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Print Name APPM 1360 Exam 1 Spring 2016 On the front of your bluebook, please write: a grading key, your name, student ID, your lecture number and instructor. This exam is worth 100 points and has 6 questions on both sides of this paper. • Submit this exam sheet with your bluebook. However, nothing on this exam sheet will be graded. Make sure all of your work is in your bluebook. • Show all work and simplify your answers! Answers with no justification will receive no points. • Please begin each problem on a new page. • No notes or papers, calculators, cell phones, or electronic devices are permitted. 1. (30 points, 10 points each) Evaluate the following integrals. Z sin θ (a) dθ cos2 θ + cos θ − 2 Z 1 (b) dt 2 t − 8t + 20 Z 1 (c) dx 2 (x − 1)3/2 Z π/4 sec2 x √ dx convergent or divergent? Fully justify your answer. 2. (10 points) Is x 0 Z 4 3. (20 points) For this problem let I = ln x2 dx. 2 (a) Estimate I using the trapezoidal approximation T2 . Simplify your answer using the approximations ln 2 ≈ 0.7 and ln 3 ≈ 1.1. (b) Now find the exact value of I. (c) How many subintervals are needed to ensure that a trapezoidal approximation of I is accurate to within 10−4 ? Simplify your answer. 4. (12 points) Consider the region enclosed by the line y = x − 1 and the parabola y 2 = 2x + 6. (a) Sketch this region. (b) Set up, but don’t evaluate, the integral to calculate the area enclosed by this region. 5. (16 points) Consider the region bounded by y = cos x and y = tan x from x = 0 to π6 . (a) Sketch this region. (b) Find the volume of the solid generated when this region is rotated about the x-axis. TURN OVER - More problems on the back! 6. (12 points) Unrelated, short answer questions. x2 + 5 . (Note: Do not try to solve for the (x − 1)2 (x2 + 1)3 coefficients. This problem is just asking about the form of the partial fraction decomposition.) 1 (b) True or False and briefly explain your answer: Let f (x) = p and p > 0 be a constant. If x Z Z (a) Write the partial fraction decomposition of ∞ 1 f (x) dx converges. f (x) dx diverges, then 0 1 (c) Z True or False and briefly explain your Z ∞answer: If f (x) is continuous for all x ∈ [0, ∞) and if ∞ f (x) dx diverges, then so does f (x) dx for all a > 0. 0 a Some Trigonometric identities 2 cos2 (x) = 1 + cos(2x) 2 sin2 (x) = 1 − cos(2x) sin(2x) = 2 sin(x) cos(x) cos(2x) = cos2 (x) − sin2 (x) Inverse Trigonometric Integral Identities Z du √ = sin−1 (u/a) + C, u2 < a2 2 2 a −u Z du 1 = tan−1 (u/a) + C 2 2 a Z a +u du 1 √ = sec−1 |u/a| + C, u2 > a2 a u u2 − a2 Midpoint Rule Z b f (x)dx ≈ Mn = ∆x[f (x̄1 ) + f (x̄2 ) + · · · + f (x̄n )] where ∆x = a |EM | ≤ b−a xi−1 + xi and x̄i = and n 2 K(b − a)3 . 24n2 Trapezoidal Rule Z b f (x)dx ≈ Tn = a ∆x b−a K(b − a)3 [f (x0 )+2f (x1 )+· · ·+2f (xn−1 )+f (xn )] where ∆x = and |ET | ≤ . 2 n 12n2