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Lennie Euler Calculus II — sample problems � 3. (1 pt) Evaluate the definite integral. 1) sinn (a x) · cos(a x) dx � e3 � ax−b 2) dx with B = 2 b a x2 − B x + c � en dx 3) 1 x · (1 + ln(x)) 4) compute area between 2 curves: integrate wrt x, then wrt y 5) solid of revolution: rotate about y = H 6) work to lift a leaking bucket � 7) � ea x · sin(b x) dx 8) a x · ln(b x) dx 1 4. (1 pt) Consider the area between the graphs x + 1y = −2 and x + 8 = y2 . This area can be computed in two different ways using integrals. First of all it can be computed as a sum of two integrals � b a � 9) xa · cos(xb ) dx with a = 2 b − 1 cubic x 2 − a2 quadratic 11) partial fractions for with x − a being a (x − a) · (quadratic) factor of the denominator quadratic 13) � � ∞ a dx √ dx 2 x · x2 + a2 with the following values: α= β= h(y) = b g(x) dx h(y) dy Either way we find that the area is y= 22) Do you believe Maclaurin series for cosine is a function? 23) Taylor polynomial for x3 centered at A 24) match: graphs of curves, parametric equations a 25) Cartesian equation for polar curve: r = b cos(θ) + c cos(θ) 1. � (1 pt) Evaluate the indefinite integral. +C. 2. � (1 pt) Evaluate the indefinite integral. 3x − 2 dx = 2 (3x − 4x + 5)6 +C. . 5. (1 pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the the line y = −3. a ; radius 1 − x3 21) coefficients c0 , . . . , c4 of Taylor series centered at A for ln(x); interval of convergence sin7 (4x) cos(4x) dx = � β α 14) match: sequence, convergence/divergence behavior 15) geometric series: evaluate if convergent 16) match: series, convergence/divergence behavior 17) 2-phase match: series, behavior, test technique 18, 19) convergence interval for a power series of convergence � c Alternatively this area can be computed as a single integral x · eb x dx if it converges (b is negative) 20) coefficients c0 , . . . , c4 of Maclaurin series for f (x) dx + with the following values: a= b= c= f (x) = g(x) = 10) partial fractions for 12) evaluate dx x(1 + ln x) 1 , y = 0, x = 2, x = 7 x4 Answer: 6. (1 pt) A bucket that weighs 3.2 pounds and a rope of negligible weight are used to draw water from a well that is 83 feet deep. The bucket is filled with 38 pounds of water and is pulled up at a rate of 2.7 feet per second, but water leaks out of a hole in the bucket at a rate of 0.15 pounds per second. Find the work done pulling the bucket to the top of the well. Your answer must include the correct units. (You may enter lbf or lb*ft for ft-lb .) Work = 1 (Show the student hint after 1 attempts: ) 14. (1 pt) Match each sequence below to statement that BEST fits it. Hint: Calculate the work done lifting the bucket (without the water) and the water (without the bucket) separately and add them to find the total work done. Find a function that describes the weight of the water in the bucket when the bucket is x feet above its starting position. 7. � (1 pt) Evaluate the indefinite integral. e4x sin(6x) dx = 8. � (1 pt) Evaluate the indefinite integral. 5x ln(4x) dx = 9. the indefinite integral. � (1 pt) �Evaluate � 5 3 x cos x dx = STATEMENTS Z. The sequence converges to zero; I. The sequence diverges to infinity; F. The sequence has a finite non-zero limit; D. The sequence diverges. +C. SEQUENCES 1. sin(n) n100 2. (1.01) n 3. ln(ln(ln(n))) 3 −5n 4. n3n−n 5 n! 5. n1000 +C. +C. Hint: First make a substitution and then use integration by parts to evaluate the integral. 6. (ln(n)) n 7. n sin ( 1n ) 8. arctan(n + 1) 10. (1 pt) Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. � 4x3 + 3x2 − 33x − 30 Consider the indefinite integral dx x2 − 9 15. (1 pt) The following series are geometric series. Determine whether each series converges or not. For the series which converge, enter the sum of the series. For the series which diverges enter ”DIV” (without quotes). Then the integrand decomposes into the form ax + b + (a) c d + x−3 x+3 (b) where a= b= c= d= Integrating term by term, we obtain that � 4x3 + 3x2 − 33x − 30 dx = x2 − 9 +C (c) n=1 ∞ 1 ∑ 3n = n=2 ∞ , 3n ∑ 42n+1 = , n=0 ∞ xe−3x dx 13. � (1 pt) Evaluate the indefinite integral. dx √ = x2 x2 + 36 , , , . 16. (1 pt) Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series diverges. +C. 12. (1 pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -1. 6 11n 10n ∑ n= n=5 11 ∞ 6n (e) ∑ n+4 = n=1 6 ∞ 10n + 3n (f) ∑ = 11n n=1 (d) 11. (1 pt) Evaluate the indefinite integral. � 8x2 − 19x − 22 dx = (x − 2)(x2 − 4) � ∞ ∞ ∑ 10n = 1. ∞ ∑ (−1) n=1 ∞ n √ n n+2 (−3)n 7 n=1 n ∞ (n + 1)(82 − 1)n 3. ∑ 82n n=1 2. +C. 2 ∑ ∞ sin(2n) n2 n=1 ∞ (−1)n 5. ∑ n=1 3n + 5 4. Answer: ∑ Note: Give your answer in interval notation 20. (1 pt) Suppose that ∞ 9 = cn xn ∑ (1 − x3 ) n=0 17. (1 pt) For each of the series below select the letter from a to c that best applies and the letter from d to k that best applies. A possible answer is af, for example. A. B. C. D. E. F. G. H. I. J. K. 1. 2. 3. 4. 5. 6. Find the following coefficients of the power series. c0 = The series is absolutely convergent. The series converges, but not absolutely. The series diverges. The alternating series test shows the series converges. The series is a p-series. The series is a geometric series. We can decide whether this series converges by comparison with a p series. We can decide whether this series converges by comparison with a geometric series. Partial sums of the series telescope. The terms of the series do not have limit zero. None of the above reasons applies to the convergence or divergence of the series. ∞ 2 + sin(n) ∑ √n n=1 ∞ cos2 (nπ) ∑ nπ n=1 ∞ (2n + 3)! ∑ (n!)2 n=1 ∞ 1 ∑ n log(4 + n) n=2 ∞ 1 ∑ n√n n=1 ∞ cos(nπ) ∑ nπ n=1 c1 = c2 = c3 = c4 = Find the radius of convergence R of the power series. R= 21. (1 pt) The Taylor series of function f (x) = ln(x) at a = 4 is given by: f (x) = ∞ ∑ cn (x − 4)n n=0 Find the following coefficients: c0 = c1 = c2 = 18. (1 pt) Find all the values of x such that the given series would converge. ∞ xn ∑ (2)n (√n + 5) n=1 c3 = The series is convergent from x = , left end included (enter Y or N): to x = , right end included (enter Y or N): Determine the interval of convergence: c4 = Note: Give your answer in interval notation � � cos 4x2 − 1 22. (1 pt) Let f (x) = . Evaluate the 6th derivax2 tive of f at x = 0. f (6) (0) = Hint: Build a Maclaurin series for f (x) from the series for cos(x). 19. (1 pt) Find all the values of x such that the given series would converge. ∞ xn ∑ ln(n + 7) n=1 3 1. 2. 3. 4. 5. 23. (1 pt) The fourth degree Taylor polynomial for f (x) = x3 centered at a = 2 is T4 (x) = c0 + c1 (x − 2) + c2 (x − 2)2 + c3 (x − 2)3 + c4 (x − 2)4 . x = 6 cos(t) + cos(4.5t); y = 6 sin(t) − sin(4.5t) x = sin(t)(3 − 2 sin(t)); y = cos(t)(3 − 2 sin(t)) x = sin(t); y = cos(t) − 2 cos(2t) x = t 3 − 5t + 2; y = 3 − t 2 x = cos(5t); y = sin(3t) Find the coefficients of this Taylor polynomial. c0 = c1 = A c2 = c3 = B C D 25. (1 pt) A curve with polar equation c4 = 13 8 sin θ + 65 cos θ represents a line. This line has a Cartesian equation of the form r= ? The function f (x) = x3 equals its fourth degree Taylor polynomial T4 (x) centered at a = 2. Hint: Graph both of them. If it looks like they are equal, then do the algebra. y = mx + b ,where m and b are constants. Give the formula for y in terms of x. For example, if the line had equation y = 2x + 3 then the answer would be 2 ∗ x + 3 . 24. (1 pt) Assume t is defined for all time. Enter the letter of the graph below which corresponds to the curve traced by the parametric equations. Think about the range of x and y, and whether there is periodicity and or symmetry. c Generated by �WeBWorK, http://webwork.maa.org, Mathematical Association of America 4 E