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1 MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources made available by the Teaching Center include: Walk-In tutoring at Broward Hall Private-Appointment, one-on-one tutoring at Broward Hall Walk-In tutoring in LIT 25 Supplemental Instruction Video resources for Math and Science classes at UF Written exam reviews and copies of previous exams The teaching center is located in the basement of Broward Hall: You can learn more about the services offered by the teaching center by visiting

2 MAC232 Exam 3 Review. The length of the parametric curve x = e t t, y = 4e t/2, t [ 8, 3] is given by the integral: A. D (e t + )dt B. 2(e t )dt E (e t + )dt C. 2(e t + )dt 3 8 4(e t + )dt 2. The polar equation r = 2 cos θ can be expressed as: A. (x ) 2 + (y + ) 2 = B. (x + ) 2 + (y ) 2 = C. (x ) 2 + y 2 = D. (x ) 2 + (y ) 2 = 4 E. (x + ) 2 + (y + ) 2 = 2 3. Graph the following polar equation: r = 5 4 cos θ 4. Calculate the Taylor series representation of e 3x centered at 3. A. e 9 3 n (x 3) n n! 3 n x n D. n! B. E. 3 n (x 3) n n! e 9 3n (x 3) n n C. e 9 (x 3) n n! 5. Write the Cartesian point (2, 2) in polar coordinates. A. (4, π 3 ) B. (4, π 6 ) C. ( π 6, 2) D. ( π 3, 2) E. (2, π 3 ) 2

3 MAC232 Exam 3 Review 6. Consider x = t 3 t 2 + t and y = 3e t. Calculate the slope and the sign (positive or negative) of the second derivative at the point (0, 3). A. Slope = 3, second derivative is negative B. Slope =, second derivative is negative C. Slope =, second derivative is negative 3 D. Slope =, second derivative is positive 3 E. Slope = 3, second derivative is positive 7. Which of the following describes the graph of the parametric curve given by the equations below? { x = 2 sin(t) 0 t 2π y = cos(t) A. An ellipse drawn clockwise B. A circle drawn counterclockwise C. A circle drawn clockwise D. An ellipse drawn counterclockwise E. None of the others 8. Find a Taylor series for ln( + 4x) centered at. ( ) n+ 4 n (x ) n A. ln n n ( ) n+ 4 n (x ) n B. ln 5 + C. ln n ( ) n+ 4 n (x ) n 5 n n! ( ) n+ 4 n x n D. n ( ) n+ 4 n (x ) n E. 5 n n! 3

4 MAC232 Exam 3 Review 9. Find the point (x, y) where the tangent line to the curve c(t) = (3t 2 2, 3t 2 + 2t) is horizontal. A. D. ( 4 9, 52 ) 9 ( 23 ) 2, 4 B. (0, 6) C. E. DNE ( 5 ) 3, 3 0. Graph the following: (a) r = cos(3θ) (b) r = sin(2θ). Find the Maclaurin series of f(x) = A. f(x) = C. f(x) = E. f(x) = nx n 3 n+ and R = 3 x and determine its radius of convergence R. (3 x) 2 B. f(x) = nx n 3 and R = 3 D.f(x) = n+ nx n 3 n+ and R = 3 nx n 3 n+ and R = 3 nx n 3 n+ and R = 3 2. Suppose that c n (x 2) n converges for x = 4 and diverges for x = 2. Which of the following must be correct? A. c n ( 2) n converges B. c n 4 n diverges C. c n 3 n converges D. c n ( ) n converges E. c n ( 5 2) n diverges 3. If f(x) = sin 2 (x), find f (02) (0). A. 02! 205! B. 02! 205! C. 02! 5! D. 02! 205! E. 205! 02! 4

5 MAC232 Exam 3 Review 4. Set up an integral for the area of the region that lies inside r = 3 sin θ and outside r = cos θ. A. Area = B. Area = C. Area = D. Area = E. Area = 2π π/3 2π π/6 2π π/6 π π/6 π π/3 π 2 ( 3 sin θ) 2 dθ π/3 2 ( 3 sin θ) 2 dθ 2 ( 3 sin θ) 2 dθ 2 ( 3 sin θ) 2 dθ 2 ( 3 sin θ) 2 dθ π/2 π/6 π π/6 π/2 π/6 π/2 π/3 2 (cos θ)2 dθ 2 (cos θ)2 dθ 2 (cos θ)2 dθ 2 (cos θ)2 dθ 2 (cos θ)2 dθ 5. Graph the following polar equation: r = 2 2 sin θ 5

6 Exam 3 Review. Find the radius of convergence and interval of convergence for the series. ( ) a. n 4 n n x n ( ) b. n (2n )2 (x ) n n!x c. n n n 3 5 (2n ) d. 2 x n (2n) 2. Suppose that c n x n converges when x = 4 and diverges when x = 6. What can be said about the convergence or divergence of the following series? a. c n b. c n 8 n c. c n ( 3) n d. c n ( ) n 9 n 3. Let S(x) = c n x n be a power series with radius of convergence R and suppose that S(x) converges when x = 2 and diverges when x = 7. Which of the following conclusions must be true? a. The series c 0 c + c 2 c 3 + converges b. R 2 c. R < 7 d. The series c 0 2c + 4c 2 8c 3 + converges e. The series c 0 + 7c + 49c c 3 + diverges f. The series c 0 + 9c 8c c 3 + diverges g. The series c 0 + 4c + 6c c 3 + diverges 4. Find a power series representation for the series and determine the radius and interval of convergence. a. f(x) = 4 2x+3 b. f(x) = x x+2 c. f(x) = x2 x 2 +6 d. f(x) = ln(5 x) e. f(x) = x 2 arctan x f. f(x) = x (+4x) 2 g. f(x) = ln( + x 4 ) h. f(x) = e 3+x2 i. g(x) = 3 x 2 +x 2

7 5. Evaluate the indefinite integral as a power series. ˆ ˆ ˆ t tan a. dt b. x 2 x ln( + x) dx c. dx d. t8 x ˆ ˆ ˆ cos x e. x 2 sin(x 2 ) dx f. dx g. arctan(x 2 ) dx h. x ˆ x ln( + x 2 ) dx ˆ cos(x 2 ) dx x 6. Find the Taylor series for f(x) centered at the given value of a. a. f(x) = 3 x, a = 8 b. f(x) = ln x, a = c. f(x) = e 4x, a = 4 d. f(x) = x, a = 6 e. g(x) = x+4, a = 4 f. h(x) = ln( + 2x), a = 4 7. Find the first four nonzero terms of the series for f(x) centered at the given value of a. a. f(x) = sin x, a = π 6 b. f(x) = cos x, a = π 2 8. Use the series to evaluate the limit x ln(+x) a. lim x 0 x 2 b. lim x 0 cos x +x e x c. lim 3 x sin x x+ 6 x 0 x 5 9. Find the sum of the series a. ( ) n x4n d. ln 2 + n! b. ( ) n π2n 6 2n (2n)! c. (ln 2)2 (ln 2)3 2! 3! +... e ! ! + 8 4! f. ( ) n 3n n5 n ( ) n xn+ 4 n 0. Approximate the value of the integrals a. 0 e x2 dx with an error no greater than b. 0 x cos xdx with an error no greater than

8 . Find the Taylor series expansion of a. f(x) = sin x centered at a = π 4. Then use the Taylor Polynomial T 2 to approximate sin(47 ). b. g(x) = 5x + e 3x centered at a = 0. Then use the Taylor Polynomial T 2 to approximate g(0.). 2. Let f(x) = sin ( ) x 2 2. Determine the value of f (50) (0). 3. Find the Taylor series for f centered at a = 3 and its radius of convergence if f (n) ( 3) = n! 3 n+. 4. Starting with the geometric series x n =,( < x < ), use differentiation to find the sum of the series n 4. n x 5. Eliminate the parameter t to find a Cartesian equation for the curve. a. x = t 2 3, y = t + 2, 3 t 3 b. x = sin t, y = cos t, 0 t 2π c. x = t, y = t d. x = t 2, y = t 3 e. x = t 2, y = ln t f. x = e t, y = e 2t 3

9 6. Match the parameteric equations with the graphs below. a. x = sin t, y = t b. x = t 2 9, y = 8t t 3 c. x = t, y = t 2 9 d. x = 4t + 2, y = 5 3t 7. Find d2 y dt 2 at the given value of t. a. x = t 3 + t, y = 7t 2 4, t = 2 b. x = t + t, y = 4 t 2, t = c. x = cos t, y = sin t, t = π 4 8. Find the points on the curve x = 6 cos t, y = sin(2t), 0 t 2π, where the tangent line is a. horizontal b. vertical 9. Find the length of the curve. a. (3t +, 9 4t), 0 t 2 b. (e t t, 4e t 2), 0 t 2 c. (3t, 4t 3 2), 0 t d. (sin 3t, cos(3t)), 0 t π 20. Which of the following are possible pairs of polar coordinates for the point with rectangular coordinates (0, 2)? a. (2, π 2 ) b. (2, 7π 2 ) c. ( 2, 3π 2 ) d. ( 2, 7π 2 ) e. ( 2, π 2 ) f. (2, 7π 2 ) 4

10 2. Find the shaded areas below. a. The shaded region inside r = 4 cos θ and outside r =. b. The shaded region of the lemniscate r 2 = cos(2θ) c. The shaded region of the cardioid r = 2 cos θ d. The shaded area enclosed by r = cos 3θ and r = 2 e. The area of the region that lies inside both curves r = cos θ and r 2 = sin θ. 5

11 22. Match the equations with the graphs a. r = 3 sin 4θ b. r 2 = 4 cos θ c. r = 2 3 sin θ d. r = + 2 cos θ e. r = 3 cos 3θ d. r = e θ 6 6

12 ˆ b 23. Given flat and polar curves of r = cos θ + sin 2θ, let be the area of the shaded region. Find a and b. a 2 (cos θ + sin 2θ)2 dθ 7