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InsallW98T2

# InsallW98T2 - Z 1 ln(1 e x dx Find bounds on the error of...

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Insall - WS98 Calculus II Exam #2 February 19, 1998 1. (18 pts.) Theorems and Techniques: (a) Complete the statement of the following theorem. If f and g are differentiable functions on some interval, then Z f ( x ) g 0 ( x ) dx = (b) Explain how to use the method of partial fractions. 2. (18 pts.) Indefinite Integrals: Evaluate the following. (a) Z 1 x ( x + 1)(2 x + 3) dx (b) Z 9 x 2 - 4 x dx 3. (16 pts.) Definite Integrals: Compute the following. (a) Z π 2 0 sin 2 (3 x ) dx (b) Z π 2 π 4 x csc 2 ( x ) dx 4. (16 pts.) Approximate Integration: Solve the following: (a) Use the Trapezoidal Rule, with 8 panels, to approximate Z 1 / 2 0 cos( e x ) dx . Find bounds on the error of the approximation. (b) Use Simpson’s Rule, with 8 panels, to approximate
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Unformatted text preview: Z 1 ln(1 + e x ) dx . Find bounds on the error of the approximation. 5. (16 pts.) Improper Integrals: For each of the following, tell whether the given integral converges or diverges, and if it converges, compute its value. (a) Z ∞ cos( x ) dx (b) Z 3 1 √ x dx 6. (16 pts.) Proofs: Prove the following: (a) Z x n e x dx = x n e x-n Z x n-1 e x dx (b) For any cubic polynomial f , given by f ( x ) = c + c 1 x + c 2 x 2 + c 3 x 3 , the Simpson’s rule approximation to R b a f ( x ) dx is exact....
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