Chapter 7

Which of the following functions grow faster than \(e^{x}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(e^{x} ?\) Which grow slower? a. \(x-3\) b. \(x^{3}+\sin ^{2} x\) c. \(\sqrt{x}\) d. \(4^{x}\) e. \((3 / 2)^{x}\) f. \(e^{x / 2}\) g. \(e^{x} / 2\) h. \(\log _{10} x\)

Gives a value of \(\sinh x\) or \(\cosh x .\) Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\sinh x=-\frac{3}{4}$$

Use \(I^{\prime}\) Hôpital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2. $$\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-4}$$

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. \(2 y^{\prime}+3 y=e^{-x}\) a. \(y=e^{-x}\) b. \(y=e^{-x}+e^{-(3 / 2) x}\) c. \(y=e^{-x}+C e^{-(3 / 2) x}\)

Solve for \(t.\) a. \(e^{-0.3 t}=27\) b. \(e^{k t}=\frac{1}{2}\) c. \(e^{(\ln 0.2) t}=0.4\)

Express the following logarithms in terms of \(\ln 2\) and \(\ln 3\) a. \(\ln 0.75\) b. \(\ln (4 / 9) \quad\) c. \(\ln (1 / 2)\) d. \(\ln \sqrt[3]{9}\) e. \(\ln 3 \sqrt{2}\) f. \(\ln \sqrt{13.5}\)

Which of the following functions grow faster than \(e^{x}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(e^{x} ?\) Which grow slower? a. \(10 x^{4}+30 x+1\) b. \(x \ln x-x\) c. \(\sqrt{1+x^{4}}\) d. \((5 / 2)^{x}\) e. \(e^{-x}\) f. \(x e^{x}\) g. \(e^{\cos x}\) h. \(e^{x-1}\)

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. \(y^{\prime}=y^{2}\) a. \(y=-\frac{1}{x}\) b. \(y=-\frac{1}{x+3}\) c. \(y=-\frac{1}{x+C}\)

Gives a value of \(\sinh x\) or \(\cosh x .\) Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\sinh x=\frac{4}{3}$$

Use \(I^{\prime}\) Hôpital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2. $$\lim _{x \rightarrow 0} \frac{\sin 5 x}{x}$$

Solve for \(t.\) a. \(e^{-0.01 t}=1000\) b. \(e^{k t}=\frac{1}{10}\) c. \(e^{(\ln 2) t}=\frac{1}{2}\)

Express the following logarithms in terms of \(\ln 5\) and \(\ln 7\) a. \(\ln (1 / 125)\) b. \(\ln 9.8\) c. \(\ln 7 \sqrt{7}\) d. \(\ln 1225\) e. \(\ln 0.056\) f. \(\quad(\ln 35+\ln (1 / 7)) /(\ln 25)\)

Which of the following functions grow faster than \(x^{2}\) as \(x \rightarrow \infty\) ? Which grow at the same rate as \(x^{2} ?\) Which grow slower? a. \(x^{2}+4 x\) b. \(x^{5}-x^{2}\) c. \(\sqrt{x^{4}+x^{3}}\) d. \((x+3)^{2}\) e. \(x \ln x\) f. \(2^{x}\) g. \(x^{3} e^{-x}\) h. \(8 x^{2}\)

Gives a value of \(\sinh x\) or \(\cosh x .\) Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\cosh x=\frac{17}{15}, \quad x>0$$

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$y=\frac{1}{x} \int_{1}^{x} \frac{e^{t}}{t} d t, \quad x^{2} y^{\prime}+x y=e^{x}$$

Use \(I^{\prime}\) Hôpital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2. $$\lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x}{7 x^{2}+1}$$

Solve for \(t.\) $$e^{\sqrt{t}}=x^{2}$$

Use the properties of logarithms to simplify the expressions. a. \(\ln \sin \theta-\ln \left(\frac{\sin \theta}{5}\right)\) b. \(\ln \left(3 x^{2}-9 x\right)+\ln \left(\frac{1}{3 x}\right)\) c. \(\frac{1}{2} \ln \left(4 r^{4}\right)-\ln 2\)

Which of the following functions grow faster than \(x^{2}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(x^{2} ?\) Which grow slower? a. \(x^{2}+\sqrt{x}\) b. \(10 x^{2}\) c. \(x^{2} e^{-x}\) d. \(\log _{10}\left(x^{2}\right)\) e. \(x^{3}-x^{2}\) f. \((1 / 10)^{x}\) g. \((1.1)^{x}\) h. \(x^{2}+100 x\)

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$y=\frac{1}{\sqrt{1+x^{4}}} \int_{1}^{x} \sqrt{1+t^{4}} d t, \quad y^{\prime}+\frac{2 x^{3}}{1+x^{4}} y=1$$

Gives a value of \(\sinh x\) or \(\cosh x .\) Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\cosh x=\frac{13}{5}, \quad x>0$$

Use \(I^{\prime}\) Hôpital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2. $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{4 x^{3}-x-3}$$

Solve for \(t.\) $$e^{\left(x^{2}\right)} e^{(2 x+1)}=e^{t}$$

Use the properties of logarithms to simplify the expressions. a. \(\ln \sec \theta+\ln \cos \theta\) b. \(\ln (8 x+4)-2 \ln 2\) c. \(3 \ln \sqrt[3]{t^{2}-1}-\ln (t+1)\)

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? a. \(\log _{3} x\) b. \(\ln 2 x\) c. \(\ln \sqrt{x}\) d. \(\sqrt{x}\) e. \(x\) f. \(5 \ln x\) g. \(1 / x\) h. \(e^{x}\)

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$2 \cosh (\ln x)$$

Use \(I^{\prime}\) Hôpital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2. $$\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=e^{-5 x}$$

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? a. \(\log _{2}\left(x^{2}\right)\) b. \(\log _{10} 10 x\) c. \(1 / \sqrt{x}\) d. \(1 / x^{2}\) e. \(x-2 \ln x\) f. \(e^{-x}\) g. \(\ln (\ln x)\) h. \(\ln (2 x+5)\)

Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \begin{array}{l} \text { Differential } \\ \text { equation } \end{array} & \begin{array}{l} \text { Initial } \\ \text { equation } \end{array} & \begin{array}{l} \text { Solution } \\ \text { candidate } \end{array} \\ \hline y^{\prime}=e^{-x^{2}}-2 x y &y(2)=0 & y=(x-2) e^{-x^{2}} \end{array}$$

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\sinh (2 \ln x)$$

Use \(I^{\prime}\) Hôpital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2. $$\lim _{x \rightarrow \infty} \frac{2 x^{2}+3 x}{x^{3}+x+1}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=e^{2 x / 3}$$

Order the following functions from slowest growing to fastest growing as \(x \rightarrow \infty\) a. \(e^{x}\) b. \(x^{x}\) c. \((\ln x)^{x}\) d. \(e^{x / 2}\)

Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \begin{array}{l} \text { Differential } \\ \text { equation } \end{array} & \begin{array}{l} \text { Initial } \\ \text { equation } \end{array} & \begin{array}{l} \text { Solution } \\ \text { candidate } \end{array} \\ \hline \begin{aligned} &x y^{\prime}+y=-\sin x\\\ &x>0 \end{aligned} &y\left(\frac{\pi}{2}\right)=0&y=\frac{\cos x}{x} \end{array}$$

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\cosh 5 x+\sinh 5 x$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=e^{5-7 x}$$

Order the following functions from slowest growing to fastest growing as \(x \rightarrow \infty\) a. \(2^{x}\) b. \(x^{2}\) c. \((\ln 2)^{x}\) d. \(e^{x}\)

Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \begin{array}{l} \text { Differential } \\ \text { equation } \end{array} & \begin{array}{l} \text { Initial } \\ \text { equation } \end{array} & \begin{array}{l} \text { Solution } \\ \text { candidate } \end{array} \\ \hline \begin{aligned} &x^{2} y^{\prime}=x y-y^{2}\\\ &x>1 \end{aligned} &y(e)=e&y=\frac{x}{\ln x} \end{array}$$

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\cosh 3 x-\sinh 3 x$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln \left(t^{3 / 2}\right)$$

True, or false? As \(x \rightarrow \infty\) a. \(\quad x=o(x)\) b. \(x=o(x+5)\) c. \(x=O(x+5)\) d. \(x=O(2 x)\) e. \(e^{x}=o\left(e^{2 x}\right)\) f. \(x+\ln x=O(x)\) g. \(\ln x=o(\ln 2 x)\) h. \(\sqrt{x^{2}+5}=O(x)\)

Solve the differential equations. $$2 \sqrt{x y} \frac{d y}{d x}=1, \quad x, y>0$$

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$(\sinh x+\cosh x)^{4}$$

Find the values. $$\sin \left(\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)\right)$$

Use l'Hôpital's rule to find the limits. $$\lim _{t \rightarrow-3} \frac{t^{3}-4 t+15}{t^{2}-t-12}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=x e^{x}-e^{x}$$

True, or false? As \(x \rightarrow \infty\) a. \(\frac{1}{x+3}=O\left(\frac{1}{x}\right)\) b. \(\frac{1}{x}+\frac{1}{x^{2}}=O\left(\frac{1}{x}\right)\) c. \(\frac{1}{x}-\frac{1}{x^{2}}=o\left(\frac{1}{x}\right)\) d. \(2+\cos x=O(2)\) g. \(\ln (\ln x)=O(\ln x)\) h. \(\ln (x)=o\left(\ln \left(x^{2}+1\right)\right)\)