Exam 3. MA 114 Exam 3 Fall Multiple Choice Questions. 1. Find the average value of the function f (x) = 2 sin x sin 2x on 0 x π. C. 0 D. 4 E.

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1 Exam 3 Multiple Choice Questions 1. Find the average value of the function f (x) = sin x sin x on x π. A. π 5 π C. E. 5. Find the volume of the solid S whose base is the disk bounded by the circle x + y = r and the parallel cross-sections perpendicular to the base are squares. A. C. 8 3 r r3 3 r3 3 r3 E.

2 3. Consider the parametric curve (x(t), y(t)) = (sin(πt), cos(πt)). At which value of t does the curve pass through (, 1)? A. t = t = 1/ C. t = 1 t = 3/ E. t =. Find the center of mass of a lamina (or thin plate) that occupies the region R = {(x, y) y x }. A. (, 1) (, 1 ) ( ) 9 C., π ( ) 8, 3π ( E., 5 ) 6 Page of 1

3 5. Consider the curve of points (x, y) that satisfy the equation y = x and lies on the parabola between the points (, ) and (3, 9). Write an integral whose value is the length of this curve. 3 A. 1 + x dx C. E x dx 1 + x dx 1 + x dx 1 + x dx 6. An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve y = 1 x, x, about the x-axis (dimensions are in feet). Find the integral that computes how many cubic feet of fuel the tank will hold. ) A. π (1 x dx ) πx (1 x dx C. E. π ) (1 x dx ) π (1 x 1 + x dx π 1 + x dx Page 3 of 1

4 7. Find dy dx for the curve given by x = tet and y = t 3e t. A. 3e t 3e t e t + te t C. (1 + t)e t E. e t + te t 3e t 3e t e t 8. Find the volume of the solid obtained by rotating the region bounded by y = x 3, y = and x = 1 about the line y = 3. 5π A. 63 3π C. 7π 3π 1 E. π Page of 1

5 9. Write out the first four terms of the Maclaurin series of f (x) if f () = 1, f () = 13, f () = 1, and f () = 3. A x 1 x + 1 x3 1 13x 1 x 1 x3 C x + 1 x + 1 x x + x + 3x 3 E x x + 3x 3 1. Using the parameterization x = a cos t, y = b sin t, t π, which of the following definite integrals represents the surface area of the region obtained by rotating the ellipse x a + y = 1 about the x-axis. b π A. πa cos t a sin t + b cos t dt C. E. π π π π πb sin t a cos t + b sin t dt πa cos t a cos t + b sin t dt πab sin t cos t a sin t + b cos t dt πb sin t a sin t + b cos t dt Page 5 of 1

6 Free Response Questions 11. Consider the function f (x) = x 3 + x + 1. (a) (5 points) Find all derivatives, f (n) (x) of f (x). f (x) = x 3 + x + 1 f (x) = 3x + 1 f (x) = 6x f (x) = 6 f (n) (x) =, n (b) (5 points) Find the Taylor series for f (x) centered at a = 1. f (1) = 3, f (1) =, f (1) = 6, f (1) = 6 T(x) = f (1) + f (1)(x 1) + f (x) (x 1) + f x (x 1) 3! 3! = 3 + (x 1) + 3(x 1) + (x 1) 3 (c) (3 points) Find the radius of convergence for this Taylor series. The radius of convergence is since the Taylor series is a polynomial. Page 6 of 1

7 1. Let R be the region bounded by y = e x, y =, x = and x = 1. (a) (5 points) Using the method of cylindrical shells write down the integral needed to find the volume generated by rotating R about the y-axis. V = πx f (x) dx = π xe x dx. (b) (5 points) Showing all of your work, evaluate this integral. π xe x dx = πe x 1 = π ( 1 1 ). e Page 7 of 1

8 13. Consider the parametric curve C given by {(t, 3 t3/ ) : t T}. (a) ( points) Write an integral for the length of the curve C. x (t) = 1 and y (t) = t 1/ = t. L = T T [x (t)] + [y (t)] dt = 1 + t dt (b) (5 points) Evaluate the integral you found to express the length as a function of T. L = T 1 + t dt = (1 + t)3/ 3 T = 3 ( ) (1 + T) 3/ 1 (c) ( points) Find the value of T for which the curve has length 1/3. ( ) (1 + T) 3/ 1 = (1 T) 3/ 1 = 7 (1 T) 3/ = 8 T = 3 Page 8 of 1

9 1. Consider the parametric curve C given by (x(t), y(t)) = (t + t, t t) for t in the real numbers. (a) (5 points) Find the tangent line to the curve at the point (x, y) = (, ). Give your answer in the form y = mx + b. dx dt = t + 1 dy dt = t 1 dy dx = dy/dt dx/dt = t 1 t + 1 m = dy dx = 1 t=1 3 y = (x ) 3 y = 3 x 3 (b) ( points) Find the point(s) (x, y) on the curve where the tangent line is horizontal. Show your work. Find where dy dt =. dy dt = t 1 = t = 1 ( ( ) ( )) ( ) The point on the curve is x 1, y 1 3 =, 1. Page 9 of 1

10 15. (5 points) Set up but do not evaluate the integral that gives the surface area of revolution generated by rotating the graph of y = x x, x 1, around the x-axis. SA = = = π f (x) 1 + [ f (x)] dx π(x x ) 1 + (1 x) dx π(x x ) x + x dx Page 1 of 1