Chapter 7

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

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

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

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=(1+2 x) e^{-2 x}$$

Show that if positive functions \(f(x)\) and \(g(x)\) grow at the same rate as \(x \rightarrow \infty,\) then \(f=O(g)\) and \(g=O(f)\)

Solve the differential equations. $$\frac{d y}{d x}=e^{x-y}$$

Prove the identities $$\begin{aligned}\sinh (x+y) &=\sinh x \cosh y+\cosh x \sinh y \\\\\cosh (x+y) &=\cosh x \cosh y+\sinh x \sinh y\end{aligned}$$ Then use them to show that a. \(\sinh 2 x=2 \sinh x \cosh x\) b. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\)

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

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow \infty} \frac{5 x^{3}-2 x}{7 x^{3}+3}$$

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

When is a polynomial \(f(x)\) of smaller order than a polynomial \(g(x)\) as \(x \rightarrow \infty ?\) Give reasons for your answer.

Solve the differential equations. $$\frac{d y}{d x}=3 x^{2} e^{-y}$$

Use the definitions of cosh \(x\) and \(\sinh x\) to show that $$\cosh ^{2} x-\sinh ^{2} x=1$$

Find the values. $$\cot \left(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow \infty} \frac{x-8 x^{2}}{12 x^{2}+5 x}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\left(9 x^{2}-6 x+2\right) e^{3 x}$$

Solve the differential equations. $$\frac{d y}{d x}=\sqrt{y} \cos ^{2} \sqrt{y}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=6 \sinh \frac{x}{3}$$

Find the limits.( If in doubt, look at the function's graph.) $$\lim _{x \rightarrow 1^{-}} \sin ^{-1} x$$

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

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

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\frac{1}{2} \sinh (2 x+1)$$

Find the limits.( If in doubt, look at the function's graph.) $$\lim _{x \rightarrow-1^{+}} \cos ^{-1} x$$

Use l'Hôpital's rule to find the limits. $$\lim _{t \rightarrow 0} \frac{\sin 5 t}{2 t}$$

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

Investigate $$\lim _{x \rightarrow \infty} \frac{\ln (x+1)}{\ln x} \text { and } \lim _{x \rightarrow \infty} \frac{\ln (x+999)}{\ln x}$$ Then use l'Hôpital's Rule to explain what you find.

Solve the differential equations. $$\sqrt{x} \frac{d y}{d x}=e^{y+\sqrt{x}}, \quad x>0$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=2 \sqrt{t} \tanh \sqrt{t}$$

Find the limits.( If in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \tan ^{-1} x$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{8 x^{2}}{\cos x-1}$$

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

Show that the value of $$\lim _{x \rightarrow \infty} \frac{\ln (x+a)}{\ln x}$$ is the same no matter what value you assign to the constant \(a\) What does this say about the relative rates at which the functions \(f(x)=\ln (x+a)\) and \(g(x)=\ln x\) grow?

Solve the differential equations. $$(\sec x) \frac{d y}{d x}=e^{y+\sin x}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=t^{2} \tanh \frac{1}{t}$$

Find the limits.( If in doubt, look at the function's graph.) $$\lim _{x \rightarrow-\infty} \tan ^{-1} x$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{\sin x-x}{x^{3}}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\theta^{3} e^{-2 \theta} \cos 5 \theta$$

Show that \(\sqrt{10 x+1}\) and \(\sqrt{x+1}\) grow at the same rate as \(x \rightarrow \infty\) by showing that they both grow at the same rate as \(\sqrt{x}\) as \(x \rightarrow \infty\).

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

Find the limits.( If in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \sec ^{-1} x$$

Use l'Hôpital's rule to find the limits. $$\lim _{\theta \rightarrow \pi / 2} \frac{2 \theta-\pi}{\cos (2 \pi-\theta)}$$

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

a. Graph the function \(f(x)=\sqrt{1-x^{2}}, 0 \leq x \leq 1 .\) What symmetry does the graph have? b. Show that \(f\) is its own inverse. (Remember that \(\sqrt{x^{2}}=x\) if \(x \geq 0.\))

Show that \(\sqrt{x^{4}+x}\) and \(\sqrt{x^{4}-x^{3}}\) grow at the same rate as \(x \rightarrow \infty\) by showing that they both grow at the same rate as \(x^{2}\) as \(x \rightarrow \infty\).

Solve the differential equations. $$\frac{d y}{d x}=\frac{e^{2 x-y}}{e^{x+y}}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln (\cosh z)$$

Use l'Hôpital's rule to find the limits. $$\lim _{\theta \rightarrow-\pi / 3} \frac{3 \theta+\pi}{\sin (\theta+(\pi / 3))}$$

a. Graph the function \(f(x)=1 / x .\) What symmetry does the graph have? b. Show that \(f\) is its own inverse.

Show that \(e^{x}\) grows faster as \(x \rightarrow \infty\) than \(x^{n}\) for any positive integer \(n,\) even \(x^{1,000,000} .\) (Hint: What is the \(n\) th derivative of \(x^{n} ?\) )

Solve the differential equations. $$y^{2} \frac{d y}{d x}=3 x^{2} y^{3}-6 x^{2}$$