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Problem 1abc Points 15 1d 5 2 10 3 10 4 15 5 15 6 15 7 10 8 5 Total 100 Score M161, Test 3, Fall 2004 NAME: SECTION: INSTRUCTOR: You may use calculators. stored in your calculators. No formulas can be 1 + cos 2θ 1 − cos 2θ sin2 θ = 2 2 ′′ Error Bounds. Suppose |f (x)| ≤ K for a ≤ x ≤ b. If ET is the error in the Trapezoidal Rule, then cos2 θ = |ET | ≤ K∆x2 (b − a) K(b − a)3 or it may be written as |E | ≤ . T 12n2 12 Suppose |f (4) (x)| ≤ K for a ≤ x ≤ b. If ES is the error involved in using Simpson’s Rules, then |ES | ≤ K(b − a)5 K∆x4 (b − a) or it may be written as |ES | ≤ . 4 180n 180 1. Determine whether the series is convergent or divergent. In either case explain why. You must justify your answers. ∞ X 1 √ (a) n+1 n=1 ∞ X (b) 1 √ 3 n +n+1 n=1 (c) ∞ X (−1)n √ n+1 n=1 (d) ∞ X 32n n5 n=4 2. Determine whether the series is absolutely convergent, conditionally convergent or divergent. In any case, explain why. ∞ X ln n (a) (−1)n n n=1 (b) ∞ X (−100)n n! n=1 3. Find the interval of convergence of the series ∞ X (x + 4)n . n3n n=1 4. The parametric equations x = 4 sin(t − π), y = 2 cos(t − π), 0 ≤ t ≤ π describe the motion of a particle in the x-y plane. (a) Sketch the path of the particle indicating the direction of motion. 4 3 2 1 0 −1 −2 −3 −4 −2 −1.5 −1 −0.5 0 x 0.5 1 1.5 2 (b) What is the equation of the tangent to the curve at the point where t = 3π/4? Draw the tangent line on the curve drawn above. (c) What are the Cartesian coordinates of the point where t = π. (d) Find the Cartesian equation for the parametric equations. 5. (a) Sketch the curve r = 1 + 2 cos θ. 4 3 2 1 0 −1 −2 −3 −4 −2 −1.5 −1 −0.5 0 x 0.5 (b) Find the area inside the large loop of the curve. (c) Find the area inside the small loop of the curve. 1 1.5 2 6. (a) Find the Cartesian equation form of the polar curve r cos θ − y sin θ = 7. (b) Find the polar equation for the Cartesian equation xy = 2. (c) Find the Cartesian equation form of the polar curve r2 = 4r sin θ. Identify the curve. 7. Use power series to solve the initial value problem y ′ + 2y = 0, y(0) = 1. 8. Find the power series representation of f (x) = 1 d2 1 2 . Hint: . = (1 − x)3 dx2 1 − x (1 − x)3