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Calc2_Test1

# Calc2_Test1 - imation for the above integral such that | e...

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MA 241, Spring 2008 Test # 1 Directions: Show all work, NO CREDIT will be given when there is no work! Answer the following using tools we have developed in this class. You only need to simplify when stated. There are 102 points possible. Write legibly and clearly mark your answers. Points may be taken off if you work is illegible. 1. Evaluate: (a) ( 8 points ) Z cot( x ) dx (b) ( 8 points ) Z x 3 / 2 ln ( x ) dx (c) ( 10 points ) 3 Z - 2 1 ( x - 2) 2 dx

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(d) ( 10 points ) Z x 3 + x 2 - 12 x + 7 x 2 + x - 12 dx (e) ( 8 points ) Z 2 x p ( x 2 - 2) 2 - 9 dx (Use table of integrals. Indicate which number you use.) 2. ( 10 points ) Determine the form of the partial fraction decomposition of the function. DO NOT determine the numerical values of the coefﬁcients. x 3 - x 2 + 5 x 3 ( x 2 + 1)( x 2 - 1)
3. ( 10 points ) Use the Comparison Theorem to determine whether the integral is convergent or divergent. Z 1 sin( x ) + 2 x - 0 . 5 dx 4. Consider the deﬁnite integral Z 9 5 1 x 2 dx (a) ( 8 points ) Use the Trapezoid Method to approximate the above integral with 4 subintervals. (b) ( 10 points ) Find the least number of subintervals for which the Trapezoid Rule provides an approx-

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Unformatted text preview: imation for the above integral such that | e T | ≤ 5 4 6(4 3 )(1200) . (Recall the Trapezoidal Rule Error Bound: Suppose f 00 ( x ) is continuous on [ a,b ] and | f 00 ( x ) | ≤ K for all x ∈ [ a,b ] then | e T | ≤ K ( b-a ) 3 12 n 2 ) 5. Let R be the region enclosed by the curves y = 1 + √ x and y = 3 + x 3 . (a) ( 4 points ) Make a clear sketch the region R labeling all curves and important points. Be sure to shade R . (b) ( 6 points ) i. Draw a typical approximating rectangle when integrating with respect to x and lable Δ x on your graph. ii. Set up the integral with respect to x that is used to ﬁnd the area of the region R . DO NOT EVALUATE! (c) ( 10 points ) i. Draw a typical approximating rectangle when integrating with respect to y and lable Δ y on your graph. ii. Set up the integral with respect to y that is used to ﬁnd the area of the region R . DO NOT EVALUATE!...
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Calc2_Test1 - imation for the above integral such that | e...

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