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? $$ \begin{array}{ll}{\text { a. }} & {x-3} & {\text { b. } x^{3}+\sin ^{2} x} \\\ {\text { c. }} & {\sqrt{x}} & {\text { d. } 4^{x}} \\ {\text { e. }} & {(3 / 2)^{x}} & {\text { f. }} {e^{x / 2}} \\ {\text { g. }} & {e^{x} / 2} & {\text { h. } \log _{10} x}\end{array} $$

Each of Exercises \(1-4\) 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. $$\sinh x=-\frac{3}{4}$$

In Exercises \(1-4,\) show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$\begin{array}{l}{2 y^{\prime}+3 y=e^{-x}} \\ {\text { a. } y=e^{-x}} \\\ {\text { c. } y=e^{-x}+C e^{-(3 / 2 x}}\end{array} \quad \text { b. } y=e^{-x}+e^{-(3 / 2) x}$$

In Exercises \(1-6,\) use l'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter \(2 .\) $$\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-4}$$

solve for \(t.\) \begin{equation}\quad \text { a. }e^{-0.3 t}=27 \quad \text { b. } e^{k t}=\frac{1}{2} \quad \text { c. } e^{(\ln 0.2) t}=0.4\end{equation}

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

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? $$ \begin{array}{ll}{\text { a. } 10 x^{4}+30 x+1} & {\text { b. } x \ln x-x} \\\ {\text { c. } \sqrt{1+x^{4}}} & {\text { d. }(5 / 2)^{x}} \\ {\text { e. }} {e^{-x}} & {\text { f. } x e^{x}} \\ {\text { g. }} e^{\cos x}& {\text { h. } e^{x-1}}\end{array} $$

Each of Exercises \(1-4\) 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. $$\sinh x=\frac{4}{3}$$

In Exercises \(1-4,\) show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$\begin{array}{l}{y^{\prime}=y^{2}} \\ {\text { a. } y=-\frac{1}{x} \quad \text { b. } y=-\frac{1}{x+3}} \\ {\text { c. } y=-\frac{1}{x+C}}\end{array}$$

In Exercises \(1-6,\) use l'Hopital'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 \(t.\) \begin{equation} \quad \text { a. }e^{-0.01 t}=1000 \quad \text { b. } e^{k t}=\frac{1}{10} \quad \text { c. } e^{(\ln 2) t}=\frac{1}{2}\end{equation}

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

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? $$ \begin{array}{ll}{\text { a. } x^{2}+4 x} & {\text { b. } x^{5}-x^{2}} \\\ {\text { c. } \sqrt{x^{4}+x^{3}}} & {\text { d. }(x+3)^{2}} \\ {\text { e. }} {x \ln x} & {\text { f. } 2^{x}} \\ {\text { g. }} x^{3} e^{-x}& {\text { h. } 8 x^{2}}\end{array} $$

Each of Exercises \(1-4\) 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. $$\cosh x=\frac{17}{15}, \quad x>0$$

In Exercises \(1-4,\) show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$y=\frac{1}{x} \int_{1}^{x} \frac{e^{i}}{t} d t, \quad x^{2} y^{\prime}+x y=e^{x}$$

In Exercises \(1-6,\) use l'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter \(2 .\) $$ \lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x}{7 x^{2}+1} $$

Use the properties of logarithms to simplify the expressions in Exercises 3 and \(4 .\) $$ \begin{array}{l}{\text { a. } \ln \sin \theta-\ln \left(\frac{\sin \theta}{5}\right) \quad \text { b. } \ln \left(3 x^{2}-9 x\right)+\ln \left(\frac{1}{3 x}\right)} \\ {\text { c. } \frac{1}{2} \ln \left(4 t^{4}\right)-\ln 2}\end{array} $$

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? $$ \begin{array}{ll}{\text { a. }} & {x^{2}+\sqrt{x}} & {\text { b. } 10 x^{2}} \\\ {\text { c. }} & {x^{2} e^{-x}} & {\text { d. } \log _{10}\left(x^{2}\right)} \\ {\text { e. }} & {x^{3}-x^{2}} & {\text { f. }(1 / 10)^{x}} \\ {\text { g. }} & {(1.1)^{x}} & {\text { h. } x^{2}+100 x}\end{array} $$

Each of Exercises \(1-4\) 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. $$\cosh x=\frac{13}{5}, \quad x>0$$

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

In Exercises \(1-6,\) use l'Hopital'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 \(t.\) \begin{equation}e^{\left(x^{2}\right)} e^{(2 x+1)}=e^{t}\end{equation}

Use the properties of logarithms to simplify the expressions in Exercises 3 and \(4 .\) $$ \begin{array}{l}{\text { a. } \ln \sec \theta+\ln \cos \theta} & {\text { b. } \ln (8 x+4)-2 \ln 2} \\ {\text { c. } 3 \ln \sqrt[3]{t^{2}-1}-\ln (t+1)}\end{array} $$

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? $$ \begin{array}{ll}{\text { a. } \log _{3} x} & {\text { b. } \ln 2 x} \\\ {\text { c. } \ln \sqrt{x}} & {\text { d. } \sqrt{x}} \\ {\text { e. } x} & {\text { f. } 5 \ln x} \\ {\text { g. } 1 / x} & {\text { h. } e^{x}}\end{array} $$

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$2 \cosh (\ln x)$$

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. $$y^{\prime}+y=\frac{2}{1+4 e^{2 x}} \quad y(-\ln 2)=\frac{\pi}{2} \quad y=e^{-x} \tan ^{-1}\left(2 e^{x}\right)$$

In Exercises \(1-6,\) use l'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter \(2 .\) $$ \lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}} $$

\begin{equation}e^{2 t}-3 e^{t}=0\end{equation}

In Exercises 5 and \(6,\) solve for \(t\) $$\ln \left(\frac{t}{t-1}\right)=2$$

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? $$ \begin{array}{ll}{\text { a. } \log _{2}\left(x^{2}\right)} & {\text { b. } \log _{10} 10 x} \\ {\text { c. } 1 / \sqrt{x}} & {\text { d. } 1 / x^{2}}\\\\{\text { e. } x-2 \ln x} & {\text { f. } e^{-x}} \\ {\text { g. } \ln (\ln x)} & {\text { h. } \ln (2 x+5)}\end{array} $$

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$\sinh (2 \ln x)$$

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. $$y^{\prime}=e^{-x^{2}}-2 x y \quad y(2)=0 \quad y=(x-2) e^{-x^{2}}$$

In Exercises \(1-6,\) use l'Hopital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter \(2 .\) $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}+3 x}{x^{3}+x+1} $$

\begin{equation}e^{-2 t}+6=5 e^{-t}\end{equation}

In Exercises 5 and \(6,\) solve for \(t\) $$\ln (t-2)=\ln 8-\ln t$$

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

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. $$\begin{array}{ll}{x y^{\prime}+y=-\sin x,} & {y\left(\frac{\pi}{2}\right)=0 \quad y=\frac{\cos x}{x}} \\ {x>0}\end{array}$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} $$

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

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln 3 x$$

In Exercises \(7-10\) , determine from its graph if the function is one-to-one. $$f(x)=\left\\{\begin{array}{ll}{3-x,} & {x<0} \\ {3,} & {x \geq 0}\end{array}\right.$$

Order the following functions from slowest growing to fastest growing as \(x \rightarrow \infty\) $$ \begin{array}{ll}{\text { a. } 2^{x}} & {\text { b. } x^{2}} \\ {\text { c. }} {(\ln 2)^{x}} & {\text { d. } e^{x}}\end{array} $$

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. $$\begin{array}{ll}{x^{2} y^{\prime}=x y-y^{2},} & {y(e)=e \quad y=\frac{x}{\ln x}} \\ {x>1}\end{array}$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5} $$

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

Find the values in Exercises \(9-12\) $$ \sin \left(\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)\right) $$

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

Solve the differential equations in Exercises \(9-22\) $$2 \sqrt{x y} \frac{d y}{d x}=1, \quad x, y>0$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{t \rightarrow-3} \frac{t^{3}-4 t+15}{t^{2}-t-12} $$

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