| utf06-004 Let $ f( x) = \frac{1}{ x}\sin x$, assume we only take $x>0$.
|
| utf06-007 Show that if $0\leq f( x) \leq 1$ for each $x\in [0,1] $ and $ f( x)$ is a continuous function, then there is some number $a$, $0\leq a \leq 1$, such that $ f( a) = a$. |
| utf07-001 True or False. If false, give a counter-example.
|
| utf07-002 Match each of the derivatives shown below with the corresponding functions. Which functions shown below are continuous? Which derivatives shown below are continuous? Explain what shapes continuous functions must have to produce jump discontinuities in the derivative and asymptotes in the derivative. Draw a continuous function whose derivative has a jump discontinuity at $x=1$ and an asymptote at $x=5$; draw the derivative. |
| utf08-001 Sketch the derivative of each of the functions below. |