Chapter 4

Find the limits. $$\lim _{x \rightarrow 0^{-}} \sin x \cdot \ln x$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\cos ^{-1}\left(x^{2}\right)$$

If the graphs of two differentiable functions \(f(x)\) and \(g(x)\) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer.

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\begin{aligned} &\int \cot ^{2} x d x\\\ &\text { (Hint: }\left.1+\cot ^{2} x=\csc ^{2} x\right) \end{aligned}$$

a. How close does the curve \(y=\sqrt{x}\) come to the point \((3 / 2,0) ?\) (Hint: If you minimize the square of the distance, you can avoid square roots.) b. Graph the distance function \(D(x)\) and \(y=\sqrt{x}\) together and reconcile what you see with your answer in part (a). GRAPH CANT COPY

Sketch the graph of a differentiable function \(y=f(x)\) through the point (1,1) if \(f^{\prime}(1)=0\) and a. \(f^{\prime}(x)>0\) for \(x<1\) and \(f^{\prime}(x)<0\) for \(x>1\) b. \(f^{\prime}(x)<0\) for \(x<1\) and \(f^{\prime}(x)>0\) for \(x>1\) c. \(f^{\prime}(x)>0\) for \(x \neq 1\) d. \(f^{\prime}(x)<0\) for \(x \neq 1\)

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow \infty} \frac{\sqrt{9 x+1}}{\sqrt{x+1}}$$

If \(|f(w)-f(x)| \leq|w-x|\) for all values \(w\) and \(x\) and \(f\) is a differentiable function, show that \(-1 \leq f^{\prime}(x) \leq 1\) for all \(x\) -values.

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(1-\cot ^{2} x\right) d x$$

a. How close does the semicircle \(y=\sqrt{16-x^{2}}\) come to the point \((1, \sqrt{3}) ?\) b. Graph the distance function and \(y=\sqrt{16-x^{2}}\) together and reconcile what you see with your answer in part (a).

Sketch the graph of a differentiable function \(y=f(x)\) that has a. a local minimum at (1,1) and a local maximum at (3,3) b. a local maximum at (1,1) and a local minimum at (3,3) c. local maxima at (1,1) and (3,3) d. local minima at (1,1) and (3,3)

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow 0^{+}} \frac{\sqrt{x}}{\sqrt{\sin x}}$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x^{2 / 3}(x+2)$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \cos \theta(\tan \theta+\sec \theta) d \theta$$

Assume that \(f\) is differentiable on \(a \leq x \leq b\) and that \(f(b)

Sketch the graph of a continuous function \(y=g(x)\) such that a. \(g(2)=2,02,\) and \(g^{\prime}(x) \rightarrow-1^{+}\) as \(x \rightarrow 2^{+}\) b. \(g(2)=2, g^{\prime}<0\) for \(x<2, g^{\prime}(x) \rightarrow-\infty\) as \(x \rightarrow 2^{-}\) \(g^{\prime}>0\) for \(x>2,\) and \(g^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 2^{+}\)

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow(\pi / 2)} \frac{\sec x}{\tan x}$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x^{2 / 3}\left(x^{2}-4\right)$$

Let \(f\) be a function defined on an interval \([a, b] .\) What conditions could you place on \(f\) to guarantee that $$\min f^{\prime} \leq \frac{f(b)-f(a)}{b-a} \leq \max f^{\prime}$$ where \(\min f^{\prime}\) and \(\max f^{\prime}\) refer to the minimum and maximum values of \(f^{\prime}\) on \([a, b]\) ? Give reasons for your answers.

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{\csc \theta}{\csc \theta-\sin \theta} d \theta$$

Sketch the graph of a continuous function \(y=h(x)\) such that a. \(h(0)=0,-2 \leq h(x) \leq 2\) for all \(x, h^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 0^{-}\) and \(h^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 0^{+}\) b. \(h(0)=0,-2 \leq h(x) \leq 0\) for all \(x, h^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 0^{-}\) and \(h^{\prime}(x) \rightarrow-\infty\) as \(x \rightarrow 0^{+}\)

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow 0^{+}} \frac{\cot x}{\csc x}$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x \sqrt{4-x^{2}}$$

Verify the formulas in Exercises by differentiation. $$\int(7 x-2)^{3} d x=\frac{(7 x-2)^{4}}{28}+C$$

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow \infty} \frac{2^{x}-3^{x}}{3^{x}+4^{x}}$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x^{2} \sqrt{3-x}$$

Verify the formulas in Exercises by differentiation. $$\int(3 x+5)^{-2} d x=-\frac{(3 x+5)^{-1}}{3}+C$$

Find the open intervals on which the function \(f(x)=a x^{2}+\) \(b x+c, a \neq 0,\) is increasing and decreasing. Describe the reasoning behind your answer.

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow-\infty} \frac{2^{x}+4^{x}}{5^{x}-2^{x}}$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll} 4-2 x, & x \leq 1 \\ x+1, & x>1 \end{array}\right.$$

Verify the formulas in Exercises by differentiation. $$\int \sec ^{2}(5 x-1) d x=\frac{1}{5} \tan (5 x-1)+C$$

Let \(f\) be differentiable at every value of \(x\) and suppose that \(f(1)=1,\) that \(f^{\prime}<0\) on \((-\infty, 1),\) and that \(f^{\prime}>0\) on \((1, \infty)\). a. Show that \(f(x) \geq 1\) for all \(x\). b. Must \(f^{\prime}(1)=0 ?\) Explain.

Determine the values of constants \(a\) and \(b\) so that \(f(x)=\) \(a x^{2}+b x\) has an absolute maximum at the point (1,2)

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow \infty} \frac{e^{x^{2}}}{x e^{x}}$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll} 3-x, & x<0 \\ 3+2 x-x^{2}, & x \geq 0 \end{array}\right.$$

Verify the formulas in Exercises by differentiation. $$\int \csc ^{2}\left(\frac{x-1}{3}\right) d x=-3 \cot \left(\frac{x-1}{3}\right)+C$$

Determine the values of constants \(a, b, c,\) and \(d\) so that \(f(x)=a x^{3}+b x^{2}+c x+d\) has a local maximum at the point (0,0) and a local minimum at the point (1,-1)

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll}-x^{2}-2 x+4, & x \leq 1 \\\\-x^{2}+6 x-4, & x>1\end{array}\right.$$

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

Locate and identify the absolute extreme values of a. \(\ln (\cos x)\) on \([-\pi / 4, \pi / 3]\) b. \(\cos (\ln x)\) on \([1 / 2,2]\)

Which one is correct, and which one is wrong? Give reasons for your answers. $$\text { a. } \lim _{x \rightarrow 3} \frac{x-3}{x^{2}-3}=\lim _{x \rightarrow 3} \frac{1}{2 x}=\frac{1}{6}$$.$$\text { b. } \lim _{x \rightarrow 3} \frac{x-3}{x^{2}-3}=\frac{0}{6}=0$$.

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll}-\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4}, & x \leq 1 \\\x^{3}-6 x^{2}+8 x, & x>1\end{array}\right.$$

Use the same-derivative argument to prove the identities a. \(\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\) b. \(\sec ^{-1} x+\csc ^{-1} x=\frac{\pi}{2}\)

Verify the formulas in Exercises by differentiation. $$\int \frac{1}{(x+1)^{2}} d x=\frac{x}{x+1}+C$$

a. Prove that \(f(x)=x-\ln x\) is increasing for \(x>1\) b. Using part (a), show that \(\ln x1\)

Which one is correct, and which one is wrong? Give reasons for your answers. $$\begin{aligned}&\text { a. } \lim _{x \rightarrow 0} \frac{x^{2}-2 x}{x^{2}-\sin }=\lim _{x \rightarrow 0} \frac{2 x-2}{2 x-\cos x}\\\&=\lim _{x \rightarrow 0} \frac{2}{2+\sin x}=\frac{2}{2+0}=1\end{aligned}$$ $$\text { b. } \lim _{x \rightarrow 0} \frac{x^{2}-2 x}{x^{2}-\sin x}=\lim _{x \rightarrow 0} \frac{2 x-2}{2 x-\cos x}=\frac{-2}{0-1}=2$$

Give reasons for your answers. Let \(f(x)=(x-2)^{2 / 3}\) a. Does \(f^{\prime}(2)\) exist? b. Show that the only local extreme value of \(f\) occurs at \(x=2\) c. Does the result in part (b) contradict the Extreme Value Theorem? d. Repeat parts (a) and (b) for \(f(x)=(x-a)^{2 / 3},\) replacing 2 by \(a\)

Verify the formulas in Exercises by differentiation. $$\int \frac{1}{x+1} d x=\ln |x+1|+C, \quad x \neq-1$$

Starting with the equation \(e^{x_{1}} e^{x_{2}}=e^{x_{1}+x_{2}},\) derived in the text, show that \(e^{-x}=1 / e^{x}\) for any real number \(x .\) Then show that \(e^{x_{1}} / e^{x_{2}}=e^{x_{1}-x_{2}}\) for any numbers \(x_{1}\) and \(x_{2}\).

Find the a$$0$$bsolute maximum and minimum values of \(f(x)=\) \(e^{x}-2 x\) on [0,1]