w04 - the interval[2 7(be careful when examining the...

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Math 152 Workshop #4 Spring 2011 1. Determine how large n has to be in order to approximate the integral R 1 0 cos( x 2 ) dx , using the Midpoint Rule, with error at most 10 - 6 . Then use this value of n to calculate the integral to this accuracy. 2. Consider the function f ( x ) = e x sin Nx on the interval [0 , 1] where N is a positive integer. (a) With a sketch or otherwise, describe the graph of this function when N = 5, N = 10, and N = 100. (b) Compute R 1 0 f ( x ) dx . Evaluate this integral when N = 5, N = 10, and N = 100. (c) What happens to the graph and to the value of the integral as N → ∞ ? Does the graph confirm the limiting behavior of the integral’s value? 3. Calculate four of the following integrals: Z x cos x 2 dx ; Z x 2 cos x 2 dx ; Z x 2 cos xdx ; Z x 2 cos 2 xdx ; Z x cos 2 xdx. Comment Most people use many parentheses and rewrite the integrands to decrease possible confusion. So x 2 cos 2 x becomes x 2 (cos x ) 2 and x 2 cos x 2 becomes x 2 cos( x 2 ) . 4. Suppose f is defined by f ( x ) = 3 e cos x . Maple pro- duced graphs of f and its first four derivatives on
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Unformatted text preview: the interval [2 , 7] (be careful when examining the derivative graphs – look carefully at the vertical scales!). The graph of f is to the right, and the graphs of the first four derivatives of f are on the back of this page. You should assume that the graphs are correct for this problem. Suppose I is the value of Z 7 2 f ( x ) dx . (a) Use the graph of f alone to estimate I . (b) Use the information in the graphs to tell how many subdivisions N are needed so that the Trapezoid Rule approximation T N will approximate I with error < 10-5 . (c) Use the information in the graphs to tell how many subdivisions N are needed so that the Simpson’s Rule approximation S N will approximate I with error < 10-5 . Graph of f Graph of f 00 Graph of f (3) Graph of f (4)...
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This note was uploaded on 03/24/2011 for the course CALCULUS 152 taught by Professor Sosa during the Spring '11 term at Rutgers.

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w04 - the interval[2 7(be careful when examining the...

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