C(x) = 0.5x^2 + 10x + 300$, where $x$ represents the number of pairs of jeans sold per week. Find the optimal number of jeans E-Denim must…
8 Multiple Choice 1 point For which values of a and b does the function $f(x) = \frac{ax}{b+x^2}$ have a critical point at $x = 2$ and $x = -2$? $a = 1, b = 4$ $a = 1, b = 1$ $a = 2, b = 2$ $a = 4, b = 2$ $a = 1, b = 3$ Clear my selection
Use the Ratio Test to determine whether the series converges absolutely or diverges. $$ \sum_{k=1}^{\infty} \frac{2^k}{3^k} $$ Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer in simplified form.) A. The series converges absolutely…
Page Mathematics 2450, Calculus 3, Final Exam Show all details. A correct answer with no work (or unreadable work) counts as zero. 1. Let the velocity vector be $v(t) = \sin t i + e^{2t} j - 2t k$, and the initial position vector be $r(0) = -i + 2j - 2k$. Compute the acceleration vector $a(t)$,…
Question 2 A company that produces cell phones has a cost function of $C = 6z^2 - 145z + 14906$, where C is the cost in dollars and z is the number of cell phones produced (in thousands). How many units of cell phone (in thousands) minimizes the cost function? $z = $ thousand phones produced…
12.2.2 Let $u(x, y) = y^3 - 3x^2y$ and $v(x, y) = x^3 - 3xy^2$. (a) Show that $\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$ on $\mathbb{R}^2$. (b) Show that $\frac{\partial^2v}{\partial x^2} + \frac{\partial^2v}{\partial y^2} = 0$ on $\mathbb{R}^2$. (c) Show that…
2. [Medium] (Laplace Transform (System)) Use Laplace transform method to find the PS of $$ \left\{ \begin{array}{l} \begin{bmatrix} x \\ y \end{bmatrix}' = \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + 25 \begin{bmatrix} \cos t \\ \sin t \end{bmatrix} \\…
The force exerted by an electric charge at the origin on a charged particle at a point $(x, y, z)$ with position vector $\mathbf{r}=\langle x, y, z\rangle$ is $\mathbf{F}(r)=\frac{K \mathbf{r}}{|\mathbf{r}|^{3}}$ where $K$ is a constant. (See this example.) Find the work done as the particle…
Problem 1. (20 points) Consider a circle A of radius 1 centered at (0,0) and another circle B of radius $0 < r < 1$ centered at (0,1). Compute the area lying inside B but outside A. Draw a picture of this configuration of circles and shade the area to be computed.
ScalcET9 15.8.0 Use spherical coordinates. Evaluate $$ \iiint_E \sqrt{x^2 + y^2 + z^2} \, dV $$ where E lies above the cone $$ z = \sqrt{x^2 + y^2} $$ and between the spheres $$ x^2 + y^2 + z^2 = 1 $$ and $$ x^2 + y^2 + z^2 = 16. $$
The acceleration function (in m/s$$^2$$) and the initial velocity v(0) (in m/s) are a(t) = 2t + 2, v(0) = -15, 0 $$ \le $$ t $$ \le $$ 5 (a) Find the velocity (in m/s) at time t. v(t) = m/s (b) Find the distance traveled (in m) during the given time interval. m
Step 5 We have found y in terms of Inverse Laplace transforms and the resulting transforms as follows. $y(t) = \frac{2}{125}\mathcal{L}^{-1}\left\{\frac{1}{s}\right\} + \frac{1}{25}\mathcal{L}^{-1}\left\{\frac{1}{s^2}\right\} - \frac{2}{125}\mathcal{L}^{-1}\left\{\frac{1}{s-5}\right\} +…
To solve the given differential equation, integrate both sides of the separated equation. We can do so as we integrate a of x with respect to x on the right side of the equation. Each integration requires a simple substitution. Let $u = 4y + 5$ and $v = 8x + 9$. Then $dy = \frac{du}{4}$ and $dx…
The heat capacity $C(T)$ of a substance is the amount of energy (in joules) required to raise the temperature of $1 \mathrm{~g}$ by $1^{\circ} \mathrm{C}$ at temperature $T$. How much energy is required to raise the temperature from $33^{\circ} \mathrm{C}$ to $73^{\circ} \mathrm{C}$ if…
View History Bookmarks Window Help buckeyelink3.osu.edu Question 19 of 25 A wheel makes 35 revolutions each second. Find its approximate velocity in radians per second. A) 110 B) 11 C) 6 D) 220 E) 35 Go to question: Go Click the Go button. (Enter' or 'Return' will take you to Question…
Populations of aphids and ladybugs are modeled by the following equations. $$ \frac{dA}{dt} = 2A - 0.01AL $$ $$ \frac{dL}{dt} = -0.5L + 0.0001AL $$ (a) Find the equilibrium solutions. smaller A-value $$(A, L) = (\square)$$ larger A-value $$(A, L) = (\square)$$ (b) Find an expression for $$…
3. The parabolic reflector of a satellite dish is 1 meter deep and 8 meters wide. The origin (0,0) is at the vertex of the parabola. (a) Write an equation that models the cross-section of the satellite dish. (b) Find the depth of the reflector at a distance of 2 meters from the center.
A line in the $xy$-plane has the equation $y = mx + 6$, where $m$ is a constant and $3 \le m \le 4$. Which of the following values could be the $x$-intercept of the line? Indicate all such values. $\Box -3$ $\Box -2$ $\Box -\frac{7}{4}$ $\Box -\frac{5}{4}$ $\Box \frac{5}{4}$ $\Box…
We are asked to find the Maclaurin series for a function involving cos(x). Recall the Maclaurin series for cos(x). $$cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$$ The same equality would be true for any variable, and in particular for $$u = \frac{1}{13}x^2$$. Therefore, the…
Business and Social Sciences Compound Interest Homework Question 7, 5.1.2 Part 1 of 2 Find the compound amount and the amount of interest earned by the deposit below. $4,000 at 3.61% compounded continuously for 5 years. What is the compound amount? $ (Do not round until the final answer. Then…
8. DETAILS MY NOTES ASK YOUR TEACHER Determine the set of points at which the function is continuous. $F(x, y) = \cos(\sqrt{1 + x - y})$ $\{(x, y)|y \ge x\}$ $\{(x, y)|y > -x\}$ $\{(x, y)|y \ge x - 1\}$ $\{(x, y)|y \le x + 1\}$ $\{(x, y)|y > x + 1\}$
2. DETAILS MY NOTES ASK YOUR TEACHER Determine whether or not $\mathbf{F}$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$. (If the vector field is not conservative, enter DNE.) $\mathbf{F}(x, y)=\left(y^{4} \cos (x)+\cos (y)\right)…
1. DETAILS MY NOTES ASK YOUR TEACHER Use Green's theorem to evaluate $\int_C \mathbf{F} \cdot d\mathbf{r}$. (Check the orientation of the curve before applying the theorem.) $\mathbf{F}(x, y) = \langle y - \cos(y), x \sin(y) \rangle$, C is the circle $(x - 6)^2 + (y + 2)^2 = 16$ oriented…
Evaluate the triple integral. $$ \iiint_E 2xy \, dV $$ where E lies under the plane $$ z = 1 + x + y $$ and above the region in the xy-plane bounded by the curves $$ y = \sqrt{x}, y = 0, $$ and $$ x = 1 $$
Question 10 10. Evaluate the definite integral $$ \int_{1}^{6} \frac{14}{z^{4}} dz $$ A) $$ \frac{1519}{324} $$ B) $$ \frac{301}{108} $$ C) $$ \frac{1519}{540} $$ D) $$ \frac{215}{217} $$ E) $$ \frac{1505}{324} $$
16. DETAILS MY NOTES ASK YOUR TEACHER Consider the following. $$ \iiint_E z \, dV $$ where E is enclosed by the paraboloid $$ z = x^2 + y^2 $$ and the plane $$ z = 16 $$ Write the above using cylindrical coordinates. (Choose $$ 0 < A \le 2\pi $$. Choose $$ 0 < B $$. Choose $$ C < 16 $$.) $$…
10 pts 5. After t weeks of practice, a typing student can type $100(1-e^{-0.2t})$ words per minute (wpm). How soon will the student type 80 wpm? Round your answer to the nearest number of weeks. A) 6 weeks B) 5 weeks C) 7 weeks D) 9 weeks E) 8 weeks
[5 Points] One of the critical points of the function $(x, y) = 2 - x^3 + xy - \frac{y^2}{6}$ is $(x,y)=(1,3)$. Use the Second Derivative Test to determine if $(x, y) = (1,3)$ is a relative maximum, relative minimum, or saddle. Show all your work for full credit.
Question 18 Consider the equation $$4x^2 - 17x - 10 = 5.$$ Find the solutions by using the quadratic formula. $$x = 0.75$$ and $$x = 5$$ $$x = -0.75$$ and $$x = 5$$ $$x = -5$$ and $$x = 0.75$$ $$x = -5$$ and $$x = -0.75$$
A volume is described as follows: 1. the base is the region bounded by $x = -y^2 + 6y + 45$ and $x = y^2 - 18y + 99$; 2. every cross section perpendicular to the y-axis is a semi-circle. Find the volume of this object. volume =
Question 20 5 pts Which of the following is NOT the result of $$ \int (\tan x) (\sec^2 x) dx $$ $$ \frac{1}{2 \cos^2 x} + C $$ $$ \frac{\sec^2 x}{2} + C $$ $$ \frac{1}{2 \sin^2 x} + C $$ $$ \frac{\tan^2 x}{2} + C $$
Solve the given initial-value problem $y'' + 4y = g(x)$, $y(0) = 1$, $y'(0) = 1$, where $g(x) = \begin{cases} \sin(x), & 0 \le x \le \pi/2 \\ 0, & x > \pi/2 \end{cases}$ $g(x)$ is discontinuous. [Hint: Solve the problem on two intervals, and then find a solution so that $y$ and $y'$ are…
b. The rectangles in the graph below illustrate a right endpoint Riemann sum for $f(x) = -\frac{x^2}{6} + 2x$ on the interval $[3, 7]$. The value of this right endpoint Riemann sum is , and this Riemann sum is [select an answer] the area of the region enclosed by $y = f(x)$, the x-axis, the…
Compute or approximate the corresponding function values and derivative values for the given area function. $G(x) = \int_{0}^{x} \sqrt{196 - t^2} dt$ In some cases, approximations will need to be done by using geometry. (Give exact answers. Use symbolic notation and fractions where…
Question 1: Classify the differential equation. $$ \frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^2 + xy = 1 $$ A) first-order, non-linear ordinary differential equation. B) second-order, non-linear ordinary differential equation. C) second-order, linear ordinary differential equation. D)…
3. (2 pts) Suppose we are maximizing some unknown quantity $f(x)$ on the domain $0 \leq x \leq 10$ and we find that $a$ is a critical point of $f$. It's possible that $a$ is the solution to our optimization problem, but it might not be. For what reasons might $a$ not be the solution? Name 4…
8. Use Green's theorem to evaluate the line integral $$I = \oint_C y^2 dx + 3xy^2 dy,$$ around the closed curve C consisting of the boundary of the triangle with vertices given in the following order (0,0) to (2,0) to (1,1) and back to (0,0). Answer:
Question 3 of 25 A solution of $$\frac{x-5}{2} + \frac{2}{x+5} = \frac{17}{4}$$ is A) $$-\frac{3}{2}$$ B) $$-3$$ C) $$4$$ D) $$0$$ E) None of the above Go to question: Go Click the Go button. (Enter or Return will take you to Question #1). Prev Next
Find the average rate of change of $f(x) = 2x^2 + 5$ over each of the following intervals. (a) From 1 to 3 (b) From 2 to 4 (c) From 3 to 6 (a) The average rate of change from 1 to 3 is
Question 14 5 pts $$ \int \sin^3 x dx = $$ $$ -\frac{\cos^3 x}{3} + \cos x + C $$ $$ \frac{\cos^3 x}{3} - \cos x + C $$ $$ \frac{\sin^4 x}{4} + C $$ $$ \frac{\cos^3 x}{3} - x + C $$
10 pts 7. Use algebra to rewrite the integrand; then integrate and simplify. $$ \int \frac{x-4}{\sqrt{x}} dx $$ A) $$ \frac{2}{3} x\sqrt{x}-4\sqrt{x}+C $$ B) $$ \frac{2}{3} x\sqrt{x}-8\sqrt{x}+C $$ C) $$ \frac{2}{3} x\sqrt{x}-12\sqrt{x}+C $$ D) $$ \frac{1}{2} x^2-12\sqrt{x}+C $$ E) $$…
Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem. $y''(\theta) + 10y(\theta)^3 = \sin \theta$; $y(0) = 0$, $y'(0) = 0$ The Taylor approximation to three nonzero terms is $y(\theta) = \boxed{\phantom{}}$ + ...
Find the volume of the wedge-shaped region (Figure 1) contained in the cylinder $x^2 + y^2 = 16$ and bounded above by the plane $z = x$ and below by the $xy$-plane. z = x z y x FIGURE 1 V =
Express this alternating sum $S$ as a simple fraction: $S = 1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} \pm \dots + \frac{1}{4782969}$. Your answer must be a simplified fraction. You may use a calculator for reducing the fraction. Answer: $S = 21523361/28697814$
Question For the series $$ \sum_{n=1}^{\infty} \frac{1}{n^2 - 5} $$, determine which of the following convergence tests is the best to use. Select the correct answer below: The integral test. The alternating series test. The ratio test. The comparison test. SUBMIT
Evaluate the following partial derivatives using the Fundamental Theorem of Calculus. For $f(x, y) = \int_{x}^{y} \sin \left(\frac{1}{t}\right) dt$, compute $f_x = $ $f_{xx} = $ For $f(x, y) = \int_{1}^{xy} \sqrt{1+t^3} dt$, compute $f_y = $ $f_{yy} = $
Question 5 5 pts The fraction $$ \frac{ax^3+bx^2+cx+d}{(x^2+1)(x^2-4)} $$ will take which form when expressed as partial fractions? $$ \frac{Ax+B}{x^2+1} + \frac{Cx+D}{x^2-4} $$ $$ \frac{A}{x^2+1} + \frac{B}{x+2} + \frac{C}{x-2} $$ $$ \frac{Ax+B}{x^2+1} + \frac{C}{x^2-4} $$ $$…
Identify the range of the exponential function $y = 10^x$ Select one: O a. $\{y|y < 0, y \in R\}$ O b. $\{y|y > 0, y \in R\}$ O c. $\{y|y \neq 0, y \in R\}$ O d. $\{y|y \in R\}$
Point-masses $m_i$ are located on the $x$-axis as shown. Find the moment $M$ of the system about the origin and the center of mass $\bar{x}$. $m_1 = 16$ $m_2 = 17$ $m_3 = 25$ $M = 186$ $\bar{x} = $
Evaluate the surface integral. $$ \iint_S xyz \, dS $$ S is the cone with parametric equations $$ x = u \cos(v), y = u \sin(v), z = u, 0 \le u \le 2, 0 \le v < \frac{\pi}{2} $$
Evaluate each definite integral. Enter either an exact answer or an answer rounded to three decimal places. (a) $$ \int_{1}^{4} (96x^{-3} - 10x + 4)dx = $$ (b) $$ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (8 \sin t + 8 \csc^2 t)dt = $$
Verify the identity. (Simplify at each step.) $$ \tan \left(\sin ^{-1}\left(\frac{x-1}{2}\right)\right)=\frac{x-1}{\sqrt{4-(x-1)^{2}}} $$ Let $$ \theta=\sin ^{-1}\left(\frac{x-1}{2}\right) \Rightarrow \sin (\theta)=\frac{x-1}{2} $$ $$ \tan \left(\sin ^{-1}\left(\frac{x-1}{2}\right)\right)=\tan…
Compute the flow of $\vec{F} = \langle x, y \rangle$ along and across $C : \vec{r}(t) = \langle 3 \cos(t), 3 \sin(t) \rangle$ for $\frac{\pi}{3} \leq t \leq 2\pi$. Flow = Next Question
The doubling period of a bacterial population is 15 minutes. At time $t = 100$ minutes, the bacterial population is 6,000. a. Determine the initial population. Bacteria b. Find the bacterial population after 4 hours. Bacteria
Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$, where C is given by the vector function $\mathbf{r}(t)$. $\mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} + xy\mathbf{k}$, $\mathbf{r}(t) = \cos(t)\mathbf{i} + \sin(t)\mathbf{j} + t\mathbf{k}$, $0 \le t \le \pi$
Find the volume of the given solid. enclosed by the paraboloid $z = 6x^2 + 4y^2$ and the planes $x = 0, y = 0, z = 0$ and $x + y = 1$.
Find the radius of convergence, R, of the series. R = $$ \sum_{n=3}^{\infty} \frac{x^{n+4}}{4n!} $$ Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I =
3. DETAILS MY NOTES ASK YOUR TEACHER Find the area of the surface. The part of the sphere $x^2 + y^2 + z^2 = 64$ that lies above the plane $z = 6$.
SCalcET9 16.2.003.MI. Evaluate the line integral, where C is the given plane curve. $$ \int_C xy^4 ds $$ C is the right half of the circle $$ x^2 + y^2 = 4 $$ oriented counterclockwise
In the formula $z = \sqrt{\frac{tx^3}{y}}$ $x$ is subjected to an increase of $2\%$ and $t$ is subjected to an decrease of $0.1\%$. Calculate, approx- imately, the percentage change needed in $y$ to ensure that $z$ remains unchanged.
12.3.2 Calculate $F'(y)$ for each of the following functions $F$. (a) $F(y) = \int_{0}^{1} \mathrm{e}^{-x^{2} y^{2}} d x$ (b) $F(y) = \int_{0}^{y} \mathrm{e}^{-x^{2} y^{2}} d x$ (c) $F(y) = \int_{0}^{y^{3}} \mathrm{e}^{-x^{2} y^{2}} d x$
Use cylindrical coordinates to evaluate $$ \iiint_E \sqrt{x^2+y^2} \, dV $$ where E is the region that lies inside the cylinder $$ x^2+y^2=16 $$ and between the planes $$ z=-4 $$ and $$ z=3 $$.
Integrate the given series expansion of $f$ term-by-term from zero to $x$ to obtain the corresponding series expansion for the indefinite integral of $f$. $f(x) = \frac{2x^7}{1+x^8} = \sum_{n=0}^{\infty} 2(-1)^n x^{8n+7}$ $\int_0^x f(x) dx = \sum_{n=0}^{\infty}$
Write the first five terms of the sequence with the given $n$th term. $a_n = 20 + \frac{3}{n} + \frac{4}{n^2}$ $a_1 = $ $a_2 = $ $a_3 = $ $a_4 = $ $a_5 = $
1. Find the derivative of each function. a. $f(x) = 13x^4 - 7x^3 + 5x^2 + 11x + 75$ [2 marks] b. $f(x) = (x^3 + 2x^2 + 4)(x^4 - 3)$ [3 marks]
13. DETAILS MY NOTES ASK YOUR TEACHER Find the first partial derivatives of the function. $f(x, y, z) = x^9yz^2 + 2yz$ $f_x(x, y, z) =$ $f_y(x, y, z) =$ $f_z(x, y, z) =$
Use the fact that $5x^2 - y^2 = c$ is a one-parameter family of solutions of the differential equation $y \frac{dy}{dx} = 5x$ to find an implicit solution of the initial-value problem $y \frac{dy}{dx} = 5x$, $y(3) = -8$.
Estimate the area under the graph of $f(x) = 9 - x^2$ over the interval $[-3, 2]$ using four approximating rectangles and right endpoints. $R_n = $ Repeat the approximation using left endpoints. $L_n = $
Calculate the integral, assuming that $f$ is integrable and $\int_{1}^{b} f(x) dx = 1 - b^{-1}$ for all $b > 0$. $\int_{1}^{6} (8f(x) - 4) dx = \boxed{}$ Enter your answer an simplified form (as an integer or fraction).
The doubling period of a bacterial population is 20 minutes. At time $t = 100$ minutes, the bacterial population is 8,000. a. Determine the initial population. Bacteria b. Find the bacterial population after 2 hours. Bacteria
Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use C for the arbitrary constant. Absorb into C as much as possible.) $$ \int (37 - e^{24x}) dx = $$
Evaluate the line integral, where C is the given curve. $$ \int_C xyz^2 ds $$ C is the line segment from $$ (-3, 3, 0) $$ to $$ (-1, 4, 4) $$
Question 1 Which of the following is the best substitution for the indefinite integral $$ \int \frac{x^3}{\sqrt{x^2-1}} dx $$ $$ x = \sin \theta $$ $$ x = \sec \theta $$ $$ u = x^2 - 1 $$ $$ x = \tan \theta $$ Question 2 5 pts 5 pts
Compute the double integral $$ \iint_{D} x d A, $$ where D is the region in the first quadrant that lies between the circles $$ x^{2}+y^{2}=1 $$ and $$ x^{2}+y^{2}=2. $$
Use the Limit Comparison Test to determine whether the series converges. $$ \sum_{k=1}^{\infty} \frac{k^4 + 1}{k^5 - 9} $$ The Limit Comparison Test with $$ \sum_{k=1}^{\infty} $$ shows that the series
19. Find the most general antiderivative of the function. (Check your answer by differentiation. Remember the constant of the antiderivative.) $g(v) = 5 \cos(v) - \frac{8}{\sqrt{1 - v^2}}$ $G(v) = $
To finish, evaluate each definite integral in the product. $$ \int_{0}^{2\pi} d\theta \int_{0}^{7} r^3 dr = \left[ \theta \right]_{0}^{2\pi} \left[ \left( \frac{\Box}{\Box} \right) \right]_{0}^{7} $$ $$ = \Box \times \Box $$
12. Find the sum of the convergent series $\sum_{n=0}^{\infty}(-1)^{n} \frac{3^{2 n}}{5^{2 n}(2 n) !}$ by using a well known function. Identify the function and explain how you obtain the sum.
Find $\frac{dw}{dt}$ by using the Chain Rule. Express your final answer in $w = \sqrt{x^2 + y^2 - z^2}$; $x = \cos s^2t$, $y = \sin s^2t$, $z = s^2t^2$
14 15 -- -- Question 5 (7 points) Listen 17 18 -- -- 3. (7 points) Use the ratio test to determine whether $\sum_{n=1}^{\infty} \frac{n!}{n^2}$ converges or diverges. formation Done
Watch on YouTube Given the graph of $f(x)$ above, $f'(1) \approx 4$ $f'(2) \approx 3$ $f'(3) \approx 0$ $f'(4) \approx -3$ $f'(5) \approx -5$ Since these values are decreasing increasing
The indefinite integral of $\frac{1}{x}$ with respect to x is: $\ln (1/x) + C, x > 0$ $e^x$ $\ln (x) + C$ $\ln (-x) + C$ if $x < 0$
17 18 -- -- Question 7 (7 points) Listen mation 5. (7 points) Determine whether $$ \sum_{n=1}^{\infty} \frac{1}{3n^{\frac{2}{3}}+1} $$ converges or diverges. Justify your answer by citing a relevant test.
3. [0 / 1 Points] Evaluate the integral. $$ \int_{0}^{2} \left( \frac{4}{5}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \right) dt $$ Need Help? Read It SUBMIT ANSWER 4. [0 / 1 Points]
Find a vector function, $$r(t)$$, that represents the curve of intersection of the two surfaces. The paraboloid $$z = 3x^2 + y^2$$ and the parabolic cylinder $$y = 3x^2$$
Evaluate the integral by making the given substitution. (Remember the constant of integration.) $$ \int xe^{-x^2} dx, u = -x^2 $$ $$ e^{-2x} $$ Remember the constant of integration.
Find the most general antiderivative of the function. (Check your answer by differentiation. Remember the constant of the antiderivative.) $f(x) = 4x + 9$ $F(x) = $ SUBMIT ANSWER
[- / 1 Points] Find the volume of the solid in the first octant bounded by the cylinder $z = 25 - x^2$ and the plane $y = 2$.
+1260 pts /1800 < Question 14 of 18 > Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use $$ \int (35 - e^{18x}) dx = $$
14 15 -- -- Question 10 (7 points) Listen 7 18 -- -- 8. (7 points) Find the interval of convergence for $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-8)^{n}}{\sqrt{n}} $$ nation Done
A bullet follows the trajectory $c(t) = (90t - 20, 150t - 4.9t^2)$. Describe this trajectory in the form $y = f(x)$. (Use symbolic notation and fractions where needed.)
Evaluate the line integral, where C is the given curve: $$ \int_C y^3 ds, C: x = t^3, y = t, 0 \le t \le 4 $$ ScalceT9 16.XP.2.001.
Use the substitution $u = x^4 - 3$ to evaluate the following indefinite integral. $$ \int \frac{x^3}{x^4 - 3}dx $$
Use the formula $$ \int \sin^{-1}(u) du = u \sin^{-1}(u) + \sqrt{1 - u^2} + C $$ to evaluate the following integral. $$ \int x \arcsin(3x^2) dx $$
Question 6 5 pts Evaluate the improper integral $$ \int_{1}^{\infty} \frac{\ln x}{x} d x $$ divergent 3 1/4 2
Find the general indefinite integral. (Remember the constant of integration. Remem $$ \int \left(e^x + \frac{1}{9x}\right) dx $$
Calculate the integral. (Express numbers in exact form. Use symbolic notation and into C as much as possible.) $$ \int \frac{5x \, dx}{x^4 + 1} = $$
6. Evaluate the line integral over the curve $C: x = \sin(t), y = \cos(t), 0 \le t \le \pi$, $$ \int_C (3x - 2y)ds. $$ Answer:
3. Find the maximum rate of change of the fucation $f$ at the point $P$ and the direction it occurs, for $f(x, y)=x^{3}+2xy+y^{4}$ and $P=(2,1)$. Answer:
Let $z = x^2 \sin y$ where $x = t^2 - \phi^2$ and $y = t^2 + \phi^2$. Evaluate $\frac{\partial^2 z}{\partial t^2}$ and $\frac{\partial^2 z}{\partial \phi^2}$.
Evaluate the double integral. $$ \iint_D 6xy^2 \, dA $$ D is enclosed by $$ x = 0 $$ and $$ x = \sqrt{4-y^2} $$
Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. $$ \lim_{x \to 0} \sin(9x) \csc(7x) $$
9. DETAILS MY NOTES ASK YOUR TEACHER Find the indicated partial derivative. $f(x, y, z) = x^{8yz}$, $f_z(e, 2, 0)$ $f_z(e, 2, 0) = $
3. (2 PTS) Show that $$F = (2xy + y^3, x^2 + 3xy^2 + 2y)$$ is conservative and find a potential function. Answer to 3
Example 2. Your Turn, p. 400 Solve. Round answers to two decimal places. a) $2^x = 2500$ b) $5^{x-3} = 1700$ c) $6^{3x+1} = 8^{x+3}$
Compute the integral: $$ \int_1^4 xe^x dx $$ Remember that you need to choose the best answer. 165.79 $$ 3e^4 $$ 163 $$ 4e $$
Use symmetry to evaluate the following integral. $$ \int_{-\pi/6}^{\pi/6} 4 \sec^2 x \, dx $$ $$ \int_{-\pi/6}^{\pi/6} 4 \sec^2 x \, dx = \boxed{} $$
Evaluate the following integral. $$ \int \frac{dx}{\sqrt{x^2 + 2x + 101}} $$ $$ \int \frac{dx}{\sqrt{x^2 + 2x + 101}} = \boxed{} $$ (Type an exact answer.)
SUBMIT ANSWER 13. [0 / 1 Points] Evaluate the integral. $$ \int_{0}^{1} (9x^e + 5e^x) dx $$ Need Help? Read It Watch It SUBMIT ANSWER
2.6. DERIVATIVES OF INVERSE FUNCTIONS Activity 2.6.2. For each function given below, find its derivative. (a) $h(x) = x^2 \ln(x)$ $= x + 2x \ln(x)$
Answer the following questions about $F(x) = -x^2 + 35x + 348$. (A) Calculate the change in $F(x)$ from $x = 10$ to $x = 15$.
Change from rectangular to cylindrical coordinates. (a) $(0, -3, 8)$ $(r, \theta, z) = (\quad)$ (b) $(-3, 3\sqrt{3}, 2)$ $(r, \theta, z) = (\quad)$
Solve the given initial-value problem. $$5y'' + y' = -4x, y(0) = 0, y'(0) = -5$$ $$y(x) =$$ Need Help? Read It Watch It
Question 10 of 18 Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use $$ \int \frac{(\tan^{-1}(x))^2}{1 + x^2} dx = $$
Question 20 (1 point) Round to the specified place. 1,745.376 (nearest hundredth) 1745.5 1745.38 1746
4) Find the interval of convergence and radius of convergence for $$ \sum_{n=0}^{\infty} \frac{(x-3)^{n+1}}{(n+1)4^{n+1}} $$. Test the endpoints of your interval to determine inclusivity.
(a) What does the equation $y = x^2$ represent as a curve in $\mathbb{R}^2$? O line O circle O ellipse O parabola O hyperbola
Question 6 (7 points) Listen 4. (7 points) Determine whether $\sum_{n=1}^{\infty} \frac{n!}{10^{4n}}$ converges or diverges. Justify your answer by citing a relevant test.
(1.8) $$ \int_{0}^{\sqrt{\frac{\pi}{4}}} (\sin 2x^2)^2 x \cos 2x^2 dx $$ (1.9) $$ \int \frac{3(e^2)^2}{(e^x)^2 + e^x - 2} dx, x > 0 $$
Find the general indefinite integral. (Remember the constant of integration.) $$ \int \left( 7x^2 + 8 + \frac{8}{x^2 + 1} \right) dx $$
1. Write down a similar series expression for the following functions: (a) $g(x) = \frac{1}{1+x}$ (b) $h(x) = \frac{1}{1-x^2}$ (c) $k(x) = \frac{1}{2-x}$
Evaluate the indefinite integral. (Remember to use absolute values where appropriate. Remember the constant of integration.) $$ \int \frac{z^8}{z^9 + 5} dz $$
Identify the decay sequence. $^{80}_{38}Sr \rightarrow ^{80}_{39}Y + \beta^{-} + \bar{\nu}_{e}$ a) Alpha b) Beta minus - c) Beta plus + d) Gamma
Find the general indefinite integral. (Use C for the constant of integration.) $$ \int \left(u^6 - 3u^5 - u^3 + \frac{4}{7}\right) du $$
20. DETAILS MY NOTES ASK YOUR TEACHER Find the limit. $$ \lim_{(x, y) \to (3, -4)} (x^2y + 2xy^2 + 6) $$
2) Calculate the following limits (You can't use the L'Hopital's rule): a) Algebraically: $$ \lim_{x \to -1} \frac{\sqrt{x+5}-2}{x^2+6x+5} $$ $$ 3x-1 $$
CURRENT OBJECTIVE Find the derivative of a natural logarithmic function Question Find the derivative of $f(x) = -2x^{-3} \cdot \ln(x + x^{-1})$.
1. Find a point in the 2-dimensional plane that has the same representation whether you represent it in polar or cartesian coordinates.
Find the limit. Use l'Hospital's Rule where appropriate. If there is a more e $$ \lim_{t \to 0} \frac{e^{4t} - 1}{\sin(t)} $$
[5 Points] Find the function $f(t)$ that satisfies the following initial value problem: $f'(t) = \frac{2}{t^2} + 4t^2$, $f(1) = 5$.
Make the given substitution to evaluate the indefinite integral. $$ \int 4(4x + 8)^7 dx, u = 4x + 8 $$
iew Find the indefinite integral $$\int \left( \sqrt[8]{x} + \sqrt[9]{x} \right) dx.$$ $$\int \left( \sqrt[8]{x} + \sqrt[9]{x} \right) dx = \square$$
9. [0 / 1 Points] Evaluate the integral. $$ \int_{1}^{4} \sqrt{x} dx $$ Need Help? Read It Watch It SUBMIT ANSWER
Evaluate the given algebraic expression for x = 5. $$2 + 9x$$ The solution is . (Type an integer.) Next
Question Evaluate the following integral. $$ \int p(p+9)^7 dp $$ $$ \int p(p+9)^7 dp = \square $$ Get more help
Express the integrand as a sum of partial fractions and evaluate the integral. $$ \int \frac{18}{x^2 + 6x} dx $$
10. Find all first partial derivatives of $f(x,y) = 2x^2 + 6xy$ 11. Find the total differential $w = \frac{3x+y}{2x+2y}$
Need Help? Read It Watch It SUBMIT ANSWER 9. [0 / 1 Points] Evaluate the integral. $$ \int_{1}^{4} \sqrt{x} dx $$ Need Help? Read It Watch It SUBMIT ANSWER
4. Use the method of determinants to find the area of the triangle with vertices at $(0,4)$, $(5,1)$, and $(3,2)$.
Problem 5. Determine whether the series is convergent or divergent. State which tests you are using. $$ \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+2n^2)^n} $$
Find the general indefinite integral. (Remember the constant of integration.) $$ \int \sec(t)(6 \sec(t) + 7 \tan(t)) dt $$
Find the indicated limit or state that it does not exist. $$ \lim_{(x, y) \to \left(\frac{25}{2}, \frac{25}{2}\right)} \frac{x+y-25}{\sqrt{x+y}-5} $$
Evaluate the indefinite integral. (Remember the constant of integration) $$ \int \left(x - \frac{1}{x^2}\right) \left(x^2 + \frac{2}{x}\right)^6 dx $$
Evaluate the following integral. $$ \int_{0}^{\frac{\pi}{6}} \frac{\cos ^{4} x}{\sqrt{1-\sin x}} d x $$ $$ \int_{0}^{\frac{\pi}{6}} \frac{\cos ^{4} x}{\sqrt{1-\sin x}} d x = $$ (Type an exact answer.)
Find the general indefinite integral. (Use C for the constant of integration.) $$ \int (2x^2 + 3x^{-2}) dx $$
Find the Arc Length of a Plane Curve over the given interval $r(t) = t^2i + 2tk$, $[0,2\pi]$ Find the limit (if it exists): $\lim_{(x,y,z)\to(-3,1,2)} \frac{xyz}{x-z}$
Use the form of the definition of the integral given in this the $$ \int_0^5 x^2 dx $$
Find the 2nd of the three cube roots of: $Z=27(\cos(2\pi/3)+i\sin(2\pi/3))$. a. $Z^{\frac{1}{3}}=2(\cos(8\pi/9)+i\sin(8\pi/9))$ b. $Z^{\frac{1}{3}}=3(\cos(8\pi/9)+i\sin(8\pi/9))$ c. $Z^{\frac{1}{3}}=2(\cos(8\pi/9)+i\sin(8\pi/9))$ d. $Z^{\frac{1}{3}}=3(\cos(8\pi/9)+i\sin(8\pi/9))$
(1.12) $$ \int \frac{x-1}{\sqrt{x^2 - 4x + 3}} dx $$ I
Find the general solutions of all the equations by transforming them into separable equations. 3. $$\frac{dy}{dx} = \sqrt{x+y+1}$$
Find the most general antiderivative or indefinite integral. $$ \int \frac{\csc 6\theta}{\csc 6\theta - \sin 6\theta} d\theta $$
B. evaluate the indefinite integral. (Use C for the constant of integration). $$ \int x^{17} \sin(x^{18}) dx $$
Question 3 5 pts Find $F'(x)$ if $F(x) = \int_{1}^{x} e^{-t} dt$ $-e^{-x} + e^{-1}$ $2xe^{-x}$ $e^{-x}$ $2te^{-t}$
Find the derivative of y with respect to x. $y = 9x^5 \arcsin(9x^5) + \sqrt{1 - 81x^{10}}$
Evaluate the indefinite integral. (Remember the constant of integration.) $$ \int y^2 (5 - y^3)^{2/3} dy $$
Use the rules for sums of powers of integers to compute the sum. $$ \sum_{j=7}^{21} j^2 $$
Use the Fundamental Theorem to evaluate the definite integral exactly. $$ \int_{1}^{3} 18x^2 dx = i $$
30. Evaluate the indefinite integral. (Remember the constant of integration.) $$ \int \cos(3 + 7t) dt $$
Find the derivative of the function. $f(x) = \ln(5x^2 - 2x + 7)$ $f'(x) = $
Evaluate the integral. (Use symbolic notation and fractions where needed. $$ \int e^{34-26t} dt = $$
a) For all $n > 1, 0 \le \frac{\sin^2(n)}{n^2} \le \frac{1}{n^2}$, and the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so by the Comparison Test, the series $\sum_{n=1}^{\infty} \frac{\sin^2(n)}{n^2}$ converges. Correct b) For all $n > 1, 0 \le \frac{\arctan(n)}{n^3} <…
Do the following with the given information. $$ \int_{0}^{1} 39 \cos \left(x^{2}\right) d x $$ (a) Find the approximations $$ T_{8} $$ and $$ M_{8} $$ for the given integral. (Round your answer to six decimal places.) $$ T_{8}= $$ $$ M_{8}= $$ (b) Estimate the errors in the approximations $$…
1. 2.26 / 5.26 Points DETAILS MY NOTES PREVIOUS ANSWERS ASK YOUR TEACHER SCalcET9 9.1.016. (a) For what values of k does the function $y = \cos(kt)$ satisfy the differential equation $25y'' = -81y$? (Enter your answers as a comma-separated list.) $k = \frac{9}{5}, -\frac{9}{5}$ (b) For those…
2. (3 pts total) We know that tangent lines approximations typically get worse the further you go from the point $(a, f(a))$. However, this is not always the case. In fact, sometimes, the tangent line can even touch the function at more than one point. Recall that the error of a tangent line…
(a) Find the approximations $T_{10}$, $M_{10}$ and $S_{10}$ for $\int_{0}^{\pi} 32 \sin(x) dx$. (Round your answers to six decimal places.) $T_{10} = 63.472753$ $M_{10} = 64.263949$ $S_{10} = 64.003506$ Find the corresponding errors $E_T$, $E_M$ and $E_S$. (Round your answers to six decimal…
Question 7 If g is differentiable at $(l, m, n)$, then the normal line to the surface given by $g(x, y, z) = 0$ at $(l, m, n)$ can be expressed as $x = l + g_x(l, m, n)t$, $y = m + g_y(l, m, n)t$, $z = n + g_z(l, m, n)t$, $t \in \mathbb{R}$ $\frac{x - l}{g_x(l, m, n)} = \frac{y - m}{g_y(l, m,…
The graph below illustrates approximating rectangles with left endpoints for $f(x) = (16/x)$ on the interval $[2, 6]$. The estimated area based on these rectangles is and this sum is an overestimate of the area of the region enclosed by $y = f(x)$, the x-axis, and the vertical lines $x = 2$ and…
Find the general solution in powers of z of the differential equation $$(z^2 - 1)y'' + 4zy' + 2y = 0$$ Assume the form $$y(z) = \sum_{n=0}^{\infty} c_n z^n$$ Then $$y'(z) = \sum_{n=1}^{\infty} n c_n z^{n-1}$$ $$y''(z) = \sum_{n=2}^{\infty} n(n-1) c_n z^{n-2}$$ $$z^2 y''(z) = \sum_{n=2}^{\infty}…
Tools Window Help signment 3.pdf Q Search ing out this form. AMAT 112 Calculus 1 Written Assignment 3 Question 1) You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16…
Which of the following equations will determine the centroidal x-axis for the beam cross sections shown in the Figure below? 0.441" S 200 x 34 A = 6.77 in2 12.5" y W 310 x 52 A = 10.3 in2 A. $$ \bar{y} = \frac{6.77 \left( 12.5 + \frac{0.441}{2} \right) + 10.3 \left( \frac{12.5}{2} \right)}{6.77…
7. Use Stokes' Theorem to evaluate: (a) 4 marks $$ \iint_S \text{curl} \vec{F} \cdot d\vec{S} $$ where $$ \vec{F}(x, y, z) = ze^y \vec{i} + x \cos y \vec{j} + xz \sin y \vec{k}, $$ S is the hemisphere $$ x^2 + y^2 + z^2 = 16, y \geq 0, $$ oriented in the direction of the positive y-axis. (b) 4…
An Energy Drink company wants to design a new can for their new product. The material to be used for the can is aluminum; and is cylindrical in shape. Design a can that requires the least amount of aluminum to be used; and can contain exactly 16 fluid ounces (28.875 cubic inches). Find the…
Find a parametrization for the curve. The upper half of the parabola x - 2 = y² Choose the correct answer below. A. x=t, y=t² + 2, t≤2 B. x=t² + 2, y=t, t≤0 C. x=t² + 2, y=t, t≥0 D. x=t, y=t² - 2, t≥0 E. x=t² - 2, y=t, t≥2 F. x=t, y=t² - 2, t≥2
2. (Chapter 17, Section 19.1) Consider the intersection R between the two circles $$x^2 + y^2 = 2$$ and $$(x - 2)^2 + y^2 = 2$$. y R x (a) Find a 2-dimensional vector field $$F = (M(x, y), N(x, y))$$ such that $$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 1$$. (3) (b) Using…
Find the approximations $T_n$, $M_n$, and $S_n$ for $n = 6$ and 12. Then compute the corresponding errors $E_T$, $E_M$, and $E_S$. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) $\int_1^4 \frac{27}{\sqrt{x}} dx$ $n$ $T_n$ $M_n$…
Previous Problem Problem List Next Problem Homework 8: Problem 1 (1 point) Use Green's Theorem to find the counterclockwise circulation of the vector field $\mathbf{F} = \langle 3xy, 5x + 2y \rangle$ along the curve $C$, where $C$ is the triangle with vertices $(0,0)$, $(2,0)$, and $(0,2)$. To…
We are asked to use polar coordinates to find the volume under the paraboloid $z = x^2 + y^2$ and above the disk $x^2 + y^2 \leq 49$. Recall that as the paraboloid is given in the form $z = f(x, y)$, the volume below $f(x, y)$ and above the given disc is given by the following. $V = \iint_D…
Evaluate the integral. $$ \int \frac{\sqrt{y^2 - 64}}{y} dy, y > 8 $$ Which substitution transforms the given integral into one that can be evaluated directly in terms of $$ \theta $$? O A. $$ y = 8 \sec \theta $$ O B. $$ y = 8 \sin \theta $$ O C. $$ y = 8 \tan \theta $$ Given the expression…
For each function below find the requested partial derivatives. Note that $f_x$ is equivalent to $\partial f/\partial x$. If $f(x,y) = x^6 y + x^3 + 3$ then $f_x =$ $f_y =$ If $f(x,t) = \ln(t) \sqrt{x}$ then $f_x =$ $f_t =$ Remember that square roots can be entered using "sqrt" (e.g. $\sqrt{2}$…
Express D as a region of type II. $$D = \{(x, y) | 0 \le y \le x, y \le x \le 3\}$$ $$D = \{(x, y) | 0 \le y \le x, 0 \le x \le y\}$$ $$D = \{(x, y) | 0 \le y \le 3, 0 \le x \le y\}$$ $$D = \{(x, y) | 0 \le y \le 3, y \le x \le 3\}$$ $$D = \{(x, y) | 0 \le y \le 3, 0 \le x \le 3\}$$ Evaluate…
The given curve is rotated about the x-axis. Set up, but do not evaluate, an integral for the area of the resulting surface by integrating (a) with respect to x and (b) with respect to y. $$x = \ln(8y + 1), \quad 0 \le y \le 1$$ (a) Integrate with respect to x. $$\int_{0}^{\ln(9)} \left( \left(…
2. A cable company determines that the number of hours of service N needed to install a new line is $N = f(d)$, where d is the distance in meters from the nearest service hub. Write mathematical expressions for the following statements. (a) The company estimates the number of hours needed and…
Compute area of the region bounded by $y = 3x^2 - 4x$ and $y = 2x$. The figure below gion(shaded) and coordinates of the intersection points of the two curves. $y$ $(2, 4)$ $x$ $-1$ $-0.5 (0, 0)$ $0.5$ $1$ $1.5$ $2$ $2.5$ $3$ $-1$ $1$ $2$ $3$ $4$ $5$
Question 4 Mark this question The equation $y = -16t^2 + 40t + 9$ represents the height of a projectile, $y$, in feet at a particular time $t$, in seconds. For what interval (or intervals) of time will the projectile's height be above 25 feet? Between 0.5 and 2 seconds Between 1 and 3…
7. [0 / 1 Points] Use part one of the fundamental theorem of calculus to find the derivative of the function. $f(x) = \int_{x}^{0} \sqrt{6 + \sec(3t)} dt$ [Hint: $\int_{x}^{0} \sqrt{6 + \sec(3t)} dt = - \int_{0}^{x} \sqrt{6 + \sec(3t)} dt$] $f'(x) = |$ Check the plus and minus signs of all…
Question Comp Use a change of variables or the table of general integration formulas to evaluat 8 $$ \int_{7}^{8} \frac{x}{\sqrt[3]{x^{2}-8}} d x $$ Click to view the table of general integration formulas. 8 $$ \int_{7}^{8} \frac{x}{\sqrt[3]{x^{2}-8}} d x = $$ (Type an exact answer.) Get more…
SCalcET9 11.10.01 Find the Maclaurin series for $f(x)$ using the definition of a Maclaurin series. [Assume that $f$ has a power series expansion. Do not show that $R_n(x) \to 0$.] $f(x) = 3^x$ $f(x) = \sum_{n=0}^{\infty} (\quad)$ Find the associated radius of convergence $R$. $R = \quad$
Determine the moment of the strip area $A_2$ about the x-axis in the Figure below. y $A_2$ 4" $A_1$ 2" $A_3$ 4" x 2" 2" 3" A. $A_2d_2 = 7 \times 4 \times 8$ in.$^3$ B. $A_2d_2 = 7 \times 4 \times 2$ in.$^3$ C. $A_2d_2 = 7 \times 4 \times 10$ in.$^3$ D. $A_2d_2 = 7 \times 4 \times 3.5$…
ASSIGNMENT #1 Classify the order, degree, and linearity of the following DE. 1. $f'''(x) + 3t^3f''(x) = \cos 5t$ 2. $y''' + 3x^2y'' + \ln y' = 0$ 3. $f(x)f''(x) + f'(x) = 0$ 4. $y''' + yy'' + 2y^2 + \sin(y'') + e^ty = 0$ 5. $3x^2y'' + 2\ln(x)y' + e^xy = 3x \cos x$ 6. $4yy''' - x^3y' + \cos y =…
Student Enrollment The enrollment at a local college has been decreasing linearly. In 2000, there where 830 students enrolled. By 2005, there were only 575 students enrolled. Determine the average rate of change of the school's enrollment during this time period, and write a sentence explaining…
An object was launched from the ground. The height $h$ of the object, in meters, above the ground $t$ seconds after it was launched can be modeled by the function $h(t) = -4.9t^2 + 29.4t$, where $0 \le t \le 6$. According to the model, for which of the following values of $t$ was the height of…
Use Stokes' Theorem to compute the counterclockwise circulation of the vector field $\mathbf{F} = \langle 4y, 1z, 4x \rangle$ along the curve $C$, where $C$ is the rectangle with vertices $(0, 0, 3)$, $(2, 0, 3)$, $(2, 5, 3)$, and $(0, 5, 3)$. To apply Stokes' Theorem, you must first find the…
Survey Final Portal Forensic Scien Flower Deliv Order Food Basic quizzes/89664/take Page Mathematics 2450, Calculus 3, Final Exam Show all details. A correct answer with no work (or unreadable work) counts as zero. 1. Let the velocity vector be $v(t) = \sin t \mathbf{i} + e^{2t} \mathbf{j} -…
Laplace Transform 1. [Medium] (Laplace Transform (Higher order)) Use Laplace transform method to find the PS of $$ \begin{cases} y''' + y'' - y' - y = 1 + \cos x + \cos 2x + e^x \\ y(0) = y'(0) = y''(0) = 0 \end{cases} $$
ourses/2387204/quizzes/5372830/take/questions/97713564 Question 6 10 pts 6. A cube of ice is melting so that the edge, x, is decreasing at the rate of 2 inches per hour. Write the equation for the volume, V, first and then determine how fast the volume of the ice is decreasing per hour at the…
Use the given graph of the function $f$ to answer the following questions. 1. Find the open interval(s) on which $f$ is concave upward. Answer (in help (intervals) ): $(0,2)U(5,8)$ 2. Find the open interval(s) on which $f$ is concave downward. Answer (in help (intervals) ): $(2,4)U(4,5)$ 3.…
1. 6 marks Verify Stokes' theorem (i.e. show that the line integral of F over curve $C = $ Double integral of curl F.$\hat{n}$ over the surface) if $F = < y, z, x >$ and S is the portion of the plane $x + y + z = 0$ cut out by the cylinder $x^2 + y^2 = 1$, and C is its boundary (an ellipse).
Consider the vector field $\vec{F}$ and the curve C below. $\vec{F}(x, y) = (7 + 4xy^2)\vec{i} + 4x^2y\vec{j}$, C is the arc of the hyperbola $y = 1/x$ from $(1, 1)$ to $(3, \frac{1}{3})$ (a) Find a potential function $f$ such that $\vec{F} = \nabla f$. $f(x, y) = $ (b) Use part (a) to…
6:22 Test 3 MULTIPLE CHOICE 1/1 CORRECT Write the equation of a line that is perpendicular to the straight line $$y = \frac{13}{11}x - \frac{19}{23}$$ that goes through the point $$(\frac{4}{9}, -\frac{5}{11})$$ Correct: $$y = -\frac{11}{13}x - \frac{101}{1287}$$ Correct answer B $$y =…
View History Bookmarks Window Help buckeyelink3.osu.edu Question 20 of 25 Solve for x: $|x - 4| > 6$. A) $-10 < x < 2$ B) $x > 10$ C) $x > 10$ or $x < -2$ D) $-2 > x > 10$ E) None of the above Go to question: Go Prev Click the Go button. ('Enter' or 'Return' will take you to Question #1). Next
Company E-Denim sells jeans. Their weekly revenue is modeled by the function $R(x) = -x^2 + 80x$. Their weekly expenses are modeled by the cost function $C(x) = 0.5x^2 + 10x + 300$, where $x$ represents the number of pairs of jeans sold per week. Find the optimal number of jeans E-Denim must…
8 Multiple Choice 1 point For which values of a and b does the function $f(x) = \frac{ax}{b+x^2}$ have a critical point at $x = 2$ and $x = -2$? $a = 1, b = 4$ $a = 1, b = 1$ $a = 2, b = 2$ $a = 4, b = 2$ $a = 1, b = 3$ Clear my selection
Use the Ratio Test to determine whether the series converges absolutely or diverges. $$ \sum_{k=1}^{\infty} \frac{2^k}{3^k} $$ Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer in simplified form.) A. The series converges absolutely…
Page Mathematics 2450, Calculus 3, Final Exam Show all details. A correct answer with no work (or unreadable work) counts as zero. 1. Let the velocity vector be $v(t) = \sin t i + e^{2t} j - 2t k$, and the initial position vector be $r(0) = -i + 2j - 2k$. Compute the acceleration vector $a(t)$,…
Question 2 A company that produces cell phones has a cost function of $C = 6z^2 - 145z + 14906$, where C is the cost in dollars and z is the number of cell phones produced (in thousands). How many units of cell phone (in thousands) minimizes the cost function? $z = $ thousand phones produced…
12.2.2 Let $u(x, y) = y^3 - 3x^2y$ and $v(x, y) = x^3 - 3xy^2$. (a) Show that $\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$ on $\mathbb{R}^2$. (b) Show that $\frac{\partial^2v}{\partial x^2} + \frac{\partial^2v}{\partial y^2} = 0$ on $\mathbb{R}^2$. (c) Show that…
2. [Medium] (Laplace Transform (System)) Use Laplace transform method to find the PS of $$ \left\{ \begin{array}{l} \begin{bmatrix} x \\ y \end{bmatrix}' = \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + 25 \begin{bmatrix} \cos t \\ \sin t \end{bmatrix} \\…
The force exerted by an electric charge at the origin on a charged particle at a point $(x, y, z)$ with position vector $\mathbf{r}=\langle x, y, z\rangle$ is $\mathbf{F}(r)=\frac{K \mathbf{r}}{|\mathbf{r}|^{3}}$ where $K$ is a constant. (See this example.) Find the work done as the particle…
Problem 1. (20 points) Consider a circle A of radius 1 centered at (0,0) and another circle B of radius $0 < r < 1$ centered at (0,1). Compute the area lying inside B but outside A. Draw a picture of this configuration of circles and shade the area to be computed.
ScalcET9 15.8.0 Use spherical coordinates. Evaluate $$ \iiint_E \sqrt{x^2 + y^2 + z^2} \, dV $$ where E lies above the cone $$ z = \sqrt{x^2 + y^2} $$ and between the spheres $$ x^2 + y^2 + z^2 = 1 $$ and $$ x^2 + y^2 + z^2 = 16. $$
The acceleration function (in m/s$$^2$$) and the initial velocity v(0) (in m/s) are a(t) = 2t + 2, v(0) = -15, 0 $$ \le $$ t $$ \le $$ 5 (a) Find the velocity (in m/s) at time t. v(t) = m/s (b) Find the distance traveled (in m) during the given time interval. m
Step 5 We have found y in terms of Inverse Laplace transforms and the resulting transforms as follows. $y(t) = \frac{2}{125}\mathcal{L}^{-1}\left\{\frac{1}{s}\right\} + \frac{1}{25}\mathcal{L}^{-1}\left\{\frac{1}{s^2}\right\} - \frac{2}{125}\mathcal{L}^{-1}\left\{\frac{1}{s-5}\right\} +…
To solve the given differential equation, integrate both sides of the separated equation. We can do so as we integrate a of x with respect to x on the right side of the equation. Each integration requires a simple substitution. Let $u = 4y + 5$ and $v = 8x + 9$. Then $dy = \frac{du}{4}$ and $dx…
The heat capacity $C(T)$ of a substance is the amount of energy (in joules) required to raise the temperature of $1 \mathrm{~g}$ by $1^{\circ} \mathrm{C}$ at temperature $T$. How much energy is required to raise the temperature from $33^{\circ} \mathrm{C}$ to $73^{\circ} \mathrm{C}$ if…
View History Bookmarks Window Help buckeyelink3.osu.edu Question 19 of 25 A wheel makes 35 revolutions each second. Find its approximate velocity in radians per second. A) 110 B) 11 C) 6 D) 220 E) 35 Go to question: Go Click the Go button. (Enter' or 'Return' will take you to Question…
Populations of aphids and ladybugs are modeled by the following equations. $$ \frac{dA}{dt} = 2A - 0.01AL $$ $$ \frac{dL}{dt} = -0.5L + 0.0001AL $$ (a) Find the equilibrium solutions. smaller A-value $$(A, L) = (\square)$$ larger A-value $$(A, L) = (\square)$$ (b) Find an expression for $$…
3. The parabolic reflector of a satellite dish is 1 meter deep and 8 meters wide. The origin (0,0) is at the vertex of the parabola. (a) Write an equation that models the cross-section of the satellite dish. (b) Find the depth of the reflector at a distance of 2 meters from the center.
A line in the $xy$-plane has the equation $y = mx + 6$, where $m$ is a constant and $3 \le m \le 4$. Which of the following values could be the $x$-intercept of the line? Indicate all such values. $\Box -3$ $\Box -2$ $\Box -\frac{7}{4}$ $\Box -\frac{5}{4}$ $\Box \frac{5}{4}$ $\Box…
We are asked to find the Maclaurin series for a function involving cos(x). Recall the Maclaurin series for cos(x). $$cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$$ The same equality would be true for any variable, and in particular for $$u = \frac{1}{13}x^2$$. Therefore, the…
Business and Social Sciences Compound Interest Homework Question 7, 5.1.2 Part 1 of 2 Find the compound amount and the amount of interest earned by the deposit below. $4,000 at 3.61% compounded continuously for 5 years. What is the compound amount? $ (Do not round until the final answer. Then…
8. DETAILS MY NOTES ASK YOUR TEACHER Determine the set of points at which the function is continuous. $F(x, y) = \cos(\sqrt{1 + x - y})$ $\{(x, y)|y \ge x\}$ $\{(x, y)|y > -x\}$ $\{(x, y)|y \ge x - 1\}$ $\{(x, y)|y \le x + 1\}$ $\{(x, y)|y > x + 1\}$
2. DETAILS MY NOTES ASK YOUR TEACHER Determine whether or not $\mathbf{F}$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$. (If the vector field is not conservative, enter DNE.) $\mathbf{F}(x, y)=\left(y^{4} \cos (x)+\cos (y)\right)…
1. DETAILS MY NOTES ASK YOUR TEACHER Use Green's theorem to evaluate $\int_C \mathbf{F} \cdot d\mathbf{r}$. (Check the orientation of the curve before applying the theorem.) $\mathbf{F}(x, y) = \langle y - \cos(y), x \sin(y) \rangle$, C is the circle $(x - 6)^2 + (y + 2)^2 = 16$ oriented…
Evaluate the triple integral. $$ \iiint_E 2xy \, dV $$ where E lies under the plane $$ z = 1 + x + y $$ and above the region in the xy-plane bounded by the curves $$ y = \sqrt{x}, y = 0, $$ and $$ x = 1 $$
Question 10 10. Evaluate the definite integral $$ \int_{1}^{6} \frac{14}{z^{4}} dz $$ A) $$ \frac{1519}{324} $$ B) $$ \frac{301}{108} $$ C) $$ \frac{1519}{540} $$ D) $$ \frac{215}{217} $$ E) $$ \frac{1505}{324} $$
16. DETAILS MY NOTES ASK YOUR TEACHER Consider the following. $$ \iiint_E z \, dV $$ where E is enclosed by the paraboloid $$ z = x^2 + y^2 $$ and the plane $$ z = 16 $$ Write the above using cylindrical coordinates. (Choose $$ 0 < A \le 2\pi $$. Choose $$ 0 < B $$. Choose $$ C < 16 $$.) $$…
10 pts 5. After t weeks of practice, a typing student can type $100(1-e^{-0.2t})$ words per minute (wpm). How soon will the student type 80 wpm? Round your answer to the nearest number of weeks. A) 6 weeks B) 5 weeks C) 7 weeks D) 9 weeks E) 8 weeks
[5 Points] One of the critical points of the function $(x, y) = 2 - x^3 + xy - \frac{y^2}{6}$ is $(x,y)=(1,3)$. Use the Second Derivative Test to determine if $(x, y) = (1,3)$ is a relative maximum, relative minimum, or saddle. Show all your work for full credit.
Question 18 Consider the equation $$4x^2 - 17x - 10 = 5.$$ Find the solutions by using the quadratic formula. $$x = 0.75$$ and $$x = 5$$ $$x = -0.75$$ and $$x = 5$$ $$x = -5$$ and $$x = 0.75$$ $$x = -5$$ and $$x = -0.75$$
A volume is described as follows: 1. the base is the region bounded by $x = -y^2 + 6y + 45$ and $x = y^2 - 18y + 99$; 2. every cross section perpendicular to the y-axis is a semi-circle. Find the volume of this object. volume =
Question 20 5 pts Which of the following is NOT the result of $$ \int (\tan x) (\sec^2 x) dx $$ $$ \frac{1}{2 \cos^2 x} + C $$ $$ \frac{\sec^2 x}{2} + C $$ $$ \frac{1}{2 \sin^2 x} + C $$ $$ \frac{\tan^2 x}{2} + C $$
Solve the given initial-value problem $y'' + 4y = g(x)$, $y(0) = 1$, $y'(0) = 1$, where $g(x) = \begin{cases} \sin(x), & 0 \le x \le \pi/2 \\ 0, & x > \pi/2 \end{cases}$ $g(x)$ is discontinuous. [Hint: Solve the problem on two intervals, and then find a solution so that $y$ and $y'$ are…
b. The rectangles in the graph below illustrate a right endpoint Riemann sum for $f(x) = -\frac{x^2}{6} + 2x$ on the interval $[3, 7]$. The value of this right endpoint Riemann sum is , and this Riemann sum is [select an answer] the area of the region enclosed by $y = f(x)$, the x-axis, the…
Compute or approximate the corresponding function values and derivative values for the given area function. $G(x) = \int_{0}^{x} \sqrt{196 - t^2} dt$ In some cases, approximations will need to be done by using geometry. (Give exact answers. Use symbolic notation and fractions where…
Question 1: Classify the differential equation. $$ \frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^2 + xy = 1 $$ A) first-order, non-linear ordinary differential equation. B) second-order, non-linear ordinary differential equation. C) second-order, linear ordinary differential equation. D)…
3. (2 pts) Suppose we are maximizing some unknown quantity $f(x)$ on the domain $0 \leq x \leq 10$ and we find that $a$ is a critical point of $f$. It's possible that $a$ is the solution to our optimization problem, but it might not be. For what reasons might $a$ not be the solution? Name 4…
8. Use Green's theorem to evaluate the line integral $$I = \oint_C y^2 dx + 3xy^2 dy,$$ around the closed curve C consisting of the boundary of the triangle with vertices given in the following order (0,0) to (2,0) to (1,1) and back to (0,0). Answer:
Question 3 of 25 A solution of $$\frac{x-5}{2} + \frac{2}{x+5} = \frac{17}{4}$$ is A) $$-\frac{3}{2}$$ B) $$-3$$ C) $$4$$ D) $$0$$ E) None of the above Go to question: Go Click the Go button. (Enter or Return will take you to Question #1). Prev Next
Find the average rate of change of $f(x) = 2x^2 + 5$ over each of the following intervals. (a) From 1 to 3 (b) From 2 to 4 (c) From 3 to 6 (a) The average rate of change from 1 to 3 is
Question 14 5 pts $$ \int \sin^3 x dx = $$ $$ -\frac{\cos^3 x}{3} + \cos x + C $$ $$ \frac{\cos^3 x}{3} - \cos x + C $$ $$ \frac{\sin^4 x}{4} + C $$ $$ \frac{\cos^3 x}{3} - x + C $$
10 pts 7. Use algebra to rewrite the integrand; then integrate and simplify. $$ \int \frac{x-4}{\sqrt{x}} dx $$ A) $$ \frac{2}{3} x\sqrt{x}-4\sqrt{x}+C $$ B) $$ \frac{2}{3} x\sqrt{x}-8\sqrt{x}+C $$ C) $$ \frac{2}{3} x\sqrt{x}-12\sqrt{x}+C $$ D) $$ \frac{1}{2} x^2-12\sqrt{x}+C $$ E) $$…
Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem. $y''(\theta) + 10y(\theta)^3 = \sin \theta$; $y(0) = 0$, $y'(0) = 0$ The Taylor approximation to three nonzero terms is $y(\theta) = \boxed{\phantom{}}$ + ...
Find the volume of the wedge-shaped region (Figure 1) contained in the cylinder $x^2 + y^2 = 16$ and bounded above by the plane $z = x$ and below by the $xy$-plane. z = x z y x FIGURE 1 V =
Express this alternating sum $S$ as a simple fraction: $S = 1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} \pm \dots + \frac{1}{4782969}$. Your answer must be a simplified fraction. You may use a calculator for reducing the fraction. Answer: $S = 21523361/28697814$
Question For the series $$ \sum_{n=1}^{\infty} \frac{1}{n^2 - 5} $$, determine which of the following convergence tests is the best to use. Select the correct answer below: The integral test. The alternating series test. The ratio test. The comparison test. SUBMIT
Evaluate the following partial derivatives using the Fundamental Theorem of Calculus. For $f(x, y) = \int_{x}^{y} \sin \left(\frac{1}{t}\right) dt$, compute $f_x = $ $f_{xx} = $ For $f(x, y) = \int_{1}^{xy} \sqrt{1+t^3} dt$, compute $f_y = $ $f_{yy} = $
Question 5 5 pts The fraction $$ \frac{ax^3+bx^2+cx+d}{(x^2+1)(x^2-4)} $$ will take which form when expressed as partial fractions? $$ \frac{Ax+B}{x^2+1} + \frac{Cx+D}{x^2-4} $$ $$ \frac{A}{x^2+1} + \frac{B}{x+2} + \frac{C}{x-2} $$ $$ \frac{Ax+B}{x^2+1} + \frac{C}{x^2-4} $$ $$…
Identify the range of the exponential function $y = 10^x$ Select one: O a. $\{y|y < 0, y \in R\}$ O b. $\{y|y > 0, y \in R\}$ O c. $\{y|y \neq 0, y \in R\}$ O d. $\{y|y \in R\}$
Point-masses $m_i$ are located on the $x$-axis as shown. Find the moment $M$ of the system about the origin and the center of mass $\bar{x}$. $m_1 = 16$ $m_2 = 17$ $m_3 = 25$ $M = 186$ $\bar{x} = $
Evaluate the surface integral. $$ \iint_S xyz \, dS $$ S is the cone with parametric equations $$ x = u \cos(v), y = u \sin(v), z = u, 0 \le u \le 2, 0 \le v < \frac{\pi}{2} $$
Evaluate each definite integral. Enter either an exact answer or an answer rounded to three decimal places. (a) $$ \int_{1}^{4} (96x^{-3} - 10x + 4)dx = $$ (b) $$ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (8 \sin t + 8 \csc^2 t)dt = $$
Verify the identity. (Simplify at each step.) $$ \tan \left(\sin ^{-1}\left(\frac{x-1}{2}\right)\right)=\frac{x-1}{\sqrt{4-(x-1)^{2}}} $$ Let $$ \theta=\sin ^{-1}\left(\frac{x-1}{2}\right) \Rightarrow \sin (\theta)=\frac{x-1}{2} $$ $$ \tan \left(\sin ^{-1}\left(\frac{x-1}{2}\right)\right)=\tan…
Compute the flow of $\vec{F} = \langle x, y \rangle$ along and across $C : \vec{r}(t) = \langle 3 \cos(t), 3 \sin(t) \rangle$ for $\frac{\pi}{3} \leq t \leq 2\pi$. Flow = Next Question
The doubling period of a bacterial population is 15 minutes. At time $t = 100$ minutes, the bacterial population is 6,000. a. Determine the initial population. Bacteria b. Find the bacterial population after 4 hours. Bacteria
Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$, where C is given by the vector function $\mathbf{r}(t)$. $\mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} + xy\mathbf{k}$, $\mathbf{r}(t) = \cos(t)\mathbf{i} + \sin(t)\mathbf{j} + t\mathbf{k}$, $0 \le t \le \pi$
Find the volume of the given solid. enclosed by the paraboloid $z = 6x^2 + 4y^2$ and the planes $x = 0, y = 0, z = 0$ and $x + y = 1$.
Find the radius of convergence, R, of the series. R = $$ \sum_{n=3}^{\infty} \frac{x^{n+4}}{4n!} $$ Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I =
3. DETAILS MY NOTES ASK YOUR TEACHER Find the area of the surface. The part of the sphere $x^2 + y^2 + z^2 = 64$ that lies above the plane $z = 6$.
SCalcET9 16.2.003.MI. Evaluate the line integral, where C is the given plane curve. $$ \int_C xy^4 ds $$ C is the right half of the circle $$ x^2 + y^2 = 4 $$ oriented counterclockwise
In the formula $z = \sqrt{\frac{tx^3}{y}}$ $x$ is subjected to an increase of $2\%$ and $t$ is subjected to an decrease of $0.1\%$. Calculate, approx- imately, the percentage change needed in $y$ to ensure that $z$ remains unchanged.
12.3.2 Calculate $F'(y)$ for each of the following functions $F$. (a) $F(y) = \int_{0}^{1} \mathrm{e}^{-x^{2} y^{2}} d x$ (b) $F(y) = \int_{0}^{y} \mathrm{e}^{-x^{2} y^{2}} d x$ (c) $F(y) = \int_{0}^{y^{3}} \mathrm{e}^{-x^{2} y^{2}} d x$
Use cylindrical coordinates to evaluate $$ \iiint_E \sqrt{x^2+y^2} \, dV $$ where E is the region that lies inside the cylinder $$ x^2+y^2=16 $$ and between the planes $$ z=-4 $$ and $$ z=3 $$.
Integrate the given series expansion of $f$ term-by-term from zero to $x$ to obtain the corresponding series expansion for the indefinite integral of $f$. $f(x) = \frac{2x^7}{1+x^8} = \sum_{n=0}^{\infty} 2(-1)^n x^{8n+7}$ $\int_0^x f(x) dx = \sum_{n=0}^{\infty}$
Write the first five terms of the sequence with the given $n$th term. $a_n = 20 + \frac{3}{n} + \frac{4}{n^2}$ $a_1 = $ $a_2 = $ $a_3 = $ $a_4 = $ $a_5 = $
1. Find the derivative of each function. a. $f(x) = 13x^4 - 7x^3 + 5x^2 + 11x + 75$ [2 marks] b. $f(x) = (x^3 + 2x^2 + 4)(x^4 - 3)$ [3 marks]
13. DETAILS MY NOTES ASK YOUR TEACHER Find the first partial derivatives of the function. $f(x, y, z) = x^9yz^2 + 2yz$ $f_x(x, y, z) =$ $f_y(x, y, z) =$ $f_z(x, y, z) =$
Use the fact that $5x^2 - y^2 = c$ is a one-parameter family of solutions of the differential equation $y \frac{dy}{dx} = 5x$ to find an implicit solution of the initial-value problem $y \frac{dy}{dx} = 5x$, $y(3) = -8$.
Estimate the area under the graph of $f(x) = 9 - x^2$ over the interval $[-3, 2]$ using four approximating rectangles and right endpoints. $R_n = $ Repeat the approximation using left endpoints. $L_n = $
Calculate the integral, assuming that $f$ is integrable and $\int_{1}^{b} f(x) dx = 1 - b^{-1}$ for all $b > 0$. $\int_{1}^{6} (8f(x) - 4) dx = \boxed{}$ Enter your answer an simplified form (as an integer or fraction).
The doubling period of a bacterial population is 20 minutes. At time $t = 100$ minutes, the bacterial population is 8,000. a. Determine the initial population. Bacteria b. Find the bacterial population after 2 hours. Bacteria
Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use C for the arbitrary constant. Absorb into C as much as possible.) $$ \int (37 - e^{24x}) dx = $$
Evaluate the line integral, where C is the given curve. $$ \int_C xyz^2 ds $$ C is the line segment from $$ (-3, 3, 0) $$ to $$ (-1, 4, 4) $$
Question 1 Which of the following is the best substitution for the indefinite integral $$ \int \frac{x^3}{\sqrt{x^2-1}} dx $$ $$ x = \sin \theta $$ $$ x = \sec \theta $$ $$ u = x^2 - 1 $$ $$ x = \tan \theta $$ Question 2 5 pts 5 pts
Compute the double integral $$ \iint_{D} x d A, $$ where D is the region in the first quadrant that lies between the circles $$ x^{2}+y^{2}=1 $$ and $$ x^{2}+y^{2}=2. $$
Use the Limit Comparison Test to determine whether the series converges. $$ \sum_{k=1}^{\infty} \frac{k^4 + 1}{k^5 - 9} $$ The Limit Comparison Test with $$ \sum_{k=1}^{\infty} $$ shows that the series
19. Find the most general antiderivative of the function. (Check your answer by differentiation. Remember the constant of the antiderivative.) $g(v) = 5 \cos(v) - \frac{8}{\sqrt{1 - v^2}}$ $G(v) = $
To finish, evaluate each definite integral in the product. $$ \int_{0}^{2\pi} d\theta \int_{0}^{7} r^3 dr = \left[ \theta \right]_{0}^{2\pi} \left[ \left( \frac{\Box}{\Box} \right) \right]_{0}^{7} $$ $$ = \Box \times \Box $$
12. Find the sum of the convergent series $\sum_{n=0}^{\infty}(-1)^{n} \frac{3^{2 n}}{5^{2 n}(2 n) !}$ by using a well known function. Identify the function and explain how you obtain the sum.
Find $\frac{dw}{dt}$ by using the Chain Rule. Express your final answer in $w = \sqrt{x^2 + y^2 - z^2}$; $x = \cos s^2t$, $y = \sin s^2t$, $z = s^2t^2$
14 15 -- -- Question 5 (7 points) Listen 17 18 -- -- 3. (7 points) Use the ratio test to determine whether $\sum_{n=1}^{\infty} \frac{n!}{n^2}$ converges or diverges. formation Done
Watch on YouTube Given the graph of $f(x)$ above, $f'(1) \approx 4$ $f'(2) \approx 3$ $f'(3) \approx 0$ $f'(4) \approx -3$ $f'(5) \approx -5$ Since these values are decreasing increasing
The indefinite integral of $\frac{1}{x}$ with respect to x is: $\ln (1/x) + C, x > 0$ $e^x$ $\ln (x) + C$ $\ln (-x) + C$ if $x < 0$
17 18 -- -- Question 7 (7 points) Listen mation 5. (7 points) Determine whether $$ \sum_{n=1}^{\infty} \frac{1}{3n^{\frac{2}{3}}+1} $$ converges or diverges. Justify your answer by citing a relevant test.
3. [0 / 1 Points] Evaluate the integral. $$ \int_{0}^{2} \left( \frac{4}{5}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \right) dt $$ Need Help? Read It SUBMIT ANSWER 4. [0 / 1 Points]
Find a vector function, $$r(t)$$, that represents the curve of intersection of the two surfaces. The paraboloid $$z = 3x^2 + y^2$$ and the parabolic cylinder $$y = 3x^2$$
Evaluate the integral by making the given substitution. (Remember the constant of integration.) $$ \int xe^{-x^2} dx, u = -x^2 $$ $$ e^{-2x} $$ Remember the constant of integration.
Find the most general antiderivative of the function. (Check your answer by differentiation. Remember the constant of the antiderivative.) $f(x) = 4x + 9$ $F(x) = $ SUBMIT ANSWER
[- / 1 Points] Find the volume of the solid in the first octant bounded by the cylinder $z = 25 - x^2$ and the plane $y = 2$.
+1260 pts /1800 < Question 14 of 18 > Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use $$ \int (35 - e^{18x}) dx = $$
14 15 -- -- Question 10 (7 points) Listen 7 18 -- -- 8. (7 points) Find the interval of convergence for $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-8)^{n}}{\sqrt{n}} $$ nation Done
A bullet follows the trajectory $c(t) = (90t - 20, 150t - 4.9t^2)$. Describe this trajectory in the form $y = f(x)$. (Use symbolic notation and fractions where needed.)
Evaluate the line integral, where C is the given curve: $$ \int_C y^3 ds, C: x = t^3, y = t, 0 \le t \le 4 $$ ScalceT9 16.XP.2.001.
Use the substitution $u = x^4 - 3$ to evaluate the following indefinite integral. $$ \int \frac{x^3}{x^4 - 3}dx $$
Use the formula $$ \int \sin^{-1}(u) du = u \sin^{-1}(u) + \sqrt{1 - u^2} + C $$ to evaluate the following integral. $$ \int x \arcsin(3x^2) dx $$
Question 6 5 pts Evaluate the improper integral $$ \int_{1}^{\infty} \frac{\ln x}{x} d x $$ divergent 3 1/4 2
Find the general indefinite integral. (Remember the constant of integration. Remem $$ \int \left(e^x + \frac{1}{9x}\right) dx $$
Calculate the integral. (Express numbers in exact form. Use symbolic notation and into C as much as possible.) $$ \int \frac{5x \, dx}{x^4 + 1} = $$
6. Evaluate the line integral over the curve $C: x = \sin(t), y = \cos(t), 0 \le t \le \pi$, $$ \int_C (3x - 2y)ds. $$ Answer:
3. Find the maximum rate of change of the fucation $f$ at the point $P$ and the direction it occurs, for $f(x, y)=x^{3}+2xy+y^{4}$ and $P=(2,1)$. Answer:
Let $z = x^2 \sin y$ where $x = t^2 - \phi^2$ and $y = t^2 + \phi^2$. Evaluate $\frac{\partial^2 z}{\partial t^2}$ and $\frac{\partial^2 z}{\partial \phi^2}$.
Evaluate the double integral. $$ \iint_D 6xy^2 \, dA $$ D is enclosed by $$ x = 0 $$ and $$ x = \sqrt{4-y^2} $$
Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. $$ \lim_{x \to 0} \sin(9x) \csc(7x) $$
9. DETAILS MY NOTES ASK YOUR TEACHER Find the indicated partial derivative. $f(x, y, z) = x^{8yz}$, $f_z(e, 2, 0)$ $f_z(e, 2, 0) = $
3. (2 PTS) Show that $$F = (2xy + y^3, x^2 + 3xy^2 + 2y)$$ is conservative and find a potential function. Answer to 3
Example 2. Your Turn, p. 400 Solve. Round answers to two decimal places. a) $2^x = 2500$ b) $5^{x-3} = 1700$ c) $6^{3x+1} = 8^{x+3}$
Compute the integral: $$ \int_1^4 xe^x dx $$ Remember that you need to choose the best answer. 165.79 $$ 3e^4 $$ 163 $$ 4e $$
Use symmetry to evaluate the following integral. $$ \int_{-\pi/6}^{\pi/6} 4 \sec^2 x \, dx $$ $$ \int_{-\pi/6}^{\pi/6} 4 \sec^2 x \, dx = \boxed{} $$
Evaluate the following integral. $$ \int \frac{dx}{\sqrt{x^2 + 2x + 101}} $$ $$ \int \frac{dx}{\sqrt{x^2 + 2x + 101}} = \boxed{} $$ (Type an exact answer.)
SUBMIT ANSWER 13. [0 / 1 Points] Evaluate the integral. $$ \int_{0}^{1} (9x^e + 5e^x) dx $$ Need Help? Read It Watch It SUBMIT ANSWER
2.6. DERIVATIVES OF INVERSE FUNCTIONS Activity 2.6.2. For each function given below, find its derivative. (a) $h(x) = x^2 \ln(x)$ $= x + 2x \ln(x)$
Answer the following questions about $F(x) = -x^2 + 35x + 348$. (A) Calculate the change in $F(x)$ from $x = 10$ to $x = 15$.
Change from rectangular to cylindrical coordinates. (a) $(0, -3, 8)$ $(r, \theta, z) = (\quad)$ (b) $(-3, 3\sqrt{3}, 2)$ $(r, \theta, z) = (\quad)$
Solve the given initial-value problem. $$5y'' + y' = -4x, y(0) = 0, y'(0) = -5$$ $$y(x) =$$ Need Help? Read It Watch It
Question 10 of 18 Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use $$ \int \frac{(\tan^{-1}(x))^2}{1 + x^2} dx = $$
Question 20 (1 point) Round to the specified place. 1,745.376 (nearest hundredth) 1745.5 1745.38 1746
4) Find the interval of convergence and radius of convergence for $$ \sum_{n=0}^{\infty} \frac{(x-3)^{n+1}}{(n+1)4^{n+1}} $$. Test the endpoints of your interval to determine inclusivity.
(a) What does the equation $y = x^2$ represent as a curve in $\mathbb{R}^2$? O line O circle O ellipse O parabola O hyperbola
Question 6 (7 points) Listen 4. (7 points) Determine whether $\sum_{n=1}^{\infty} \frac{n!}{10^{4n}}$ converges or diverges. Justify your answer by citing a relevant test.
(1.8) $$ \int_{0}^{\sqrt{\frac{\pi}{4}}} (\sin 2x^2)^2 x \cos 2x^2 dx $$ (1.9) $$ \int \frac{3(e^2)^2}{(e^x)^2 + e^x - 2} dx, x > 0 $$
Find the general indefinite integral. (Remember the constant of integration.) $$ \int \left( 7x^2 + 8 + \frac{8}{x^2 + 1} \right) dx $$
1. Write down a similar series expression for the following functions: (a) $g(x) = \frac{1}{1+x}$ (b) $h(x) = \frac{1}{1-x^2}$ (c) $k(x) = \frac{1}{2-x}$
Evaluate the indefinite integral. (Remember to use absolute values where appropriate. Remember the constant of integration.) $$ \int \frac{z^8}{z^9 + 5} dz $$
Identify the decay sequence. $^{80}_{38}Sr \rightarrow ^{80}_{39}Y + \beta^{-} + \bar{\nu}_{e}$ a) Alpha b) Beta minus - c) Beta plus + d) Gamma
Find the general indefinite integral. (Use C for the constant of integration.) $$ \int \left(u^6 - 3u^5 - u^3 + \frac{4}{7}\right) du $$
20. DETAILS MY NOTES ASK YOUR TEACHER Find the limit. $$ \lim_{(x, y) \to (3, -4)} (x^2y + 2xy^2 + 6) $$
2) Calculate the following limits (You can't use the L'Hopital's rule): a) Algebraically: $$ \lim_{x \to -1} \frac{\sqrt{x+5}-2}{x^2+6x+5} $$ $$ 3x-1 $$
CURRENT OBJECTIVE Find the derivative of a natural logarithmic function Question Find the derivative of $f(x) = -2x^{-3} \cdot \ln(x + x^{-1})$.
1. Find a point in the 2-dimensional plane that has the same representation whether you represent it in polar or cartesian coordinates.
Find the limit. Use l'Hospital's Rule where appropriate. If there is a more e $$ \lim_{t \to 0} \frac{e^{4t} - 1}{\sin(t)} $$
[5 Points] Find the function $f(t)$ that satisfies the following initial value problem: $f'(t) = \frac{2}{t^2} + 4t^2$, $f(1) = 5$.
Make the given substitution to evaluate the indefinite integral. $$ \int 4(4x + 8)^7 dx, u = 4x + 8 $$
iew Find the indefinite integral $$\int \left( \sqrt[8]{x} + \sqrt[9]{x} \right) dx.$$ $$\int \left( \sqrt[8]{x} + \sqrt[9]{x} \right) dx = \square$$
9. [0 / 1 Points] Evaluate the integral. $$ \int_{1}^{4} \sqrt{x} dx $$ Need Help? Read It Watch It SUBMIT ANSWER
Evaluate the given algebraic expression for x = 5. $$2 + 9x$$ The solution is . (Type an integer.) Next
Question Evaluate the following integral. $$ \int p(p+9)^7 dp $$ $$ \int p(p+9)^7 dp = \square $$ Get more help
Express the integrand as a sum of partial fractions and evaluate the integral. $$ \int \frac{18}{x^2 + 6x} dx $$
10. Find all first partial derivatives of $f(x,y) = 2x^2 + 6xy$ 11. Find the total differential $w = \frac{3x+y}{2x+2y}$
Need Help? Read It Watch It SUBMIT ANSWER 9. [0 / 1 Points] Evaluate the integral. $$ \int_{1}^{4} \sqrt{x} dx $$ Need Help? Read It Watch It SUBMIT ANSWER
4. Use the method of determinants to find the area of the triangle with vertices at $(0,4)$, $(5,1)$, and $(3,2)$.
Problem 5. Determine whether the series is convergent or divergent. State which tests you are using. $$ \sum_{n=1}^{\infty} \frac{n^{2n}}{(1+2n^2)^n} $$
Find the general indefinite integral. (Remember the constant of integration.) $$ \int \sec(t)(6 \sec(t) + 7 \tan(t)) dt $$
Find the indicated limit or state that it does not exist. $$ \lim_{(x, y) \to \left(\frac{25}{2}, \frac{25}{2}\right)} \frac{x+y-25}{\sqrt{x+y}-5} $$
Evaluate the indefinite integral. (Remember the constant of integration) $$ \int \left(x - \frac{1}{x^2}\right) \left(x^2 + \frac{2}{x}\right)^6 dx $$
Evaluate the following integral. $$ \int_{0}^{\frac{\pi}{6}} \frac{\cos ^{4} x}{\sqrt{1-\sin x}} d x $$ $$ \int_{0}^{\frac{\pi}{6}} \frac{\cos ^{4} x}{\sqrt{1-\sin x}} d x = $$ (Type an exact answer.)
Find the general indefinite integral. (Use C for the constant of integration.) $$ \int (2x^2 + 3x^{-2}) dx $$
Find the Arc Length of a Plane Curve over the given interval $r(t) = t^2i + 2tk$, $[0,2\pi]$ Find the limit (if it exists): $\lim_{(x,y,z)\to(-3,1,2)} \frac{xyz}{x-z}$
Use the form of the definition of the integral given in this the $$ \int_0^5 x^2 dx $$
Find the 2nd of the three cube roots of: $Z=27(\cos(2\pi/3)+i\sin(2\pi/3))$. a. $Z^{\frac{1}{3}}=2(\cos(8\pi/9)+i\sin(8\pi/9))$ b. $Z^{\frac{1}{3}}=3(\cos(8\pi/9)+i\sin(8\pi/9))$ c. $Z^{\frac{1}{3}}=2(\cos(8\pi/9)+i\sin(8\pi/9))$ d. $Z^{\frac{1}{3}}=3(\cos(8\pi/9)+i\sin(8\pi/9))$
(1.12) $$ \int \frac{x-1}{\sqrt{x^2 - 4x + 3}} dx $$ I
Find the general solutions of all the equations by transforming them into separable equations. 3. $$\frac{dy}{dx} = \sqrt{x+y+1}$$
Find the most general antiderivative or indefinite integral. $$ \int \frac{\csc 6\theta}{\csc 6\theta - \sin 6\theta} d\theta $$
B. evaluate the indefinite integral. (Use C for the constant of integration). $$ \int x^{17} \sin(x^{18}) dx $$
Question 3 5 pts Find $F'(x)$ if $F(x) = \int_{1}^{x} e^{-t} dt$ $-e^{-x} + e^{-1}$ $2xe^{-x}$ $e^{-x}$ $2te^{-t}$
Find the derivative of y with respect to x. $y = 9x^5 \arcsin(9x^5) + \sqrt{1 - 81x^{10}}$
Evaluate the indefinite integral. (Remember the constant of integration.) $$ \int y^2 (5 - y^3)^{2/3} dy $$
Use the rules for sums of powers of integers to compute the sum. $$ \sum_{j=7}^{21} j^2 $$
Use the Fundamental Theorem to evaluate the definite integral exactly. $$ \int_{1}^{3} 18x^2 dx = i $$
30. Evaluate the indefinite integral. (Remember the constant of integration.) $$ \int \cos(3 + 7t) dt $$
Find the derivative of the function. $f(x) = \ln(5x^2 - 2x + 7)$ $f'(x) = $
Evaluate the integral. (Use symbolic notation and fractions where needed. $$ \int e^{34-26t} dt = $$
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