Chapter 7

Evaluate the integrals. $$\int_{1 / 2}^{1} \frac{6 d t}{\sqrt{3+4 t-4 t^{2}}}$$

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

L'Hópital's Rule does not help with the limits. 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 derivative of \(y\) with respect to the given independent variable. $$y=x^{3} \log _{10} x$$

Evaluate the integrals. $$\int \frac{d y}{y^{2}-2 y+5}$$

Find the area between the curves \(y=\ln x\) and \(y=\ln 2 x\) from \(x=1\) to \(x=5\)

L'Hópital's Rule does not help with the limits. 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 derivative of \(y\) with respect to the given independent variable. $$y=\log _{3} r \cdot \log _{9} r$$

Evaluate the integrals. $$\int \frac{d y}{y^{2}+6 y+10}$$

Find the area between the curve \(y=\tan x\) and the \(x\) -axis from \(x=-\pi / 4\) to \(x=\pi / 3\)

L'Hópital's Rule does not help with the limits. 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 derivative of \(y\) with respect to the given independent variable. $$y=\log _{3}\left(\left(\frac{x+1}{x-1}\right)^{\ln 3}\right)$$

Evaluate the integrals. $$\int_{1}^{2} \frac{8 d x}{x^{2}-2 x+2}$$

The region in the first quadrant bounded by the coordinate axes, the line \(y=3,\) and the curve \(x=2 / \sqrt{y+1}\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid.

L'Hópital's Rule does not help with the limits. Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow 0^{+}} \frac{x}{e^{-1 / x}}$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{5} \sqrt{\left(\frac{7 x}{3 x+2}\right)^{\ln 5}}$$

Evaluate the integrals. $$\int_{2}^{4} \frac{2 d x}{x^{2}-6 x+10}$$

The region between the curve \(y=\sqrt{\cot x}\) and the \(x\) -axis from \(x=\pi / 6\) to \(x=\pi / 2\) is revolved about the \(x\) -axis to generate a solid. Find the volume of the solid.

Show that if a function \(f\) is defined on an interval symmetric about the origin (so that \(f\) is defined at \(-x\) whenever it is defined at \(x\) ), then $$f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$ Then show that \((f(x)+f(-x)) / 2\) is even and that \((f(x)-\) \(f(-x)) / 2\) is odd.

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

Find the derivative of \(y\) with respect to the given independent variable. $$y=\theta \sin \left(\log _{7} \theta\right)$$

Evaluate the integrals. $$\int \frac{x+4}{x^{2}+4} d x$$

The region between the curve \(y=1 / x^{2}\) and the \(x\) -axis from \(x=1 / 2\) to \(x=2\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid.

Derive the formula \(\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})\) for all real \(x\) Explain in your derivation why the plus sign is used with the square root instead of the minus sign.

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

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{7}\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right)$$

Evaluate the integrals. $$\int \frac{t-2}{t^{2}-6 t+10} d t$$

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. \(\lim _{x \rightarrow 0^{+}} x \ln x=\lim _{x \rightarrow 0^{+}} \frac{\ln x}{(1 / x)}\) \(=\lim _{x \rightarrow 0^{+}} \frac{(1 / x)}{\left(-1 / x^{2}\right)}=\lim _{x \rightarrow 0^{+}}(-x)=0\)

Skydiving If a body of mass \(m\) falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity \(t\) s into the fall satisfies the differential equation $$m \frac{d v}{d t}=m g-k v^{2}$$ where \(k\) is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is short enough so that the variation in the air's density will not affect the outcome significantly.) a. Show that $$\boldsymbol{v}=\sqrt{\frac{m g}{k}} \tanh (\sqrt{\frac{g k}{m}} t)$$ satisfies the differential equation and the initial condition that \(v=0\) when \(t=0\) b. Find the body's limiting velocity, lim \(_{t \rightarrow \infty} v\) c. For a \(75 \mathrm{kg}\) skydiver \((m g=735 \mathrm{N}),\) with time in seconds and distance in meters, a typical value for \(k\) is \(0.235 .\) What is the diver's limiting velocity?

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{10} e^{x}$$

Evaluate the integrals. $$\int \frac{x^{2}+2 x-1}{x^{2}+9} d x$$

Find the lengths of the following curves. a. \(y=\left(x^{2} / 8\right)-\ln x, \quad 4 \leq x \leq 8\) b. \(x=(y / 4)^{2}-2 \ln (y / 4), \quad 4 \leq y \leq 12\)

Accelerations whose magnitudes are proportional to displacement Suppose that the position of a body moving along a coordinate line at time \(t\) is a. \(s=a \cos k t+b \sin k t\) b. \(s=a \cosh k t+b \sinh k t\) Show in both cases that the acceleration \(d^{2} s / d t^{2}\) is proportional to \(s\) but that in the first case it is directed toward the origin, whereas in the second case it is directed away from the origin.

Find the derivative of \(y\) with respect to the given independent variable. $$y=\frac{\theta 5^{\theta}}{2-\log _{5} \theta}$$

Evaluate the integrals. $$\int \frac{t^{3}-2 t^{2}+3 t-4}{t^{2}+1} d t$$

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.

Volume A region in the first quadrant is bounded above by the curve \(y=\cosh x,\) below by the curve \(y=\sinh x,\) and on the left and right by the \(y\) -axis and the line \(x=2,\) respectively. Find the volume of the solid generated by revolving the region about the \(x\) -axis.

Find the derivative of \(y\) with respect to the given independent variable. $$y=3^{\log _{2} t}$$

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

a. Find the centroid of the region between the curve \(y=1 / x\) and the \(x\) -axis from \(x=1\) to \(x=2\). Give the coordinates to two decimal places. b. Sketch the region and show the centroid in your sketch.

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

Volume The region enclosed by the curve \(y=\operatorname{sech} x,\) the \(x\) -axis, and the lines \(x=\pm \ln \sqrt{3}\) is revolved about the \(x\) -axis to generate a solid. Find the volume of the solid.

Find the derivative of \(y\) with respect to the given independent variable. $$y=3 \log _{8}\left(\log _{2} t\right)$$

Evaluate the integrals. $$\int \frac{d x}{(x-2) \sqrt{x^{2}-4 x+3}}$$

a. Find the center of mass of a thin plate of constant density covering the region between the curve \(y=1 / \sqrt{x}\) and the \(x\) -axis from \(x=1\) to \(x=16\) b. Find the center of mass if, instead of being constant, the density function is \(\delta(x)=4 / \sqrt{x}\)

a. Estimate the value of $$\lim _{x \rightarrow \infty}(x-\sqrt{x^{2}+x})$$ by graphing \(f(x)=x-\sqrt{x^{2}+x}\) over a suitably large interval of \(x\) -values. b. Now confirm your estimate by finding the limit with I'Hôpital's Rule. As the first step. multiply \(f(x)\) by the fraction \((x+\sqrt{x^{2}+x}) /(x+\sqrt{x^{2}+x})\) and simplify the new numerator.

Arc length Find the length of the graph of \(y=(1 / 2) \cosh 2 x\) from \(x=0\) to \(x=\ln \sqrt{5}\).

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{2}\left(8 t^{\ln 2}\right)$$

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

Use a derivative to show that \(f(x)=\ln \left(x^{3}-1\right)\) is one-to-one.