Problem Points Score 1 25 2ab 30 2c 19 Section: 3 30 Instructor: 4 30 5 30 6 36 ∑ 200 M161, Final, Spring 2009 Name: Time: 110 minutes. You may not use calculators on this exam d 1 1 1 d d asin(x) = √ acos(x) = − √ atan(x) = , , , dx dx 1 + x2 1 − x2 1 − x 2 dx d d sec(x) = sec(x) tan(x), tan(x) = sec2 (x), sin(2x) = 2 sin(x) cos(x) dx dx ln xdx = x ln x − x + C sec(x)dx = ln(sec(x) + tan(x)) + C 1 + cos(2x) 1 − cos(2x) tan2 (x) + 1 = sec2 (x) cos2 (x) = sin2 (x) = 2 2 ∫ ∫ Taylor series of the function f (x) about x = a: ∞ f ′ (a) f ′′ (a) f (n) (a) f (a) + (x − a) + (x − a)2 + ⋯ = ∑ (x − a)n 1! 2! n! n=0 Multiple Choice Answer Block A a b c d e D a b c d e B a b c d e E a b c d e C a b c d e F a b c d e 1) Sketch the polar curve r = 2 sin(2θ) in the following diagram: 5 4 3 2 1 0 −5 −4 −3 −2 −1 −1 0 1 2 3 4 5 −2 −3 −4 −5 b) Determine the area inside this curve. (You may — and should — use without proof that this curve has a 4-fold symmetry.) 2) a) b) ∫ ln(x) dx x4 (3x − 1) ∫ 9 − 2x + 3x dx 2 c) ∫ x3 √ dx 1 − x2 3) a) Determine all complex 3rd roots of 8i. b) Determine a complex number z such that e z = −1 + i 4) Determine, for example by appropriate manipulation of known power series, or by calculating the Taylor series about a = 0, power series for the following functions centered at 0. Collect sums of series into a single infinite sum. Your solution should make clear what steps were performed with series of what functions. If you use “+⋯” notation without indicating the generic form of terms, you should give at least the first four nonzero terms of the series to indicate the pattern of coefficients. x 1 dt(= arctanh(x)) for ∣x∣ < 1 a) 0 1 − t2 ∫ b) 3x − 3 . (Hint: Partial Fractions) (x + 1)(x − 2) 5) Solve the following initial value problem using power series. y ′ + 2x y = 2x, Identify the function from the power series solution. y(0) = 2 6) The following multiple choice problems will be graded correct answer only. You do not need to show work, and no partial credit will be given. Record your answer in the answer block on the front page. Answers given on these pages will not be scored. You also may tear off these pages and do not need to hand them in. It is strongly recommended that you work out the problems until the correct answer is uniquely determined and don’t just try to solve them by “intuition” or “guessing” – doing so is likely to result in a wrong pick. Each correct anwer is worth 6 points, each incorrect answer is counted as 0 points. (Unanswered questions are 1 point, questions in which more than one answer is ticked are considered to have been answered wrongly.) A) a B) An equation for the tangent to the graph of y = arcsin(x/2) at the origin is x − 2y = 0 b x−y=0 c x=0 Let g be the function given by g(x) = y=0 d ∫ 0 x e πx − 2y = 0 sin(t 2 )dt for −1 ≤ t ≤ 3. On which of the following intervals is g one-to-one. √ √ a √ −1 < x < 0 √b 0 < x < 2 π c 0 < x < π e π<x<2 π d 1 < x < π2 Which of the following three improper integrals are convergent C) A. ∫ 1 ∞ x 1.001 dx ln(x) B. ∫ 1 ∞ ln(x) dx x 1.001 C. ∫ 1 ∞ x 10 dx 2x Integrals A,B and C are convergent Integrals A and C are convergent, B is divergent Integrals B is convergent, A and C are divergent Integrals C is convergent, A and B are divergent Integrals B and C are convergent, A is divergent a b c d e a Which of the following statements about the convergence of 1 dx is true? x ln(x) It converges, because lim F(x) is finite for an antiderivative F(x) of the integrand b c 1 It converges by limit comparison with x 5/4 It diverges, because lim F(x) = ∞ for an antiderivative F(x) of the integrand d e It diverges by limit comparison with x1 We cannot decide on convergence using methods from the course D) ∫ 2 ∞ x→∞ x→∞ If 1 =/ z is a complex number such that E) z 20 18 = 1, then ∑ z n = n=0 a F) a c −1 b − 1 z c 0 d 1 e 1 z Which of the following is a power series expression for x2 ? 1 − x2 1 + x 2 + x 4 + x 6 + x 8 + ⋯ b x 2 + 2x 3 + 3x 4 + 4x 5 + ⋯ x2 + x3 + x4 + x5 + ⋯ d x2 + x4 + x6 + x8 + ⋯ e x2 − x4 + x6 − x8 + ⋯ If you are done and have time left, check your answers on all the problems. Is in each problem clear, what your answer is? Did you tick the correct boxes on the multiple choice questions?