Chapter 4

Let \(f(x)=|x| .\) Show that \(f(-2)=f(2)\), but there is no number \(c\) in \((-2,2)\) such that \(f^{\prime}(c)=0 .\) Does this result contradict Rolle's Theorem? Explain.

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ g(x)=1 /\left(x^{2}+1\right) $$

A Norman window is a window in the shape of a rectangle with a semicircle attached at the top (Figure 4.53). Assuming that the perimeter of the window is 12 feet, find the dimensions that allow the maximum amount of light to enter.

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=\frac{1}{2} x, f\left(\frac{1}{2}\right)=-1 $$

Find all critical numbers of the given function. $$ f(z)=z|z+3| $$

Find the given limit. $$ \lim _{x \rightarrow \infty} \tan \frac{1}{x} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ f(x)=x+\frac{1}{x} $$

Suppose \(\left|f^{\prime}(x)\right| \leq M\) for \(a \leq x \leq b\). Using the Mean Value Theorem, prove that \(|f(b)-f(a)| \leq M(b-a)\), so that \(f(a)-M(b-a) \leq f(b) \leq f(a)+M(b-a)\).

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ f(x)=x /\left(16+x^{3}\right) $$

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=x^{2}, f(0)=-5 $$

Find all critical numbers of the given function. $$ f(x)=x^{2} e^{x} $$

Find the given limit. $$ \lim _{x \rightarrow-\infty} x \tan \frac{1}{x} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ f(x)=x \sqrt{x^{2}-4} $$

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ f(x)=\sqrt{|x|+1} $$

At 3 P.M. an oil tanker traveling west in the ocean at 15 kilometers per hour passes the same point as a luxury liner that arrived at the same spot at 2 P.M. while traveling north at 25 kilometers per hour. At what time were the ships closest together?

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=-\frac{3}{2} x^{2}, f(-1)=-\frac{1}{2} $$

Find all critical numbers of the given function. $$ f(x)=1 /\left(e^{x}-1\right) $$

Find the given limit. $$ \lim _{x \rightarrow \infty} e^{-x} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ f(x)=\sin 2 x $$

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ f(x)=x \sqrt{1-x^{2}} $$

A horse breeder plans to set aside a rectangular region of 1 square kilometer for horses and wishes to build a wooden fence to enclose the region. Since one side of the region will run along a well-traveled highway, the breeder decides to make that side more attractive, using wood that costs three times as much per meter as the wood for the other sides. What dimensions will minimize the cost of the fence?

Halley's Law states that the barometric pressure \(p(t)\) in inches of mercury at \(t\) miles above sea level is given by $$ p(t) \approx 29.92 e^{-0.2 t} \quad \text { for } t \geq 0 $$ Find the barometric pressure a. at sea level b. 5 miles above sea level c. 10 miles above sea level

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=\cos x, f(\pi / 3)=1 $$

Find all critical numbers of the given function. $$ f(x)=x \ln x $$

Find the given limit. $$ \lim _{x \rightarrow-\infty}(x-1) \sin \frac{1}{x-1} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ f(x)=\cos \frac{1}{2} x-1 $$

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ f(x)=\frac{x^{2}+x+1}{x^{2}-x+1} $$

A manufacturer wishes to produce rectangular containers with square bottoms and tops, each container having a capacity of 250 cubic inches. If the material used for the top and the bottom costs twice as much per square inch as the material for the sides, what dimensions will minimize the cost?

Assume that the air pressure \(p(x)\) in pounds per square foot at \(x\) feet above sea level is given by $$ p(x) \approx 2140 e^{-0.000035 x} \quad \text { for } x \geq 0 $$ and that an airplane is losing altitude at the rate of 20 miles per hour. At what rate is the air pressure just outside the plane increasing when the plane is 2 miles above sea level?

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=\sec \frac{x}{2} \tan \frac{x}{2}, f(\pi / 2)=2 $$

Find all critical numbers of the given function. $$ f(x)=\ln \left(2 x+e^{-x}\right) $$

Find the given limit. $$ \lim _{x \rightarrow-\infty} e^{-1 / x} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ f(x)=\sec x $$

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ k(x)=\cos x+\frac{1}{2} x $$

Let \(p, q\), and \(r\) be positive constants with \(q

For an operation a dog is anesthetized with sodium pentobarbitol, which is eliminated exponentially from the blood stream. Assume that of any sodium pentobarbitol in the blood stream, half is eliminated in 5 hours. Assume also that to anesthetize a dog, 20 milligrams of sodium pentobarbitol are required for each kilogram of body mass. What single dose of sodium pentobarbitol would be required to anesthetize a 10 -kilogram dog for half an hour?

Find all extreme values (if any) of the given function on the given interval. Determine at which numbers in the interval these values occur. $$ f(x)=x^{2}-x ;[0,2] $$

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=e^{x}, f(0)=10 $$

Find the given limit. $$ \lim _{x \rightarrow \infty} \ln (1 / x) $$

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ k(x)=\sin x-\frac{\sqrt{3}}{2} x $$

If a sum of \(S\) dollars is invested at \(p\) percent interest and interest is compounded continuously, then the amount \(A(t)\) of money accumulated after \(t\) years is given by $$ A(t)=S e^{p t / 100} \quad \text { for } t \geq 0 $$ In terms of \(S\), how much money will there be after 10 years, if the interest rate is 6 percent?

Find all extreme values (if any) of the given function on the given interval. Determine at which numbers in the interval these values occur. $$ g(x)=1 / x ;(0,3] $$

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=\frac{1}{x}, f(e)=-3 $$

Find the given limit. $$ \lim _{x \rightarrow \infty} \frac{1}{\ln x} $$

Find all inflection points (if any) of the graph of the function. Then sketch the graph of the function. $$ f(x)=(x+2)^{3} $$

Use Rolle's Theorem to show that there is a solution of the equation \(\tan x=1-x\) in \((0,1)\). (Hint: Let \(f(x)\) \(=(x-1) \sin x\), and find \(f(0), f(1)\), and \(\left.f^{\prime}(x) .\right)\)

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ k(x)=\sin \left(\frac{x^{2}}{1+x^{2}}\right) $$

Find all extreme values (if any) of the given function on the given interval. Determine at which numbers in the interval these values occur. $$ f(t)=-1 /(2 t) ;(0, \infty) $$

Determine all functions \(f\) satisfying the given conditions. $$ f^{\prime \prime}(x)=0 \text { (Hint: Use Theorem 4.6 twice.) } $$

Find the given limit. $$ \lim _{x \rightarrow \infty} e^{x} \ln x $$