Suppose $f(3)=2$, $f^{\prime}(3)=\frac{1}{2}$, and $f^{\prime}(x)>0$ and $f^{\prime \prime}(x)<0$ for all $x$. Is it possible that $f^{\prime}(2)=\frac{1}{3}$? Why?

Produce graphs of $f$ that reveal all the important aspects of the curve. In particular, you should use graphs of $f^{\prime}$ and $f^{\prime \prime}$ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$$f(x)=\dfrac{x}{x^3-x^2-4 x+1}$$

Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value.

$$\lim _{x \rightarrow 0} \frac{5^x-4^x}{3^x-2^x}$$

Suppose $f(3)=2$, $f^{\prime}(3)=\frac{1}{2}$, and $f^{\prime}(x)>0$ and $f^{\prime \prime}(x)<0$ for all $x$. Sketch a possible graph for $f$.

Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.

A company estimates that the marginal cost (in dollars per item) of producing $x$ items is $1.92-0.002 x$. If the cost of producing one item is $562, find the cost of producing 100 items.

Find the absolute maximum and absolute minimum values of $f(x)=x-\ln x$ on the interval $\left[\frac{1}{2}, 2\right]$.

Use the following graph to state the absolute and local maximum and minimum values of the function.

A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at , average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales?

$f^{\prime}(x)>0$ if $x \neq 2, \quad f^{\prime \prime}(x)>0$ if $x<2$, $f^{\prime \prime}(x)<0$ if $x>2, f$ has inflection point $(2,5)$, $\lim _{x \rightarrow \infty} f(x)=8, \lim _{x \rightarrow-\infty} f(x)=0$

A cylindrical tank with radius 5 m is being filled with water at a rate of $3 \mathrm{~m}^3 / \mathrm{min}$. How fast is the height of the water increasing?

Find the limit. Use l'Hospital's Rule where appropriate. If there is an applicable alternate method to l'Hospital's Rule, consider using it instead. If l'Hospital's Rule does not apply, explain why.

$$\displaystyle\lim _{x \rightarrow-\infty} x \ln \left(1-\frac{1}{x}\right)$$

For $x_1=5$, determine graphically what happens if Newton's method is used for the function whose graph is shown.

Two balls are thrown upward from the edge of the cliff. The first is thrown with a speed of 48 ft/s and the other is thrown a second later with a speed of 24 ft/s. Do the balls ever pass each other?