| utf15-004 Show that the sum of a positive number and its reciprocal must be at least 2. |
| utf16-001 A certain function $ f( x)$ defined for all $x$ has $f^{''}(0)=0$, $f^{''}(x)>0$ for $x>0$, and $f^{''}(x)<0$ for $x<0$. The number of critical points, number of local maxima, number of local minima, and number of roots of $ f( x)$ are all tabulated. Give all possible such tabulations.{({NOTE:} this problem appeared on Dr. Hamrick's 408C exam in 1988.)} |
| utf16-002 For the following functions:
&a) f( x) = x^4 + 4x & &b) f( x) = \frac{x}{ x^2 + 4}\\ &c) f( x) = \frac{(1+x)\sqrt {1+x^2} }{ x} & &d) f( x) = \frac{x^2 }{ x^2 -4} \\ &e) f( x) = \frac{x^2 }{ \sqrt {x^2 -2}} & &f) f( x) = x^3 + \frac{3}{ x}\\ &g) f( x) = 2\cos x \sin x -x, \text{on} [0,2\pi] & &h) 2\tan x - \sec^2 x, \text{on} (0,\frac{\pi}{ 2}) \end{alignat*} |
| utf17-001 Early in $1492$, Crist\'{o}bal Col\'{o}n was commissioned by King Ferdinand and Queen Isabella of Spain to journey west to reach the Orient. On September 6, 1492, Columbus (Col\'{o}n's name in Latin) left the Canary Islands to make history. Not knowing exactly which direction he should head, Columbus manages to get his ship moving east or west according to the function $x(t)=(t-36)^3 (100-t)$. We pick up the action at $t=0$ and follow him as $t\to\infty$. (Note that $t$ is in days, $x(t)$ is in meters, and that we ignore the reality that Columbus sailed southwestward, not just east and west.)
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| utf20-003 For each of the following solve the indefinite integral, describe your technique, then write two more integrals which would use the same technique to solve. \begin{alignat*}{2} &a) \int{\frac{1 }{ \sqrt {2x+1}} \sin {( \sqrt {2x+1} )}} & &b) \int {( \sqrt {\tan x} + ( \tan x )^\frac{1 }{ 3} ) \sec^2 x} \\ &c) \int {\cos^2 {4x}} & &d) \int {x^{-\frac{2}{ 3}}\cos {x^\frac{1}{ 3}} ( 1+ \sin^2 {x^\frac{1 }{ 3}} )} \\ &e) \int \frac{( \tan^\frac{1}{ 3} {\sqrt x} ) ( 1+\tan^2 {\sqrt x}) }{\sqrt x} & &f) \int {( 1 - \cos^2 x )^{-1}} \\ \end{alignat*} |