Chapter 4

Only one of these calculations is correct. Which one? Why are the others wrong? Give reasons for your answers. a. \(\lim _{x \rightarrow 0^{+}} x \ln x=0 \cdot(-\infty)=0\) b. \(\lim _{x \rightarrow 0^{+}} x \ln x=0 \cdot(-\infty)=-\infty\) c. \(\lim _{x \rightarrow 0^{+}} x \ln x=\lim _{x \rightarrow 0^{-}} \frac{\ln x}{(1 / x)}=\frac{-\infty}{\infty}=-1\). d. \(\lim _{x \rightarrow 0^{+}} x \ln x=\lim _{x \rightarrow 0^{-}} \frac{\ln x}{(1 / x)}\). \(=\lim _{x \rightarrow 0^{+}} \frac{(1 / x)}{\left(-1 / x^{2}\right)}=\lim _{x \rightarrow 0^{+}}(-x)=0\).

Give reasons for your answers. Let \(f(x)=\left|x^{3}-9 x\right|\) a. Does \(f^{\prime}(0)\) exist? b. Does \(f^{\prime}(3)\) exist? c. Does \(f^{\prime}(-3)\) exist? d. Determine all extrema of \(f\)

Verify the formulas in Exercises by differentiation. $$\int x e^{x} d x=x e^{x}-e^{x}+C$$

Show that \(\left(e^{x_{1}}\right)^{x_{2}}=e^{x_{1} x_{2}}=\left(e^{x_{2}}\right)^{x_{1}}\) for any numbers \(x_{1}\) and \(x_{2}\).

Find all values of \(c\) that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval. a. \(f(x)=x, \quad g(x)=x^{2}\) \((a, b)=(-2,0)\) b. \(f(x)=x, \quad g(x)=x^{2}\) \((a, b)\) arbitrary c. \(f(x)=x^{3} / 3-4 x, \quad g(x)=x^{2}, \quad(a, b)=(0,3)\)

Find the absolute maximum value of \(f(x)=x^{2} \ln (1 / x)\) and say where it is assumed.

Find a value of \(c\) that makes the function $$f(x)=\left\\{\begin{array}{ll} \frac{9 x-3 \sin 3 x}{5 x^{3}}, & x \neq 0 \\\c, & x=0\end{array}\right.$$. continuous at \(x=0 .\) Explain why your value of \(c\) works.

If an even function \(f(x)\) has a local maximum value at \(x=c,\) can anything be said about the value of \(f\) at \(x=-c ?\) Give reasons for your answer.

a. Prove that \(e^{x} \geq 1+x\) if \(x \geq 0\). b. Use the result in part (a) to show that $$ e^{x} \geq 1+x+\frac{1}{2} x^{2} $$

For what values of \(a\) and \(b\) is $$\lim _{x \rightarrow 0}\left(\frac{\tan 2 x}{x^{3}}+\frac{a}{x^{2}}+\frac{\sin b x}{x}\right)=0 ?$$

If an odd function \(g(x)\) has a local minimum value at \(x=c,\) can anything be said about the value of \(g\) at \(x=-c ?\) Give reasons for your answer.

Show that increasing functions and decreasing functions are one to-one. That is, show that for any \(x_{1}\) and \(x_{2}\) in \(I, x_{2} \neq x_{1}\) implies \(f\left(x_{2}\right) \neq f\left(x_{1}\right)\)

We know how to find the extreme values of a continuous function \(f(x)\) by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers.

Verify the formulas in Exercises by differentiation. $$\int\left(\sin ^{-1} x\right)^{2} d x=x\left(\sin ^{-1} x\right)^{2}-2 x+2 \sqrt{1-x^{2}} \sin ^{-1} x+C$$

$$\text { Find } \lim _{x \rightarrow \infty}(\sqrt{x^{2}+1}-\sqrt{x})$$.

The function $$V(x)=x(10-2 x)(16-2 x), \quad 0

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int x \sin x \, d x=\frac{x^{2}}{2} \sin x+C\) b. \(\int x \sin x \, d x=-x \cos x+C\) c. \(\int x \sin x \, d x=-x \cos x+\sin x+C\)

0/0 Form Estimate the value of $$\lim _{x \rightarrow 1} \frac{2 x^{2}-(3 x+1) \sqrt{x}+2}{x-1}$$ by graphing. Then confirm your estimate with I'Hópital's Rule.

Consider the cubic function $$f(x)=a x^{3}+b x^{2}+c x+d$$ a. Show that \(f\) can have \(0,1,\) or 2 critical points. Give examples and graphs to support your argument. b. How many local extreme values can \(f\) have?

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{\sec ^{3} \theta}{3}+C\) b. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \tan ^{2} \theta+C\) c. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \sec ^{2} \theta+C\)

This exercise explores the difference between the limit $$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x^{2}}\right)^{x} $$ and the limit $$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$$. a. Use I'Hôpital's Rule to show that \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e\) T b. Graph $$f(x)=\left(1+\frac{1}{x^{2}}\right)^{x} \text { and } g(x)=\left(1+\frac{1}{x}\right)^{x}$$ together for \(x \geq 0 .\) How does the behavior of \(f\) compare with that of \(g\) ? Estimate the value of \(\lim _{x \rightarrow \infty} f(x)\). c. Confirm your estimate of \(\lim _{x \rightarrow \infty} f(x)\) by calculating it with I'Hópital's Rule.

The height of a body moving vertically is given by $$s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}, \quad g>0$$ with \(s\) in meters and \(t\) in seconds. Find the body's maximum height.

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int(2 x+1)^{2} d x=\frac{(2 x+1)^{3}}{3}+C\) b. \(\int 3(2 x+1)^{2} d x=(2 x+1)^{3}+C\) c. \(\int 6(2 x+1)^{2} d x=(2 x+1)^{3}+C\)

Peak alternating current Suppose that at any given time \(t\) (in seconds) the current \(i\) (in amperes) in an alternating current circuit is \(i=2 \cos t+2 \sin t .\) What is the peak current for this circuit (largest magnitude)?

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int \sqrt{2 x+1} d x=\sqrt{x^{2}+x+C}\) b. \(\int \sqrt{2 x+1} d x=\sqrt{x^{2}+x}+C\) c. \(\int \sqrt{2 x+1} d x=\frac{1}{3}(\sqrt{2 x+1})^{3}+C\)

Then find the extreme values of the function on the interval and say where they occur. $$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$$

Right, or wrong? Give a brief reason why. $$ \int \frac{-15(x+3)^{2}}{(x-2)^{4}} d x=\left(\frac{x+3}{x-2}\right)^{3}+C $$

Use limits to find horizontal asymptotes for each function. $$\text { a. } y=x \tan \left(\frac{1}{x}\right)$$ .$$\text { b. } y=\frac{3 x+e^{2 x}}{2 x+e^{3 x}}$$

Then find the extreme values of the function on the interval and say where they occur. $$g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7$$

Right, or wrong? Give a brief reason why. $$ \int \frac{x \cos \left(x^{2}\right)-\sin \left(x^{2}\right)}{x^{2}} d x=\frac{\sin \left(x^{2}\right)}{x}+C $$

$$\text { Find } f^{\prime}(0) \text { for } f(x)=\left\\{\begin{array}{ll}e^{-1 / x^{2}}, & x \neq 0 \\\0, & x=0\end{array}\right.$$.

Then find the extreme values of the function on the interval and say where they occur. $$h(x)=|x+2|-|x-3|, \quad-\infty

Which of the following graphs shows the solution of the initial value problem $$ \frac{d y}{d x}=2 x, \quad y=4 \text { when } x=1 ? $$ Give reasons for your answer. (GRAPHS CAN'T COPY)

The continuous extension of \((\sin x)^{x}\) to \([0, \pi]\) a. Graph \(f(x)=(\sin x)^{x}\) on the interval \(0 \leq x \leq \pi .\) What value would you assign to \(f\) to make it continuous at \(x=0 ?\) b. Verify your conclusion in part (a) by finding \(\lim _{x \rightarrow 0^{+}} f(x)\) with I'Hôpital's Rule. c. Returning to the graph, estimate the maximum value of \(f\) on \([0, \pi] .\) About where is max \(f\) taken on? -d. Sharpen your estimate in part (c) by graphing \(f^{\prime}\) in the same window to see where its graph crosses the \(x\) -axis. To simplify your work, you might want to delete the exponential factor from the expression for \(f^{\prime}\) and graph just the factor that has a

Then find the extreme values of the function on the interval and say where they occur. $$k(x)=|x+1|+|x-3|, \quad-\infty

The function \((\sin x)^{\tan x}\). a. Graph \(f(x)=(\sin x)^{\tan x}\) on the interval \(-7 \leq x \leq 7 .\) How do you account for the gaps in the graph? How wide are the gaps? b. Now graph \(f\) on the interval \(0 \leq x \leq \pi .\) The function is not defined at \(x=\pi / 2,\) but the graph has no break at this point. What is going on? What value does the graph appear to give for \(f\) at \(x=\pi / 2 ?\) (Hint: Use I'Hópital's Rule to find lim \(f^{-}\) as \(\left.x \rightarrow(\pi / 2)^{-} \text {and } x \rightarrow(\pi / 2)^{+} .\right)\) c. Continuing with the graphs in part (b), find max \(f\) and min \(f\) as accurately as you can and estimate the values of \(x\) at which they are taken on.

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25]$$

Solve the initial value problems in Exercises. $$\frac{d y}{d x}=2 x-7, \quad y(2)=0$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=-x^{4}+4 x^{3}-4 x+1, \quad[-3 / 4,3]$$

Solve the initial value problems in Exercises. $$\frac{d y}{d x}=10-x, \quad y(0)=-1$$.

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=x^{2 / 3}(3-x), \quad[-2,2]$$

Solve the initial value problems in Exercises. $$\frac{d y}{d x}=\frac{1}{x^{2}}+x, \quad x > 0 ; \quad y(2)=1$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=2+2 x-3 x^{2 / 3}, \quad[-1,10 / 3]$$

Solve the initial value problems in Exercises. $$\frac{d y}{d x}=9 x^{2}-4 x+5, \quad y(-1)=0$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=\sqrt{x}+\cos x, \quad[0,2 \pi]$$

Solve the initial value problems in Exercises. $$\frac{d y}{d x}=3 x^{-2 / 3}, \quad y(-1)=-5$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]$$

Solve the initial value problems in Exercises. $$\frac{d y}{d x}=\frac{1}{2 \sqrt{x}}, \quad y(4)=0$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=\pi x^{2} e^{-3 x / 2}, \quad[0,5]$$

Solve the initial value problems in Exercises. $$\frac{d s}{d t}=1+\cos t, \quad s(0)=4$$