Exam 2: Derivatives

For the function f whose graph is shown, state the following. $\lim _{x \rightarrow-4} f(x)$

Find the limit. $\lim _{x \rightarrow 0^{+}} \tan ^{-1}\left(\frac{2}{x}\right)$

Use a graph to find a number $\delta$ such that $\left|\sin x-\frac{1}{2}\right|<0.4$ whenever $\left|x-\frac{\pi}{6}\right|<\delta$ . Round your answer to the nearest thousandth.

Evaluate the limit. $\lim _{x \rightarrow 3}\left(\frac{x^{3}-7}{x^{2}-2}\right)$

At what point is the function $f(x)=|4-x|$ not differentiable.

Use the definition of the limit to find values of $\delta$ that corresponds to $\varepsilon=0.75$ . $\lim _{x \rightarrow 1}\left(4+x-3 x^{3}\right)=2$ Round your answer to the nearest thousandth.

If an equation of the tangent line to the curve $y=f(x)$ at the point where $a=3 \text { is } y=4 x-9 \text {, }$ $\text { find } f(3) \text { and } f^{\prime}(3) \text {. }$

Find the limit by evaluating the derivative of a suitable function at an appropriate value of x. (Hint: Use the definition of the derivative.) $\lim _{h \rightarrow 0} \frac{2(5+h)^{2}+5(5+h)-75}{h}$

By graphing the function $f(x)=\frac{(\cos x-\cos 6 x)}{x^{2}}$ and zooming in toward the point where the graph crosses the y-axis, estimate the value of $\lim _{x \rightarrow 0} f(x)$

Which of the given functions is discontinuous?

Find a function g that agrees with f for $x \neq 81$ and is continuous on . $f(x)=\frac{9-\sqrt{x}}{81-x}$

Evaluate the limit. $\lim _{x \rightarrow 1}(x+4)^{3}\left(x^{2}-9\right)$

A machinist is required to manufacture a circular metal disk with area 1000 $\mathrm{cm}^{2}$ . If the machinist is allowed an error tolerance of $\pm 15 \mathrm{~cm}^{2}$ in the area of the disk, how close to the ideal radius must the machinist control the radius? Round your answer to the nearest hundred thousandth.

Use the graph of the function to find each limit. $\lim _{x \rightarrow-2^{-}} f(x)$ , $\lim _{x \rightarrow-2^{+}} f(x)$ , $\lim _{x \rightarrow-2} f(x)$

Write the expression as a derivative of a function of x. $\lim _{h \rightarrow 0} \frac{\frac{1}{19(x+h)}+7 \sqrt{x+h}-\frac{1}{19 x}-7 \sqrt{x}}{h}$

Find an equation of the tangent line to the curve $y=x^{3}-2 x$ at the point $(2,6)$ .

How close to -8 do we have to take x so that $\frac{1}{(x+8)^{2}}>100 ?$

If a cylindrical tank holds $10000$ gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume of water remaining in the tank after t minutes as $V(t)=10000\left(1-\frac{1}{60} t\right)^{2}, 0 \leq t \leq 60$ Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t.

Sketch the graph of the function f and evaluate $\lim _{x \rightarrow-3^{+}} f(x)$ $f(x)=\left\{\begin{array}{ll}x+5, & \text { if } x \leq-3 \\-2 x-1, & \text { if } x>-3\end{array}\right.$

If $1 \leq f(x) \leq x^{2}+6 x+6$ for all x, find $\lim _{x \rightarrow-1} f(x)$ .