Chapter 3

Express the given quantity as a single logarithm. $$ \frac{1}{3} \ln (x+2)^{3}+\frac{1}{2}\left[\ln x-\ln \left(x^{2}+3 x+2\right)^{2}\right] $$

If \(f^{\prime}\) is continuous, \(f(2)=0,\) and \(f^{\prime}(2)=7,\) evaluate $$\lim _{x \rightarrow 0} \frac{f(2+3 x)+f(2+5 x)}{x}$$

Show that if \(a \neq 0\) and \(b \neq 0\) , then there exist numbers \(\alpha\) and \(\beta\) such that \(a e^{x}+b e^{-x}\) equals either \(\alpha \sinh (x+\beta)\) or \(\alpha \cosh (x+\beta)\) . In other words, almost every function of the form \(f(x)=a e^{x}+b e^{-x}\) is a shifted and stretched hyperbolic sine or cosine function.

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=\sqrt{x} e^{x^{2}-x}(x+1)^{2 / 3} $$

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

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=x^{x} $$

If \(f^{\prime}\) is continuous, use l'Hospital's Rule to show that $$\quad \lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}=f^{\prime}(x)$$ Explain the meaning of this equation with the aid of a diagram.

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=x^{\cos x} $$

If \(f^{\prime \prime}\) is continuous, show that $$\quad \lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}}=f^{\prime \prime}(x)$$

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=(\cos x)^{x} $$

Suppose that the graph of \(y=\log _{2} x\) is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 \(\mathrm{ft} ?\)

$$f(x)=\left\\{\begin{array}{ll}{e^{-1 / x^{2}}} & {\text { if } x \neq 0} \\\ {0} & {\text { if } x=0}\end{array}\right.$$ (a) Use the definition of derivative to compute \(f^{\prime}(0)\) (b) Show that \(f\) has derivatives of all orders that are defined on \(\mathbb{R} .[\)Hint. . First show by induction that there is a polynomial \(p_{n}(x)\) and a nonnegative integer \(k_{n}\) such that \(f^{(n)}(x)=p_{n}(x) f(x) / x^{k_{n}}\) for \(x \neq 0 . ]\)

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=\sqrt{x}^{x} $$

Compare the functions \(f(x)=x^{0.1}\) and \(g(x)=\ln x\) by graphing both \(f\) and \(g\) in several viewing rectangles. When does the graph of \(f\) finally surpass the graph of \(g\) ?

$$f(x)=\left\\{\begin{array}{ll}{|x|^{x}} & {\text { if } x \neq 0} \\ {1} & {\text { if } x=0}\end{array}\right.$$ (a) Show that \(f\) is continuous at 0 . (b) Investigate graphically whether \(f\) is differentiable at 0 by zooming in several times toward the point \((0,1)\) on the graph of \(f .\) (c) Show that \(f\) is not differentiable at \(0 .\) How can you reconcile this fact with the appearance of the graphs in part (b)?

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=(\tan x)^{1 / x} $$

\(51-60=\) Use logarithmic differentiation or an alternative method to find the derivative of the function. $$ y=(\sin x)^{\ln x} $$

$$ y^{\prime} \text { if } e^{x^{2} y}=x+y $$

(a) What are the domain and range of \(f ?\) (b) What is the \(x\) -intercept of the graph of \(f ?\) (c) Sketch the graph of \(f .\) $$ f(x)=\ln x+2 $$

$$\begin{array}{l}{\text { Find an equation of the tangent line to the curve }} \\ {x e^{y}+y e^{x}=1 \text { at the point }(0,1) .}\end{array}$$

(a) What are the domain and range of \(f ?\) (b) What is the \(x\) -intercept of the graph of \(f ?\) (c) Sketch the graph of \(f .\) $$ f(x)=\ln (x-1)-1 $$

Solve each equation for \(x\) $$ \text { (a) }e^{7-4 x}=6 \quad \text { (b) } \ln (3 x-10)=2 $$

Find \(y^{\prime}\) if \(y=\ln \left(x^{2}+y^{2}\right)\)

Find \(y^{\prime}\) if \(x^{y}=y^{x}\)

Solve each equation for \(x\) $$ \text { (a) }\ln \left(x^{2}-1\right)=3 \quad \text { (b) } e^{2 x}-3 e^{x}+2=0 $$

Solve each equation for \(x\) $$ \text { (a) }2^{x-5}=3 \quad \text { (b) } \ln x+\ln (x-1)=1 $$

The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a Point on such a spring is $$s(t)=2 e^{-1.5 t} \sin 2 \pi t$$ where \(s\) is measured in centimeters and \(t\) in seconds. Find the velocity after \(t\) seconds and graph both the position and velocity functions for 0\(\leqslant t \leqslant 2\) .

Under certain circumstances a rumor spreads according to the equation $$p(t)=\frac{1}{1+a e^{-k t}}$$ where \(p(t)\) is the proportion of the population that knows the rumor at time \(t\) and \(a\) and \(k\) are positive constants. (a) Find \(\lim _{t \rightarrow \infty} p(t)\) . (b) Find the rate of spread of the rumor. (c) Graph \(p\) for the case \(a=10, k=0.5\) with \(t\) measured in hours. Use the graph to estimate how long it will take for 80\(\%\) of the population to hear the rumor.

Solve each equation for \(x\) $$ \text { (a) }\ln (\ln x)=1 \quad \text { (b) } e^{a x}=C e^{b x}, \text { where } a \neq b $$

Solve each inequality for \(x .\) $$ \text { (a) }\ln x<0 \quad \text { (b) } e^{x}>5 $$

Show that the function \(y=A e^{-x}+B x e^{-x}\) satisfies th differential equation \(y^{\prime \prime}+2 y^{\prime}+y=0\)

Solve each inequality for \(x .\) $$ \text { (a) }1

For what values of \(r\) does the function \(y=e^{r x}\) satisfy the equation \(y^{\prime \prime}+5 y^{\prime}-6 y=0 ?\)

(a) Find the domain of \(f(x)=\ln \left(e^{x}-3\right)\) (b) Find \(f^{-1}\) and its domain.

If \(f(x)=e^{2 x},\) find a formula for \(f^{(n)}(x)\)

(a) What are the values of \(e^{\ln 3 w}\) and \(\ln \left(e^{300}\right) ?\) (b) Use your calculator to evaluate \(e^{\ln 3100}\) and \(\ln \left(e^{300}\right) .\) What do you notice? Can you explain why the calculator has trouble?

Find the thousandth derivative of \(f(x)=x e^{-x}\)

Find the limit. $$ \lim _{x \rightarrow 3^{+}} \ln \left(x^{2}-9\right) $$

Find a formula for \(f^{(n)}(x)\) if \(f(x)=\ln (x-1)\)

Find the limit. $$ \lim _{x \rightarrow 2} \log _{5}\left(8 x-x^{4}\right) $$

Find \(\frac{d^{3}}{d x^{9}}\left(x^{8} \ln x\right)\)

Find the limit. $$ \lim _{x \rightarrow 0} \ln (\cos x) $$

Find the limit. $$ \lim _{x \rightarrow+0^{+}} \ln (\sin x) $$

Evaluate \(\lim _{x \rightarrow \pi} \frac{e^{\sin x}-1}{x-\pi}\)

Find the limit. $$ \lim _{x \rightarrow \infty}\left[\ln \left(1+x^{2}\right)-\ln (1+x)\right] $$

Find the limit. $$ \lim _{x \rightarrow \infty}[\ln (2+x)-\ln (1+x)] $$

Graph the function \(f(x)=\sqrt{x^{3}+x^{2}+x+1}\) and explain why it is one-to-one. Then use a computer algcbra system to find an cxplicit expression for \(f^{-1}(x) .\) (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by $$ Q(t)=Q_{0}\left(1-e^{-t / a}\right) $$ (The maximum charge capacity is \(Q_{0}\) and \(t\) is measured in seconds.) (a) Find the inverse of this function and explain its mcaning. (b) How long does it take to recharge the capacitor to 90\(\%\) of capacity if \(a=2 ?\)

Let \(a>1 .\) Prove, using precise definitions, that $$ \text { (a) }\lim _{x \rightarrow-\infty} a^{x}=0 \quad \text { (b) } \lim _{x \rightarrow \infty} a^{x}=\infty $$

(a) If we shift a curve to the left, what happens to its reflection about the line \(y=x ?\) In view of this geometric principle, find an expression for the inverse of \(g(x)=f(x+c),\) where \(f\) is a one-to-one function. (b) Find an expression for the inverse of \(h(x)=f(c x)\) , where \(c \neq 0\)