HW9sol - ) = 1 2+ t 4 , and by FTC1, g ( x ) = F ( x 2 )-F...

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M408C, Homework 9 — Solution Instructor: Guoliang Wu 1. If f ( x ) = x 2 - 2 x, 0 x 3, evaluate the Riemann sum with n = 6, taking the sample points to be the middle endpoints of sub-intervals. Illustrate with a diagram. Solution: The graph f ( x ) = ( x - 1) 2 - 1 is as follows. The Riemann sum, with n = 6 and the sample points 1 / 4 , 3 / 4 , 5 / 4 , 7 / 4 , 9 / 4 , 11 / 4, is given by the net sum of the areas of the rectangles: M 6 = 1 2 f (1 / 4) + 1 2 f (3 / 4) + 1 2 f (5 / 4) + 1 2 f (7 / 4) + 1 2 f (9 / 4) + 1 2 f (11 / 4) = 71 16 . 2. Find the derivative g ( x ) of the function g ( x ) = x 2 tan x 1 2 + t 4 dt. 1
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Solution: Let F ( t ) be an antiderivative of 1 2+ t 4 , then F ( t
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Unformatted text preview: ) = 1 2+ t 4 , and by FTC1, g ( x ) = F ( x 2 )-F (tan x ). Therefore, by chain rule, g ( x ) = F ( x 2 )2 x-F (tan x ) sec 2 x = 2 x 2 + x 8-sec 2 x 2 + tan 4 x 3. Evaluate the integral / 4 1 1 + cos 2 cos 2 d. Solution: / 4 1 1 + cos 2 cos 2 d = / 4 1 ( 1 cos 2 + 1 ) d = / 4 1 sec 2 + 1 d = tan + ] / 4 1 = (1 + / 4)-(tan 1 + 1) = / 4-tan 1 . 2...
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HW9sol - ) = 1 2+ t 4 , and by FTC1, g ( x ) = F ( x 2 )-F...

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