Chapter 7

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

Use the definitions of the hyperbolic functions to find each of the following limits. a. \(\lim _{x \rightarrow \infty} \tanh x\) b. \(\lim _{x \rightarrow-\infty} \tanh x\) c. \(\lim _{x \rightarrow \infty} \sinh x\) d. \(\lim _{x \rightarrow-\infty} \sinh x\) e. \(\lim _{x \rightarrow \infty} \operatorname{sech} x\) f. \(\lim _{x \rightarrow \infty} \operatorname{coth} x\) g. \(\lim _{x \rightarrow 0^{+}} \operatorname{coth} x\) h. \(\lim _{x \rightarrow 0^{-}} \operatorname{coth} x\) i. \(\lim _{x \rightarrow-\infty} \operatorname{csch} x\)

Find the derivative of \(y\) with respect to the given independent variable. $$y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$$

Evaluate the integrals. $$\int \frac{e^{\cos ^{-1} x} d x}{\sqrt{1-x^{2}}}$$

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'Hôpital's Rule.

Hanging cables Imagine a cable, like a telephone line or TV cable, strung from one support to another and hanging freely. The cable's weight per unit length is a constant \(w\) and the horizontal tension at its lowest point is a vector of length \(H\). If we choose a coordinate system for the plane of the cable in which the \(x\) -axis is horizontal, the force of gravity is straight down, the positive \(y\) -axis points straight up, and the lowest point of the cable lies at the point \(y=H / w\) on the \(y\) -axis (see accompanying figure), then it can be shown that the cable lies along the graph of the hyperbolic cosine Such a curve is sometimes called a chain curve or a catenary, the latter deriving from the Latin catena, meaning "chain." a. Let \(P(x, y)\) denote an arbitrary point on the cable. The next accompanying figure displays the tension at \(P\) as a vector of length (magnitude) \(T\), as well as the tension \(H\) at the lowest point \(A .\) Show that the cable's slope at \(P\) is b. Using the result from part (a) and the fact that the horizontal tension at \(P\) must equal \(H\) (the cable is not moving), show that \(T=w y .\) Hence, the magnitude of the tension at \(P(x, y)\) is exactly equal to the weight of \(y\) units of cable.

Evaluate the integrals in Exercises \(83-92.\) $$\int 5^{x} d x$$

Evaluate the integrals. $$\int \frac{\left(\sin ^{-1} x\right)^{2} d x}{\sqrt{1-x^{2}}}$$

Solve the initial value problems $$\frac{d y}{d x}=1+\frac{1}{x}, \quad y(1)=3$$

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 l'Hôpital's Rule to show that $$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$$ 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'Hôpital's Rule.

Evaluate the integrals in Exercises \(83-92.\) $$\int \frac{3^{x}}{3-3^{x}} d x$$

Evaluate the integrals. $$\int \frac{\sqrt{\tan ^{-1} x} d x}{1+x^{2}}$$

Solve the initial value problems $$\frac{d^{2} y}{d x^{2}}=\sec ^{2} x, \quad y(0)=0 \quad \text { and } \quad y^{\prime}(0)=1$$

Show that $$\lim _{k \rightarrow \infty}\left(1+\frac{r}{k}\right)^{k}=e^{r}$$

Area Show that the area of the region in the first quadrant enclosed by the curve \(y=(1 / a) \cosh a x,\) the coordinate axes, and the line \(x=b\) is the same as the area of a rectangle of height \(1 / a\) and length \(s,\) where \(s\) is the length of the curve from \(x=0\) to \(x=b .\) Draw a figure illustrating this result.

Evaluate the integrals in Exercises \(83-92.\) $$\int_{0}^{1} 2^{-\theta} d \theta$$

Evaluate the integrals. $$\int \frac{d y}{\left(\tan ^{-1} y\right)\left(1+y^{2}\right)}$$

Instead of approximating \(\ln x\) near \(x=1,\) we approximate \(\ln (1+x)\) near \(x=0\) We get a simpler formula this way. a. Derive the linearization \(\ln (1+x) \approx x\) at \(x=0\) b. Estimate to five decimal places the error involved in replacing \(\ln (1+x)\) by \(x\) on the interval [0,0.1] c. Graph \(\ln (1+x)\) and \(x\) together for \(0 \leq x \leq 0.5 .\) Use different colors, if available. At what points does the approximation of \(\ln (1+x)\) seem best? Least good? By reading coordinates from the graphs, find as good an upper bound for the error as your grapher will allow.

Given that \(x>0\), find the maximum value, if any, of \(\begin{array}{lll}\text { a. } x^{1 / x} & \text { b. } x^{1 / x^{2}} & \text { c. } x^{1 / x^{n}} \text { ( } n \text { a positive integer) }\end{array}\) d. Show that \(\lim _{x \rightarrow \infty} x^{1 / x^{n}}=1\) for every positive integer \(n\).

Evaluate the integrals in Exercises \(83-92.\) $$\int_{-2}^{0} 5^{-\theta} d \theta$$

Evaluate the integrals. $$\int \frac{d y}{\left(\sin ^{-1} y\right) \sqrt{1-y^{2}}}$$

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

Evaluate the integrals in Exercises \(83-92.\) $$\int_{1}^{\sqrt{2}} x 2^{\left(x^{2}\right)} d x$$

Evaluate the integrals. $$\int_{\sqrt{2}}^{2} \frac{\sec ^{2}\left(\sec ^{-1} x\right) d x}{x \sqrt{x^{2}-1}}$$

a. Graph \(y=\sin x\) and the curves \(y=\ln (a+\sin x)\) for \(a=2,4,8,20,\) and 50 together for \(0 \leq x \leq 23\) b. Why do the curves flatten as \(a\) increases? (Hint: Find an a-dependent upper bound for \(\left|y^{\prime}\right| .\) )

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

Evaluate the integrals in Exercises \(83-92.\) $$\int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} d x$$

Evaluate the integrals. $$\int_{2 / \sqrt{3}}^{2} \frac{\cos \left(\sec ^{-1} x\right) d x}{x \sqrt{x^{2}-1}}$$

Does the graph of \(y=\sqrt{x}-\ln x, x>0,\) have an inflection point? Try to answer this question (a) by graphing, (b) by using calculus.

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 l'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 zero.

Evaluate the integrals in Exercises \(83-92.\) $$\int_{0}^{\pi / 2} 7^{\cos t} \sin t d t$$

Evaluate the integrals. $$\int \frac{1}{\sqrt{x}(x+1)\left(\left(\tan ^{-1} \sqrt{x}\right)^{2}+9\right)} d x$$

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 1 'Hôpital'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.

Evaluate the integrals in Exercises \(83-92.\) $$\int_{0}^{\pi / 4}\left(\frac{1}{3}\right)^{\tan t} \sec ^{2} t d t$$

Evaluate the integrals. $$\int \frac{e^{x} \sin ^{-1} e^{x}}{\sqrt{1-e^{2 x}}} d x$$

Evaluate the integrals in Exercises \(83-92.\) $$\int_{2}^{4} x^{2 x}(1+\ln x) d x$$

Find the limits. $$\lim _{x \rightarrow 0} \frac{\sin ^{-1} 5 x}{x}$$

Find the limits. $$\lim _{x \rightarrow 1^{+}} \frac{\sqrt{x^{2}-1}}{\sec ^{-1} x}$$

Evaluate the integrals in Exercises \(93-106.\) $$\int 3 x^{\sqrt{3}} d x$$

Find the limits. $$\lim _{x \rightarrow \infty} x \tan ^{-1} \frac{2}{x}$$

Evaluate the integrals in Exercises \(93-106.\) $$\int x^{\sqrt{2}-1} d x$$

Find the limits. $$\lim _{x \rightarrow 0} \frac{2 \tan ^{-1} 3 x^{2}}{7 x^{2}}$$

Evaluate the integrals in Exercises \(93-106.\) $$\int_{0}^{3}(\sqrt{2}+1) x^{\sqrt{2}} d x$$

Find the limits. $$\lim _{x \rightarrow 0} \frac{\tan ^{-1} x^{2}}{x \sin ^{-1} x}$$

Evaluate the integrals in Exercises \(93-106.\) $$\int_{1}^{e} x^{(\ln 2)-1} d x$$

Find the limits. $$\lim _{x \rightarrow \infty} \frac{e^{x} \tan ^{-1} e^{x}}{e^{2 x}+x}$$

Evaluate the integrals in Exercises \(93-106.\) $$\int \frac{\log _{10} x}{x} d x$$

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

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

Evaluate the integrals in Exercises \(93-106.\) $$\int_{1}^{4} \frac{\ln 2 \log _{2} x}{x} d x$$