| utf20-007 Consider $f(x)=2x-x^2$ on the interval $ [ 0,2 ] $.
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| utf21-006 Let $ f( x)$ be differentiable on $ [0,1] $, the second derivative exists for all $x \in [0,1] $ and $ f'( 0) = 0, $ $ f'( 1) = 0, $ $ f( 0) = 0, $ and $ f( 1) = 1.$ Show that for some $a \in ( 0,1), | { f''( a)}| \geq 4.$ |
| utf22-001 The following integrals all require the same $\underline {\text {trick}}$ to solve using u-substitution. Solve the integrals and describe the trick. \begin{alignat*}{3} &a) \int {x {\sqrt{x+1}}} & &b) \int {\frac{x+3 }{ {\sqrt {x+1}}}} & &c) \int {t^2 ( 2t-1 )^{-7} dt} \\ &d) \int {t^3 ( t^2+6 )^\frac{1 }{ 2} dt} & &e) \int {x^8 ( x^3+1 )^{-\frac{17 }{ 2}}} \end{alignat*} |