Practice problems from old exams for math 132 William H. Meeks III

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1 Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are meant to give you an idea of the kind and varieties of questions that were asked within the time limit of that particular tests. In addition, the scope, length and format of these old exams might change from year to year. Users beware! These are NOT templates upon which future exams are based, so don t expect your exam to contain problems exactly like the ones presented here. Especially note that the Exam 3 material on improper integrals has been moved to Exam 2 (see the end of the Exam 2 problems). Practice problems for Exam.. Evaluate the following integrals algebraically. Show all your steps. (a) (ln x) 3 x 8 (x x ) 3 x (c) 3 ( + x)x 2. Set up a definite integral for the volume of the solid obtained by rotating about the x-axis the region bounded by y = 25 x 2 and y = 3. Do NOT evaluate the integral. 3. Sketch the region bounded by the curves y = cos(x), y = sin(x), x = π 4, and evaluate its area. 4. Suppose f(x) is a continuous function on the interval [ 4, 4] and 4 4 f(x) = 7, f(x) =. Evaluate, showing all your steps, (a) 4 f(x) 2 f(x2 )x 5. If F (x) = ln(x) e t2 dt (a) Find F (x). Find F (e). Justify your answers! 4 f(x) = 7,

2 PRACTICE PROBLEMS FOR EXAM (a) Approximate the area under the curve y = x 2 from 2 to 3 i. Using a Riemann sum with 4 intervals and left endpoints. ii. Using a Riemann sum with 4 intervals and right endpoints. Which of the two is a better approximation? Justify your answer! 7. A particle moves along the x-axis in a straight line with velocity v(t) = t 2 t 6 for t 4 (measured in ft/sec). (a) Determine the total displacement of the particle between the times t = and t = 4. Determine the total distance that the particle traveled between t = and t = (a) Consider the function f(x) = 6 x 2. Approximate the area under this curve between x = and x = 4, using i. four rectangles and left endpoints; ii. four rectangles and right endpoints; iii. Can you state whether each of the above answers constitutes an overestimate, or an under-estimate of the area and briefly explain why? Evaluate the definite integral 2 4 x 2 2 by interpreting it in terms of areas. 9. (a) Find a function f such that i. f (x) = 6x + 2x 2 ii. f() = 5, f () = 3. If 5 f(x) = 3, 8 5 f(x) = 7, 2 g(x) = 9, and 8 2 g(x) = 7, find (2f(x) + 3g(x)). 8. (a) For the function: i. find F (x) ii. find F ( π 2 ). F (x) = cos x t 2 + e t dt,

3 PRACTICE PROBLEMS FOR EXAM. 3 Evaluate the general indefinite integral: (sin(x) + x x + x 8 + 7e 2x). (a) Evaluate the general indefinite integral x ln 3 (x) Compute, showing all your work, the definite integral 3 + 4x + x 2 2. (a) Find the area enclosed between the curves y = 4 x 2 and y = x 2 4, by sketching it, then setting up and evaluating an appropriate definite integral. Find the volume of the solid obtained by rotating the region between the curves y = x and y = x 4, around the y = line. 3. Use appropriate substitutions to evaluate the following integrals. For each, clearly indicate the substitution you are using. (a) 3 x ex2 tan 3 r sec 2 r dr (c) t 5 t (a) Find the exact value of 3 ( + 2x) geometrically- by interpreting it as an area. Approximate the same integral 3 ( + 2x) by a Riemann sum that uses 4 equal-length subintervals and right-hand endpoints as the sample points. (Show the individual terms of the Riemann sum before you calculate the value of the sum.) 5. An object is moving along a line, with velocity v(t) = t 2 4t + 3 feet/second, until it comes to a stop at time t = 3 seconds. Use definite integrals to answer the following questions:

4 PRACTICE PROBLEMS FOR EXAM. 4 (a) How far away is the object at time t = 3 from where it was at time t =? How many feet did the object travel since time t = until time t = 3? 6. Starting at time t = minutes, a metal bar is cooling at the rate r(t) = 98e. t C/min. How much will the temperature of the bar have changed by the end of an hour? (Use a definite integral to obtain your answer.) 7. Express each of the following as a definite integral. However, do not evaluate that integral. (a) The area of the region enclosed by the graphs of y = 3x and y = x 2. The volume of the solid obtained by rotating that region around the y-axis. 8. Determine the following derivatives. Briefly justify each answer! ( d (a) ) d t sin( 3 e t + t 32 ) d t ( d π d t t sin( 3 ) e x + x 32 ) d x ( 25 (c) sin( 3 ) e t + t 32 ) d t d d t π 9. The continuous functions f and g have the properties: g(x) = 2, 8 5 [2 f(x) + g(x)] = 2, 5 f(x) = 3. (a) Find the value of 8 f(x). If also then find the value of 5 f(2x). 2. Let f(x) = x 2 +. (a) Approximate the definite integral f(x) d x = 5, 2 f(x) by the Riemann sum that uses 4 subintervals and right endpoints as the sample points. Show the individual terms of the sum before you calculate the value of the sum.

5 PRACTICE PROBLEMS FOR EXAM. 5 Draw a picture of the graph of f(x). On the graph, draw the rectangles that correspond to the Riemann sum in (a). 2. Water is being pumped into a storage tank at a rate of r(t) = 6e t 5 liters per second where time t is measured in seconds. How much water is pumped into the tank during the entire time interval 3 t 4? Use a definite integral to arrive at your answer. 22. Sketch the region enclosed by the curves y = x and y 2 = x + 2 and then determine its area. 23. Use appropriate substitutions to evaluate the following integrals. For each, clearly indicate the substitution you are using. Show all you work. (You may use your calculator to check your answer.) (a) t 2 t dt arctan x x 2 + Note: arctan is the inverse function of tan x.) 24. A particle moves along the x-axis with velocity v(t) = t 3 3t for t 5. The particle is at the origin at time. (a) What is the total distance that the particle traverses - the odometer distance - between times t = and t = 5? What is the x-coordinate of the particle at time t = 5? 25. The plane region in the first quadrant that is enclosed by the curves y =, y = sin x, and x = π is rotated around the line y = to form a solid. (a) Find a formula for the area A(x) of cross-sections of the solid formed by planes perpendicular to the axis of rotation. Express the volume of the solid as a definite integral. Do NOT evaluate that integral! 26. Find the following and justify your answers. (a) The value of ( ) d e x2 + x

6 PRACTICE PROBLEMS FOR EXAM. 6 For x >, the derivative of the function F (x) = x 27. (a) Find the general indefinite integral: ln(x) x e ln t dt. Evaluate the integral: 3 x(x x ) 28. An animal population is increasing at a rate given by the function f(t) = e 2t animal/year. What is the total change in animal population between the fifth (t = 5) and the seventh (t = 7) year? 29. If (a) find F (x) find F (π). F (x) = sin x te t dt, 3. Estimate the area under the graph of f(x) = x 2 + 2, between and 4, (a) Using 4 rectangles and left endpoints. rectangles. Plot the relevant graph showing the Is your estimate in (a) and underestimate or an overestimate? (c) Find the actual area. 3. The velocity of a particle i given by v(t) = 6 3t m/sec. What is the total displacement of the particle between t = sec and t = 3 sec? What is the total distance it has traveled between these times? 32. A particle (that can move along the x-axis) at t = sec is at x = m, with speed v = m/sec. From that moment onwards the particle has a constant acceleration a = 5 m/sec 2. At what distance x (in meters) will the particle have a speed of v = m/sec? 33. Sketch the region enclosed by the curves y = x and y = x 2 2 and evaluate its area.

7 PRACTICE PROBLEMS FOR EXAM Find the volume of the solid obtained by rotating the region bounded by y = x and y = x, between x = and x = 2 around the x-axis. 35. An object is moving along a line. At each time t, its velocity v(t) is given by v(t) = t 2 2 t 3. Find the total distance traveled by the object from time t = to time t = Use the method of substitution to evaluate each of the following and indicate what substitution you are using. If the integral is a definite integral, give an exact value and not a numerical approximation. (a) sin x cos(cos x) (c) e ( + ln x) 2 x + x 2 x 37. Suppose the derivative f of f is continuous, f() = 2, and What is the value of f(4), and why? Evaluate each of the following expressions: d ) (a) ( x 6 sin 2 x + 4e x ( d x ) t 6 sin 2 t + 4e t dt (c) d 2 ( 8 4 ) x 6 sin 2 x + 4e x 39. (a) Find the points at which the curves f (x) = 7. y 2 = x and 3y x = 2 intersect. Then sketch the two curves and indicate the region enclosed between them. Calculate the area of that region enclosed between the two curves.

8 PRACTICE PROBLEMS FOR EXAM. 8 The following 5 problems were the Midterm exam questions for fall A particle moves along the x-axis in a straight line with velocity v(t) = 3(t 2 2t 3) = 3t 2 6t 9 for t 4 (measured in ft/sec). (a) Determine the total displacement of the particle between the times t = and t = 4. Determine the total distance that the particle traveled between t = and t = Consider the function f(x) = 4 x 2. (a) Approximate the area under this curve between x = 2 and x = 2, using i. four rectangles with base length and left endpoints; ii. four rectangles with base length and right endpoints; Find the exact area. 42. Use appropriate substitutions to evaluate the following integrals. For each, clearly indicate the substitution you are using. (a) (c) 3 cos x e sin x x 2 + ln x x t 3 t 2 + dt 43. Determine the following derivatives. Briefly justify each answer! ( ) (a) sin(t 2 + ln t) d t (c) d d t d d t d d t ( ) t 2 sin(x 2 + ln x) 5 ( 25 π ) sin(t 2 + ln t) dt

9 2 PRACTICE PROBLEMS FOR INTEGRAL EXAM Express each of the following as a definite integral. However, do not evaluate that integral. (a) The area of the bounded region enclosed by the graphs of y = x and y = x 2. The volume solid obtained by rotating that region around x-axis. (c) The volume of the solid obtained by rotating that region around the y-axis. 2 Practice problems for Integral Exam 2. Aside from the most basic relations such as tan x = sin(θ) cos(θ) know the following trig identities: and sec x =, you should cos(θ) cos 2 (θ) + sin 2 (θ) =. sec 2 (θ) tan 2 (θ) =. cos 2 (θ) = + cos(2θ) 2 sin 2 (θ) = cos(2θ) 2 sin(2θ) = 2 sin(θ) cos(θ) cos(2θ) = cos 2 (θ) sin 2 (θ) = 2 cos 2 (θ) = 2 sin 2 (θ) Find the general antiderivative of each of the following. You must show the details of techniques that you use. No calculator! Note that the improper integral problems from Exam 3 have been recently moved to the end of the Exam 2 problems.. 2. x cos(x 2 + 2) sin(x 2 + 2) cos x + sin x t ln t dt xe x2 sin 2 x

10 2 PRACTICE PROBLEMS FOR INTEGRAL EXAM cos 4 x sin x tan x sec x ln(x 2 ) x cos 2 x tan x sec 2 x (e 5x + sin(3x) ) ( ) 2e x sec(x) tan(x) x 2 x + x 2 + x sin 2 (x) cos 3 (x) x 4 ln(x) ( x + 4) 5 x x 2 e x x 3 x 2 + x ( sin 2 (x 2 ) ) x + x x x 2 + e x x

11 2 PRACTICE PROBLEMS FOR INTEGRAL EXAM cos 3 (θ) sin 4 (θ) dθ x ln(x) x 2 cos(2x) x 2 cos(x) sin(x)e sin(x) (4x 5 + 3e 4x x + 7 sec 2 (x) 33) tan 2 (x) sec 2 (x) e x 5 + e x x ln(x) x sec 2 (x) sin 2 (x) cos 3 (x) sin 2 (x) cos 2 (x) x 3 x 2 + x 5 25 x 2 (cos x + 5 sin x) ( x + x ) 2

12 2 PRACTICE PROBLEMS FOR INTEGRAL EXAM x + 4 xe 2x sin x + cos 2 x x + 3 x 2 3x + 2 x 2 sin x 2x 4 x 2 tan 3 x sec 5 x e 2x e x 2 (x x + sin 2x) cos(ln x) x x 2 ln x x 2 x + x 3 4x x 2 x 6 cos x + sin x x 2 cos x 53. x 2 4 x 2

13 2 PRACTICE PROBLEMS FOR INTEGRAL EXAM 2. 3 tan 3 x 54. cos 3 x e x 56. Evaluate: ( x + e x + 3 sec 2 x ) 57. Evaluate: sin x + cos 2 x 58. Evaluate: ln(ln x) x 59. Evaluate: x 2 cos x 6. Evaluate: 3x 9 x 2 6. Evaluate: tan 3 x cos x 62. Evaluate: x x 2 x Determine whether each improper integral converges and, if it does, find its value. (a) 2 x e x2 (x ) 3 The following problems contain some improper integrals which have been moved from Exam 3 to Exam 2.

14 2 PRACTICE PROBLEMS FOR INTEGRAL EXAM Evaluate the definite integral x x Evaluate the indefinite integral x 2 e 3x 66. Evaluate the definite integral π Examine whether the improper integral π 4 sin 5 x cos 3 x ln(x) x 3 converges of diverges. If it converges, evaluate it. 68. Examine whether the improper integral x 2 converges or diverges. If it converges, evaluate it. 69. Determine whether the improper integral converges and, if so, find its value: 3 (a) (c) (d) 2 2 xe x2 x( + x (x ) 2 x 4 x 2 The following 9 problems are from Midterm 2 Spring 27.

15 2 PRACTICE PROBLEMS FOR INTEGRAL EXAM Evaluate: ( x + e x + 3 sec 2 x ) 7. Evaluate: sin x + cos 2 x 72. Evaluate: ln(ln x) x 73. Evaluate: x 2 cos x 74. Evaluate: 3x 9 x Evaluate: tan 3 x cos x 76. Evaluate: x x 2 x Determine whether each improper integral converges and, if it does, find its value. (a) 2 x e x2 (x ) Determine whether each sequence {a n } n= converges and, if it does, find its limit. (a) a n = 6n 2 5n + a n = 4 + ( )n n 3 +

16 3 PRACTICE PROBLEMS FOR EXAM Practice problems for Exam 3.. Determine whether each sequence {a n } n= converges and, if it does, find its limit. (a) a n = 6n 2 5n + a n = 4 + ( )n n Does the series converge? Why or why not? (Name any test you use and show that the conditions needed for that test are satisfied.) (a) n= (c) n= 3n 7 8n 7 +6n 2 5n 2 n 3 +2n n 4 7 (d) n= ( )n n3 n 4n 5 +n Find the interval of convergence of the power series. n= (x ) n (n + )3 n 4. Find a power series representation for f(x) = 4 +2x around a = ; write the power series using sigma (Σ) notation. State for which x the power series actually has sum f(x). 5. The sequence {S n } n= of partial sums of the series n= a n is given by S n = n 5 + n for n =, 2, 3,... (a) Does the series n= a n converge? If so, to what sum? Use the formula for S n to find the values of a and a 2. Then find an explicit formula in for a n in terms of n. 6. Consider the power series n= ( ) n (n + )! xn. (a) Determine the radius of convergence of this power series.

17 3 PRACTICE PROBLEMS FOR EXAM 3. 7 Show that this power series converges for x = 3. (c) How many terms do you need to use to approximate the infinite series in so that the error is less than.? 7. Find the first three terms of the Taylor series for the function f(x) = x ln(x 2 + ) centered around a =. 8. (a) Starting with the MacLaurin series for e x (which you may just write down), obtain a power series representation of e x2. Use your answer to (a) to express e x2 as the sum of an infinite series. (c) Find a bound on the error if the first three terms of the series you obtained in were used to approximate e x2. (Do not actually calculate the approximation and do not use any value for this integral that your calculator might give.) 9. Find the exact value of the repeating decimal.5 = as a rational number.. Does the series converge? Why or why not? (Name any test you use and show that the conditions needed for that test actually hold.) (a) ( ) n n= n 4 n= 3n n (c) n= (d) n= n 2 + 7n 2 +3n 5 n 4 + n 4 +3n+5. (a) Approximate the sum of the series n= n 5 by the sum of its first three terms. Round your answer to six decimal places. Without using any value for the actual sum of the series, find an upper bound for the error involved in that approximation. Round your answer to six decimal places. 2. Find the radius of convergence and the interval of convergence of the power series n= (x 3) n n 2 n. 3. Find the first 4 terms in the Taylor series for f(x) = sin x about π (a) Find a power series representation for the function +x 4. Use summation Σ-notation to express your answer. Use (a) to express 2 +x 4 as the sum of an infinite series.

18 3 PRACTICE PROBLEMS FOR EXAM Using the root test, find whether the series is convergent or divergent: n= ( n 2 ) n + 2 2n 2 6. For each of the following two methods, set up a sum that, if you did the arithmetic, would approximate x 2 + : (a) The trapezoidal rule with n = 6 subintervals. Simpson s rule with n = 6 subintervals. 7. Determine whether the sequence {a n } n= converges and, if so, find its limit: (a) a n = 3n2 + 4n 5 + 6n 2 a n = sin n (c) a n = n 2 (d) a n = 4n+ 7 2n () + ( )n n The parts of this question are not related. (a) Use a relevant series to find the exact value of the repeating decimal.96 = as a rational number. For a certain series a n, its partial sums s n are given by: n= s n = Determine whether the series 5n n + 4 (n =, 2, 3,... ) a n converges and, if so, find its sum. n= 9. Does the series converge? Why or why not? (a) n=2 n n

19 3 PRACTICE PROBLEMS FOR EXAM 3. 9 (c) n= n=2 ( ) n+ n4 + 2 n 4 + n 3 n (ln n) 2 2. (a) Approximate the sum of n= 3 ( ) n n 2 by using its first four terms. Round your answer to 4 decimal places (to the right of the decimal point). Find an upper bound on the error of the approximation that you obtained in (a). Justify why the method you use is appropriate. 2. (a) Find the radius of convergence of the power series n= (x 3) n n 4 n. Now find the interval of convergence of the same power series. 22. Find the first 4 nonzero terms of the Taylor series for f(x) = x ln x about x =. 23. (a) Express Use (a) to express + x 5 as the sum of a power series in x. Use notation..5 as the sum of a series of numbers. + x5 /2 (c) Use to approximate with an error less than.. + x5 (Do not use your calculator to evaluate this integral!) The following 6 problems are from Midterm 3 Spring The parts of this question are not related. (a) Use a relevant series to find the exact value of the repeating decimal.96 = as a rational number.

20 3 PRACTICE PROBLEMS FOR EXAM 3. 2 For a certain series a n, its partial sums s n are given by: n= s n = Determine whether the series 5n n + 4 (n =, 2, 3,... ) a n converges and, if so, find its sum. n= 25. Does the series converge? Why or why not? (a) (c) n=2 n= n=2 n n ( ) n+ n4 + 2 n 4 + n 3 n (ln n) (a) Approximate the sum of n= 3 ( ) n n 2 by using its first four terms. Round your answer to 4 decimal places (to the right of the decimal point). Find an upper bound on the error of the approximation that you obtained in (a). Justify why the method you use is appropriate. 27. (a) Find the radius of convergence of the power series n= (x 3) n n 4 n. Now find the interval of convergence of the same power series. 28. Find the first 4 nonzero terms of the Taylor series for f(x) = x ln x about x =. 29. (a) Express Use (a) to express + x 5 as the sum of a power series in x. Use notation..5 as the sum of a series of numbers. + x5 /2 (c) Use to approximate with an error less than.. + x5 (Do not use your calculator to evaluate this integral!)

21 4 PRACTICE PROBLEMS FOR FINAL EXAM. 2 4 Practice problems for Final Exam.. Determine whether each of the statements below is true or false. If a statement is false, briefly explain why it is false. (a) The most general antiderivative of x 2 + sin(x) is x3 3 cos(x). b a f(x) = a b f(x). (c) If f is continuous on [a, b], then (d) If f is even, then a a b f(x) =. a f(x) = F F (a) where F = f. 2. Evaluate the indefinite integral, showing all the intermediate steps of the calculation: + x 2 x 5 3. Evaluate the indefinite integral, showing all the intermediate steps of the calculation: x cos(πx) 4. Examine whether the improper integral 4x. is convergent or divergent. Evaluate it, if it is convergent. Show all the intermediate steps of the calculation. 5. Is the series n=2 ( ) n n 2 convergent? Is it absolutely convergent? You should clearly explain and justify your answer. 6. Is the following series convergent or divergent? n= 3 n 6 + n 2 + n 4 n + 3 You should clearly explain and justify your answer.

22 4 PRACTICE PROBLEMS FOR FINAL EXAM Use the integral test to verify that the following series is divergent: n=2 n log(n) 8. Convert the decimal number into a fraction using the geometric series. Show all the details of the calculation. 9. Determine whether the series n= 3 n (2n)! is convergent or divergent. You should clearly explain and justify your answer.. Determine the interval of x for which the series 5 n (x 2) n n= is convergent. You should clearly explain and justify your answer.. Approximate 2 ln( + x 3 x) in two ways: (a) By the Riemann sum with n = 4 subintervals and right endpoints. By using the Trapezoidal Rule with n = (a) Calculate the area of the region bounded by x = y and x = y 2 2. Set up - but do not evaluate - an integral for the volume of the solid obtained by rotating around the horizontal line y = 2 the region bounded by the curves y =, y = x 3, and x =. 3. Use the relevant techniques of integration to evaluate: (a) x cos x x + x 2 + 5x Does the series converge? Why or why not? (a) 2 n+ 3 n n= n

23 4 PRACTICE PROBLEMS FOR FINAL EXAM. 23 n=3 n(ln n) 2 5. Given that f is a continuous function with 2 f(x) = 3, 5 f(x) = 3, 5 f(x) = 9, 2 f(x) = 5, find the value of 2 6. Find the derivative: xf(x 2 + ). 7. Find the interval of convergence for ( ) d x 2 + e cos t dt n= (x ) n n 2 n. 8. Find the coefficient of x 4 in the Taylor series for f(x) = ln(cos x) about x =. 9. Consider the curve having polar equation r = 2 sin θ. (a) Write parametric equations for this curve using θ as the parameter: { x = y = Use either the polar equation or your parametric equations to obtain an equation for the tangent line to this curve at the point where θ = π 2. (c) Use either the polar equation or your parametric equations to express the length of this curve, for θ 2π, as a definite integral. (You need not evaluate that integral.) 2. The Maclaurin series representation for sin x is: sin x = ( ) k x 2k+ (2k + )! k= (a) Use this series to find a power series for g(x) = sin(x 2 ). Either show at least six non-zero terms or else give your answer using Σ-notation. Find a series that converges to terms or else give your answer using Σ-notation. sin(x 2 ). Either show at least six non-zero

24 4 PRACTICE PROBLEMS FOR FINAL EXAM. 24 (c) Obtain an upper bound on the error in approximating the exact value of 2. Approximate sin(x 2 ) by the sum of the first three non-zero terms from. this without actually finding a numerical value for either the sum of those three terms or the definite integral.) e x2 in each of the following ways: (a) By forming the Riemann sum with n = 4 subintervals and left endpoints as sample points. By using Simpson s Rule with n = 4 subintervals. 22. (a) Find the area of the bounded region that is enclosed by the curves y = x 2 9 and y = 9 x 2. Find the volume of the solid obtained by rotating around the x-axis the region bounded by the curve y = tan x and the lines y = and x = π Does the infinite series n=2 converge? Why or why not? n(ln n) Find the interval of convergence of the power series n= (x ) n (n + ) 3 n. 25. The rate r at which people become ill with the flu at time t days after an epidemic begins is given by r = te t 2 people per day. How many people become ill with the flu during the first days of this epidemic? 26. A rumor is spreading across campus. The time t(p), in hours, by which the fraction p of all students have heard the rumor is given by t(p) = p. 4.8 x( x). How long does it take for 9% of all the students to have heard the rumor? 27. Find a power series representation for f(x) = 4 +2x around a = ; write the power series using sigma Σ-notation. State for which x the power series actually has sum f(x). 28. Calculate the degree 3 Taylor polynomial T 3 (x) of g(x) = x around a =. 29. Find the values of: (Do

25 4 PRACTICE PROBLEMS FOR FINAL EXAM. 25 (a) (c) d ( + t 3) dt dt ( d ) + t 3 dt d ( ) x 2 + t 3 dt 3. As the parameter t increases forever, starting at t =, the curve with parametric equations { x = e t cos t, y = e t sin t spirals inward toward the origin, getting ever closer to the origin (but never actually reaching) as t. Find the length of this spiral curve. 3. (a) Starting with the Maclaurin series for e x (which you may just write down), obtain a power series representation of e x2. Use your answer to (a) to express e x2 as the sum of an infinite series. (c) Find a bound on the error if the first three (nonzero) terms of the series you obtained in were used to approximate e x2 /4. (Do not actually calculate the approximation and do not use any value for this integral that your calculator might give!) 32. Express the parametric curve x = 3 + cos t y = + sin t in Cartesian xy-coordinates. Also, compute its slope dy at t = π Compute the length of the curve x = e t cos t, y = e t sin t, t π. 34. (a) Consider the curve given in polar coordinates by the expression r = 2 + cos θ i. Sketch this curve. ii. Find the area enclosed by the curve.

26 4 PRACTICE PROBLEMS FOR FINAL EXAM. 26 Find the vertical tangents to the curve r = 2 + cos θ [find both the angle θ at which they occur and the actual equation of the vertical tangent]. To do this, find the slope dy of the polar curve and identify for which θ s the slope becomes infinite. 35. Find the area of the region bounded y = x 2 + 8x and y = 2x Find k so that = k e 4t dt 37. Let C be the curve described by x(t) = 2+2 cos t and y(t) = 4+4 sin 2 t for t π. (a) Find ( dy ). Find the x and y of the point where C has a horizontal tangent line. (c) Find an equation of the line tangent to C at (3, 7). 38. Let C be the curve described by x(t) = 2 cos 2 t and y = 3 sin 2 t for t π. (a) Draw a graph of C with a window: 2 x 4 and y 4. Eliminate the parameter t. That is, find an equation that describes C with only variables x and y. (c) A particle travels along C from time to time π. Find the distance that particle travels, i.e. the arc length of C from t = to t = π. (d) Find the distance from the initial point (x(), y()) to the terminal point ((x(π), y(π)), i.e. the displacement of the particle. 39. Find the area inside of r = 2 and to the right of r = sec θ. 4. Find a power series that represents the function: (a) 2x +x 2 ln( + x 2 ). [Hint: n= ( )n x 2n = +x 2 ] 4. Write down the MacLaurin series for (a) e x cos x

27 4 PRACTICE PROBLEMS FOR FINAL EXAM Let F (x) = x sin(t3 ) dt. Find the Maclaurin series of 43. Suppose that f (n) () = ( 2 3) n (n!). (a) Find the Maclaurin series of f(x). Find the radius of convergence. d(f (x)). [Hint: sin z = z2k+ k= ( )k (2k+)! ] 44. (a) Using the Taylor s polynomial (with a = ) for the sin x with a 5 th degree term compute an approximation of sin 2. Using Taylor s inequality find an upper bound for your answer in (a). 45. A particle moves on a straight line with initial velocity 4 m/s and acceleration given by a(t) = m/s 2. (a) Find the total distance traveled by the particle during the first 6 seconds. Find the total displacement of the particle during the first 6 seconds. 46. (a) Sketch the area enclosed by the curves y = x 2 4 and y = 4 x 2. Evaluate this area. 47. Evaluate the integral (show your work): π 2 sin 5 (x) cos 2 (x) 48. Find the points on the parametric curve x(t) = t(t 2 3), y(t) = 3(t 2 3) where the tangent is horizontal and the points where it is vertical. 49. Find the values of x for which the series converges. Find the sum of the series for those values of x ( ) ln x 2n n= [Hint: Reshape it into a geometric series.] 5. Use the integral test to show convergence and divergence of the following series: 2 (a) The convergence of n= + n 2

28 4 PRACTICE PROBLEMS FOR FINAL EXAM. 28 The divergence of n= ln(n) n 5. Find the interval and radius of convergence of the power series. ( ) n 3 n (x ) n n 52. (a) Evaluate the following indefinite integral as a power series: x 2 e x2 i= Why does the series have an infinite radius of convergence? { y = x 53. Find the area bounded by 3 x y = 3x 54. If f (x) = e 2x and f() = 5, find f(). 55. If f (x) for 5 x 5 and if the area bounded by y = f (x), x = 5, x = 5 and y = is 29, find: (a) f(5) if f( 5) =. 5 5 f (x) 56. Find the radius of convergence of (a) n= n= 4n x 3 n 4 3 n (n!) x2n 57. Determine if the following series are absolutely convergent: (a) n= 4n+ (2n+)! n= 4n 3 n (c) n= 4 n 2 n 58. Determine if converges or diverges. 5 x

29 4 PRACTICE PROBLEMS FOR FINAL EXAM Find the radius of convergence of 6. Let f(x) = +x 2. (a) Find a power series, n= n 3 n xn. a n x n whose sum is equal to f(x) for x <. [Hint: n= Use infinite geometric series.] Find a power series whose sum is tan x for x <. [Hint: 6. F (x) = tan x.] 3x 3 e t2 dt, (a) Find F (x). F () (c) F () { x = e 62. Let C be the curve described by t + t 2 y = sin t + 2t 3 (a) Find dy. Find ds dt (s stands for arc length). (c) Find the length of the arc from t = to t =. (d) Find an equation of the line tangent to the curve C at (, 3)(t = ) 63. Show that 4n 3 2n Find e x e 2x +. is divergent. 65. The parts of this question are not related! + x 2 = (a) Approximate the integral by using the Trapezoidal Rule with + x3 n = 4 subintervals. Round your answer to 4 decimal places. Starting with the Maclaurin series for e x, find a power series expansion of x 2 e x.

30 4 PRACTICE PROBLEMS FOR FINAL EXAM The acceleration (in m/s 2 ) at time t of a particle moving along a straight line is given by a(t) = 2t. (a) Determinate the velocity v(t) if v() = 2(m/s). Calculate the total distance the particle travels - not the displacement - over the time interval t Use techniques of symbolic integration to evaluate: (a) arcsin x (Hint: Try integration by parts.) e x (a) Find an equation of the tangent line at the point where t = to the curve having parametric equations { x = e 2t, y = t ln t. Write parametric equations for the curve that has polar equation r = 3 cos θ: { x = y = (c) Set up and evaluate a definite integral in order to find the area of the region enclosed by the polar curve r = 3 cos θ. 69. (a) Set up (but do not yet evaluate) an improper integral whose value would be the volume of the solid obtained by rotating around the x axis the region y x, x. Now evaluate that improper integral. 7. Do the following series converge? Why or why not? (a) n= n= 2 n + n ln n ln(n 2 + ) 7. (a) Express +x 3 as the sum of a power series in x. Use -notation.

31 4 PRACTICE PROBLEMS FOR FINAL EXAM. 3 /2 Use (a) to express as the sum of a series of numbers. Either use + x3 - notation or else give at least the first five terms of the series. The following problems are from Final Exam Spring The parts of this question are not related! (a) Approximate the integral by using the Trapezoidal Rule with + x3 n = 4 subintervals. Round your answer to 4 decimal places. Starting with the Maclaurin series for e x, find a power series expansion of x 2 e x. 73. The acceleration (in m/s 2 ) at time t of a particle moving along a straight line is given by a(t) = 2t. (a) Determinate the velocity v(t) if v() = 2(m/s). Calculate the total distance the particle travels - not the displacement - over the time interval t Use techniques of symbolic integration to evaluate: (a) arcsin x (Hint: Try integration by parts.) e x (a) Find an equation of the tangent line at the point where t = to the curve having parametric equations { x = e 2t, y = t ln t. Write parametric equations for the curve that has polar equation r = 3 cos θ: { x = y = (c) Set up and evaluate a definite integral in order to find the area of the region enclosed by the polar curve r = 3 cos θ. 76. (a) Set up (but do not yet evaluate) an improper integral whose value would be the volume of the solid obtained by rotating around the x axis the region y x, x. Now evaluate that improper integral.

32 4 PRACTICE PROBLEMS FOR FINAL EXAM Do the following series converge? Why or why not? (a) n= n= 2 n + n ln n ln(n 2 + ) 78. (a) Express as the sum of a power series in x. Use -notation. +x 3 /2 Use (a) to express as the sum of a series of numbers. Either use + x3 - notation or else give at least the first five terms of the series.