prex1sol

# prex1sol - Math 115 — Practice Exam 1 Solutions 1(a By...

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Unformatted text preview: Math 115 — Practice Exam 1 Solutions 1. (a) By the Fundamental Theorem of Calculus and the chain rule, d dx f ( x ) = 2 x 1- ( x 2 ) 3 + 1 1- x 3 (b) Observe the derivative simplifies to 2 x + 1 + x 3 1- x 6 . The denominator is positive through- out the domain, but the numerator is negative for x near- 1. Taking x =- 1 2 suffices, for there the numerator is (- 1 2 ) 3 < 0. 2. (a) Using the method of cylindrical shells, the distance to the origin is x and the height of the cylinder is ( x- 1) 2 . Then the volume is integraldisplay 2 2 πx ( x- 1) 2 dx = 2 π integraldisplay 2 ( x- 2 x 2 + x 3 ) dx = 4 π 3 . (b) Using the method of cross-sections, the cross-section at x is a disc of radius 2+( x- 1) 2 . Then the volume is integraldisplay 2 π (2 + ( x- 1) 2 ) 2 dx 3. (a) Here is a sketch of the region. (b) Area = integraltext- 1 ( x + 1) dx + integraltext 1 (1- x 2 ) dx = ( x 2 2 + x ) |- 1 + ( x- x 3 3 ) | 1 = 7 6 ....
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prex1sol - Math 115 — Practice Exam 1 Solutions 1(a By...

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