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Patrick A.

Calculus 3

12 hours ago

Writing A salesperson earns a 3$\%$ bonus on weekly sales over $\$ 5000$ . $$\begin{array}{l}{g(x)=0.03 x} \\ {h(x)=x-5000}\end{array}$$ a. Explain what each function above represents. b. Which composition, $(h \circ g)(x)$ or $(g \circ h)(x),$ represents the weekly bonus? Explain.

Angela H.

Calculus 3

12 hours ago

[T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $A(8,0,0)$, $B(8,18,0)$, $C(0,18,8)$ and $D(0,0,8)$ (see the following figure). a. Find the general form of the equation of the plane that contains the solar panel by using points A, B, and C, and show that its normal vector is equivalent to $\overrightarrow{A B} \times \overrightarrow{A D}$ b. Find parametric equations of line $L_{1}$ that passes through the center of the solar panel and has direction vector $\mathbf{s}=\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}, \quad$ which points toward the position of the Sun at a particular time of day. c. Find symmetric equations of line $L_{2}$ that passes through the center of the solar panel and is perpendicular to it. d. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines $L_{1}$ and $L_{2}$

Savannah C.

Calculus 3

13 hours ago

Find a parametric representation for the surface. The part of the hyperboloid $ 4x^2 - 4y^2 - z^2 = 4 $ that lies in front of the $ yz $-plane

Patrick W.

Calculus 3

13 hours ago

Use Exercise 22 to find the centroid of the triangle with vertices $ (0, 0) $, $ (a, 0) $, and $ (a, b) $, where $ a > 0 $ and $ b > 0 $.

Laura P.

Calculus 3

13 hours ago

For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the function $$ f(x, y) = -(x^2 - 1)^2 - (x^2y - x - 1)^2 $$ has only two critical points, but has local maxima at both of them. Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible.

Alyssa P.

Calculus 3

13 hours ago

The surfaces $ \rho = 1 + \frac{1}{5} \sin m \theta \sin n \phi $ have been used as models for tumors. The " bumpy sphere " with $ m = 6 $ and $ n = 5 $ is shown. Use a computer algebra system to find the volume it encloses.

Patrick P.

Calculus 3

13 hours ago

Where does the helix $ r(t) = \langle \cos \pi t, \sin \pi t, t \rangle $ intersect the paraboloid $ z = x^ 2 + y^2 $? What is the angle of intersection between the helix and the paraboloid? (This is the angle between the tangent vector to the curve and the tangent plane to the paraboloid.)

Aaron H.

Calculus 3

13 hours ago

The temperature $ T $ in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point $ (1, 2, 2) $ is $ 120^\circ $. (a) Find the rate of change of $ T $ at $ (1, 2, 2) $ in the direction toward the point $ (2, 1, 3) $. (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin.

Patrick N.

Calculus 3

13 hours ago

Evaluate the triple integral. $ \iiint_T xz\ dV $, where $ T $ is the solid tetrahedron with vertices $ (0, 0, 0) $, $ (1, 0, 1) $, $ (0, 1, 1) $, and $ (0, 0, 1) $

Karen M.

Calculus 3

13 hours ago

Evaluate the triple integral. $ \iiint_E 6xy\ 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 $

Stanley B.

Calculus 3

13 hours ago

A region $ R $ is shown. Decide whether to use polar coordinates or rectangular coordinates and write $ \iint_R f(x, y)\ dA $ as an iterated integral, where $ f $ is an arbitrary continuous function on $ R $.

Jason J.

Calculus 3

13 hours ago

Describe the level surfaces of the function. $ f(x, y, z) = x^2 + 3y^2 + 5z^2 $

Alex T.

Calculus 3

13 hours ago

If $ V(x, y) $ is the electric potential at a point $ (x, y) $ in the $ xy $-plane, then the level curves of $ V $ are called $\textit{equipotential curves} $ because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if $ V(x, y) = c/\sqrt{r^2 - x^2 - y^2} $, where $ c $ is a positive constant.

Patrick S.

Calculus 3

13 hours ago

A contour map for a function $ f $ is shown. Use it to estimate the values of $ f(-3, 3) $ and $ f(3, -2) $. What can you say about the shape of the graph?

Austin W.

Calculus 3

13 hours ago

Evaluate the iterated integral. $ \displaystyle \int_0^1 \int_0^{s^2} \cos (s^3)\ dt ds $

Canary P.

Calculus 3

13 hours ago

If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions $ r_1 (t) = \langle t^2, 7t - 12, t^2 \rangle $ $ r_2 (t) = \langle 4t - 3, t^2, 5t - 6 \rangle $ for $ t \ge 0 $. Do the particles collide?

Jack R.

Calculus 3

13 hours ago

Prove the property of cross products (Theorem 11). Property 3: $ a \times (b + c) = a \times b + a \times c $

Brian Y.

Calculus 3

13 hours ago

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $ a = 8i - j + 4k , b = 4j + 2k $

Thomas H.

Calculus 3

13 hours ago

A boatman wants to cross a canal that is 3 km wide and wants to land at a point 2 km upstream from his starting point. The current in the canal flows at 3.5 km/h and the speed of his boat is 13 km/h. (a) In what direction should he steer? (b) How long will the trip take?

Richard E.

Calculus 3

13 hours ago

Find a unit vector that has the same direction as the given vector. $ -5i + 3j - k $

Brady G.

Calculus 3

13 hours ago

Show that the equation represents a sphere, and find its center and radius. $ x^2 + y^2 + z^2 + 8x - 6y + 2z + 17 = 0 $

Nicholas W.

Calculus 3

1 day, 12 hours ago

Using either logarithms or a graphing calculator, find the time required for each initial amount to be at least equal to the final amount. $\$ 8000,$ deposited at 3$\%$ compounded quarterly, to reach at least $\$ 23,000$

Nate R.

Calculus 3

1 day, 12 hours ago

Evaluate the surface integral. $ \displaystyle \iint_S x^2yz \, dS $, $ S $ is the part of the plane $ z = 1 + 2x + 3y $ that lies above the rectangle $ [0, 3] \times [0, 2] $

Anthony R.

Calculus 3

1 day, 12 hours ago

(a) Set up, but do not evaluate, a double integral for the area of the surface with parametric equations $ x = au \cos v $, $ y = bu \sin v $, $ z = u^2 $, $ 0 \leqslant u \leqslant 2 $, $ 0 \leqslant v \leqslant 2\pi $. (b) Eliminate the parameters to show that the surface is an elliptic paraboloid and set up another double integral for the surface area. (c) Use the parametric equations in part (a) with $ a = 2 $ and $ b = 3 $ to graph the surface. (d) For the case $ a = 2 $, $ b = 3 $, use a computer algebra system to find the surface area correct to four decimal places.

Patrick F.

Calculus 3

1 day, 12 hours ago

Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter $p$ is a square.

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