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Derrick T.

Calculus 2 / BC

2 hours ago

A boy starts from walking North at a speed of 1.5m/s and a girl starts walking West at the same point P at the same time at a speed of 2.0m/s. At what rate is the distance between the boy and the girl increasing 6s later?

Derrick T.

Calculus 2 / BC

3 hours ago

Sketch the graph of the function f(x) = xe^x

Notreal N.

Calculus 2 / BC

11 hours ago

The figure shows a graph of $ r $ as a function of $ \theta $ in Cartesian coordinates. Use it to sketch the corresponding polar curves.

Patrick N.

Calculus 2 / BC

11 hours ago

Differential equations have been used extensively in the study of drug dissolution for patients given oral medication. One such equation is the Weibull equation for the concentration c(t) of the drug: $ \frac {dc}{dt} = \frac {k}{t^b} (c_a - c) $ where $ k $ and $ c_a $ are positive constants and $ 0 < b < 1. $ Verify that $ c(t) = c_a (1 - e^{-at^{1-b}}) $ is a solution of the Weibull equation for $ 1 > 0, $ where $ a = k/(1 - b). $ What does the differential equation say about how drug dissolution occurs?

Timothy A.

Calculus 2 / BC

11 hours ago

Evaluate the integral. $ \displaystyle \int_0^\pi x \sin x \cos x dx $

Jack B.

Calculus 2 / BC

11 hours ago

Evaluate the integral. $ \displaystyle \int t^2 \sin \beta t dt $

Janet R.

Calculus 2 / BC

11 hours ago

A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?

Cynthia S.

Calculus 2 / BC

11 hours ago

When a particle is located a distance $ x $ meters from the origin, a force of $ \cos (\frac{\pi x}{3}) $ newtons acts on it. How much work is done in moving the particle from $ x = 1 $ to $ x = 2 $? Interpret your answer by considering the work done from $ x = 1 $ to $ x = 1.5 $ and from $ x = 1.5 $ to $ x = 2 $.

Timothy H.

Calculus 2 / BC

11 hours ago

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. $ y = e^{-x^2} $ , $ y = 0 $ , $ x = 0 $ , $ x = 1 $

Samuel M.

Calculus 2 / BC

11 hours ago

Find the volume of the described solid $ S $. The base of $ S $ is an elliptical region with boundary curve $ 9x^2 + 4y^2 = 36 $. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

Alyssa P.

Calculus 2 / BC

11 hours ago

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. $ y = e^{-x^2} $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $ (a) About the x-axis (b) About $ y = -1 $

James S.

Calculus 2 / BC

1 day, 10 hours ago

In Exercises $5-12,$ use Newton's Law of Cooling. Frank's automobile engine runs at $100^{\circ} \mathrm{C}$ . On a day when the outside temperature is $21^{\circ} \mathrm{C},$ he turns off the ignition and notes that five minutes later, the engine has cooled to $70^{\circ} \mathrm{C}$ . \begin{equation}\begin{array}{l}{\text { (a) Determine the engine's cooling constant } k \text { . }} \\ {\text { (b) What is the formula for } y(t) ?} \\ {\text { (c) When will the engine cool to } 40^{\circ} \mathrm{C} \text { ? }}\end{array}\end{equation}

John E.

Calculus 2 / BC

1 day, 10 hours ago

Sketch the curve with the given polar equation by first sketching the graph of $ r $ as a function of $ \theta $ in Cartesian coordinates. $ r = 3\cos 3\theta $

Alexis C.

Calculus 2 / BC

1 day, 10 hours ago

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $ 2 < r < 3 $, $ \quad 5\pi/3 \leqslant \theta \leqslant 7\pi/3 $

Amanda G.

Calculus 2 / BC

1 day, 10 hours ago

Use series to approximate the definite integral to within the indicated accuracy. $ \int^{1/2}_0 x^3 \arctan x dx $ $ \text { (four decimal places)} $

Cynthia P.

Calculus 2 / BC

1 day, 10 hours ago

Find the Taylor series for $ f(x) $ centered at the given value of $ a. $ [Assume that $ f $ has a power series expansion. Do not show that $ R_n (x) \to 0.$] Also find the associated radius of convergence. $ f(x) = \sqrt x $ $ a = 16 $

Patrick N.

Calculus 2 / BC

1 day, 10 hours ago

The Fresnel function $ S(x) = \displaystyle \int_0^x \sin \left(\frac{1}{2} \pi t^2 \right) dt $ was discussed in Example 5.3.3 and is used extensively in the theory of optics. Find $ S(x) dx $. [Your answer will involve $ S(x) $.]

Jamie D.

Calculus 2 / BC

1 day, 10 hours ago

A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of $ h $ centimeters. Find the volume of water in the bowl.

Juan S.

Calculus 2 / BC

1 day, 10 hours ago

Refer to the figure and find the volume generated by rotating the given region about the specified line. $ \Re_1 $ about $ AB $

Justin W.

Calculus 2 / BC

4 days, 11 hours ago

(a) You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning. (b) You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

Pamela O.

Calculus 2 / BC

4 days, 11 hours ago

A point is graphed in polar form. Find its rectangular coordinates.

Drew M.

Calculus 2 / BC

4 days, 11 hours ago

Match the polar equations with the graphs labeled I-VI. Give reasons for your choices. (Don't use a graphing device.) (a) $ r = \ln \theta $, $ \; 1 \leqslant \theta \leqslant 6\pi \qquad $ (b) $ r = \theta^2 $, $ \; 0 \leqslant \theta \leqslant 8\pi $ (c) $ r = \cos 3\theta \qquad $ (d) $ r = 2 + \cos 3\theta $ (e) $ r = \cos(\theta/2) \qquad $ (f) $ r = 2 + \cos(3\theta/2) $

Christopher H.

Calculus 2 / BC

4 days, 11 hours ago

Solve the differential equation. $ \frac {du}{dt} = \frac {1 + t^4}{ut^2 + u^4t^2} $

Randall C.

Calculus 2 / BC

4 days, 11 hours ago

Evaluate the integral. $\int \frac{x-1}{x^{2}+2 x} d x$

Samantha G.

Calculus 2 / BC

4 days, 11 hours ago

Find the volume common to two spheres, each with radius $ r $, if the center of each sphere lies on the surface of the other sphere.

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