Chapter 9

a. Suppose \(f\) is continuous at \(a\) and \(g\) is discontinuous at \(a\). Is the sum \(f+g\) discontinuous at \(a ?\) Explain. b. Suppose \(f\) and \(g\) are both discontinuous at \(a\). Is the sum \(f+g\) necessarily discontinuous at \(a ?\) Explain.

Find the indicated limits, if they exist. \(\lim _{x \rightarrow-\infty} \frac{3 x^{3}+x^{2}+1}{x^{3}+1}\)

THURSTONE LEARNING MoDEL Psychologist L. L. Thurstone suggested the following relationship between learning time \(T\) and the length of a list \(n\) : $$ T=f(n)=A n \sqrt{n-b} $$ where \(A\) and \(b\) are constants that depend on the person and the task. a. Compute \(d T / d n\) and interpret your result. b. For a certain person and a certain task, suppose \(A=4\) and \(b=4\). Compute \(f^{\prime}(13)\) and \(f^{\prime}(29)\) and interpret your results.

SHORTAGE OF NURSES The projected number of nurses (in millions) from the year 2000 through 2015 is given by $$ N(t)=\left\\{\begin{array}{ll} 1.9 & \text { if } 0 \leq t<5 \\ -0.0004 t^{2}+0.038 t+1.72 & \text { if } 5 \leq t \leq 15 \end{array}\right. $$ where \(t=0\) corresponds to 2000 . The projected number of nursing jobs (in millions) over the same period is $$ J(t)=\left\\{\begin{array}{ll} -0.0002 t^{2}+0.032 t+2 & \text { if } \quad 0 \leq t<10 \\ -0.0016 t^{2}+0.12 t+1.26 & \text { if } 10 \leq t \leq 15 \end{array}\right. $$ a. Find the rule for the function \(G=J-N\) giving the gap between the supply and the demand of nurses from 2000 through 2015 . b. How fast was the gap between the supply and the demand of nurses changing in \(2008 ?\) In \(2012 ?\)

a. Suppose \(f\) is continuous at \(a\) and \(g\) is discontinuous at \(a\). Is the product \(f g\) necessarily discontinuous at \(a\) ? Explain. b. Suppose \(f\) and \(g\) are both discontinuous at \(a\). Is the product \(f g\) necessarily discontinuous at \(a\) ? Explain.

OIL SPILLS In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the area polluted is a circle and that its radius is increasing at a rate of \(2 \mathrm{ft} / \mathrm{sec}\), determine how fast the area is increasing when the radius of the circle is \(40 \mathrm{ft}\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, given an example to show why it is false. If \(f\) is differentiable, then $$ \frac{d}{d x}\left[\frac{f(x)}{x^{2}}\right]=\frac{f^{\prime}(x)}{2 x} $$

In Exercises 77 and 78 , determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) and \(g\) are differentiable, then $$ \frac{d}{d x}[2 f(x)-5 g(x)]=2 f^{\prime}(x)-5 g^{\prime}(x) $$

In Exercises 77-80, (a) show that the function \(f\) is continuous for all values of \(x\) in the interval \([a, b]\) and (b) prove that \(f\) must have at least one zero in the interval \((a, b)\) by showing that \(f(a)\) and \(f(b)\) have opposite signs. \(f(x)=x^{2}-6 x+8 ; a=1, b=3\)

(a) show that the function \(f\) is continuous for all values of \(x\) in the interval \([a, b]\) and (b) prove that \(f\) must have at least one zero in the interval \((a, b)\) by showing that \(f(a)\) and \(f(b)\) have opposite signs. \(f(x)=2 x^{3}-3 x^{2}-36 x+14 ; a=0, b=1\)

Find the indicated limits, if they exist. \(\lim _{x \rightarrow \infty} \frac{4 x^{4}-3 x^{2}+1}{2 x^{4}+x^{3}+x^{2}+x+1}\)

TRAFFIC FLow Opened in the late \(1950 \mathrm{~s}\), the Central Artery in downtown Boston was designed to move 75,000 vehicles a day. The number of vehicles moved per day is approximated by the function $$ x=f(t)=6.25 t^{2}+19.75 t+74.75 \quad(0 \leq t \leq 5) $$ where \(x\) is measured in thousands and \(t\) in decades, with \(t=0\) corresponding to the beginning of \(1959 .\) Suppose the average speed of traffic flow in mph is given by $$ S=g(x)=-0.00075 x^{2}+67.5 \quad(75 \leq x \leq 350) $$ where \(x\) has the same meaning as before. What was the rate of change of the average speed of traffic flow at the beginning of \(1999 ?\) What was the average speed of traffic flow at that time? Hint: \(S=g[f(t)]\).

Extend the product rule for differentiation to the following case involving the product of three differentiable functions: Let \(h(x)=u(x) v(x) w(x)\) and show that \(h^{\prime}(x)=\) \(u(x) v(x) w^{\prime}(x)+u(x) v^{\prime}(x) w(x)+u^{\prime}(x) v(x) w(x)\) Hint: Let \(f(x)=u(x) v(x), g(x)=w(x)\), and \(h(x)=f(x) g(x)\) and apply the product rule to the function \(h\).

Prove the power rule (Rule 2) for the special case \(n=3\). Hint: Compute \(\lim _{h \rightarrow 0}\left[\frac{(x+h)^{3}-x^{3}}{h}\right]\).

(a) show that the function \(f\) is continuous for all values of \(x\) in the interval \([a, b]\) and (b) prove that \(f\) must have at least one zero in the interval \((a, b)\) by showing that \(f(a)\) and \(f(b)\) have opposite signs. \(f(x)=x^{3}-2 x^{2}+3 x+2 ; a=-1, b=1\)

HoTEL OccuPANCY RATES The occupancy rate of the allsuite Wonderland Hotel, located near an amusement park, is given by the function $$ r(t)=\frac{10}{81} t^{3}-\frac{10}{3} t^{2}+\frac{200}{9} t+60 \quad(0 \leq t \leq 12) $$ where \(t\) is measured in months, with \(t=0\) corresponding to the beginning of January. Management has estimated that the monthly revenue (in thousands of dollars/month) is approximated by the function $$ R(r)=-\frac{3}{5000} r^{3}+\frac{9}{50} r^{2} \quad(0 \leq r \leq 100) $$ where \(r\) is the occupancy rate. a. Find an expression that gives the rate of change of Wonderland's occupancy rate with respect to time. b. Find an expression that gives the rate of change of Wonderland's monthly revenue with respect to the occupancy rate. c. What is the rate of change of Wonderland's monthly revenue with respect to time at the beginning of January? At the beginning of July? Hint: Use the chain rule to find \(R^{\prime}(r(0)) r^{\prime}(0)\) and \(R^{\prime}(r(6)) r^{\prime}(6)\).

Prove the quotient rule for differentiation (Rule 6 ). Hint: Let \(k(x)=f(x) / g(x)\) and verify the following steps: a. \(\frac{k(x+h)-k(x)}{h}=\frac{f(x+h) g(x)-f(x) g(x+h)}{h g(x+h) g(x)}\) b. By adding \([-f(x) g(x)+f(x) g(x)]\) to the numerator and simplifying, show that $$ \begin{aligned} \frac{k(x+h)-k(x)}{h}=& \frac{1}{g(x+h) g(x)} \\ & \times\left\\{\left[\frac{f(x+h)-f(x)}{h}\right] \cdot g(x)\right.\\\ &\left.-\left[\frac{g(x+h)-g(x)}{h}\right] \cdot f(x)\right\\} \\ \text { c. } k^{\prime}(x)=\lim _{h \rightarrow 0} \frac{k(x+h)-k(x)}{h} & \\ =\frac{g(x) f^{\prime}(x)-f(x) g^{\prime}(x)}{[g(x)]^{2}} \end{aligned} $$

(a) show that the function \(f\) is continuous for all values of \(x\) in the interval \([a, b]\) and (b) prove that \(f\) must have at least one zero in the interval \((a, b)\) by showing that \(f(a)\) and \(f(b)\) have opposite signs. \(f(x)=2 x^{5 / 3}-5 x^{4 / 3} ; a=14, b=16\)

EFFECT OF HousING STARTS ON JoBS The president of a major housing construction firm claims that the number of construction jobs created is given by $$ N(x)=1.42 x $$ where \(x\) denotes the number of housing starts. Suppose the number of housing starts in the next \(t\) mo is expected to be $$ x(t)=\frac{7 t^{2}+140 t+700}{3 t^{2}+80 t+550} $$ million units/year. Find an expression that gives the rate at which the number of construction jobs will be created \(t\) mo from now. At what rate will construction jobs be created 1 yr from now?

In Exercises 81-82, use the intermediate value theorem to find the value of \(c\) such that \(f(c)=M\). \(f(x)=x^{2}-4 x+6\) on \([0,3] ; M=4\)

ToxIc WASTE A city's main well was recently found to be contaminated with trichloroethylene, a cancer-causing chemical, as a result of an abandoned chemical dump leaching chemicals into the water. A proposal submitted to city council members indicates that the cost, measured in millions of dollars, of removing \(x \%\) of the toxic pollutant is given by $$ C(x)=\frac{0.5 x}{100-x} \quad(0

DEMAND FOR PCs The quantity demanded per month, \(x\), of a certain make of personal computer (PC) is related to the average unit price, \(p\) (in dollars), of PCs by the equation $$ x=f(p)=\frac{100}{9} \sqrt{810,000-p^{2}} $$ It is estimated that \(t\) mo from now, the average price of a PC will be given by $$ p(t)=\frac{400}{1+\frac{1}{8} \sqrt{t}}+200 \quad(0 \leq t \leq 60) $$ dollars. Find the rate at which the quantity demanded per month of the PCs will be changing 16 mo from now.

Use the intermediate value theorem to find the value of \(c\) such that \(f(c)=M\). \(f(x)=x^{2}-x+1\) on \([-1,4] ; M=7\)

DEMAND FOR WATCHES The demand equation for the Sicard wristwatch is given by $$ x=f(p)=10 \sqrt{\frac{50-p}{p}} \quad(0

Use the method of bisection (see Example 6 ) to find the root of the equation \(x^{5}+2 x-7=0\) accurate to two decimal places.

The average cost/disc in dollars incurred by Herald Records in pressing \(x\) DVDs is given by the average cost function $$ \bar{C}(x)=2.2+\frac{2500}{x} $$ Evaluate \(\lim _{x \rightarrow \infty} \bar{C}(x)\) and interpret your result.

CRUISE SHIP BookINGS The management of Cruise World, operators of Caribbean luxury cruises, expects that the percentage of young adults booking passage on their cruises in the years ahead will rise dramatically. They have constructed the following model, which gives the percentage of young adult passengers in year \(t\) : $$ p=f(t)=50\left(\frac{t^{2}+2 t+4}{t^{2}+4 t+8}\right) \quad(0 \leq t \leq 5) $$ Young adults normally pick shorter cruises and generally spend less on their passage. The following model gives an approximation of the average amount of money \(R\) (in dollars) spent per passenger on a cruise when the percentage of young adults is \(p\) : $$ R(p)=1000\left(\frac{p+4}{p+2}\right) $$ Find the rate at which the price of the average passage will be changing 2 yr from now.

Use the method of bisection (see Example 6 ) to find the root of the equation \(x^{3}-x+1=0\) accurate to two decimal places.

The concentration of a certain drug in a patient's bloodstream \(t \mathrm{hr}\) after injection is given by $$ C(t)=\frac{0.2 t}{t^{2}+1} $$ \(\mathrm{mg} / \mathrm{cm}^{3}\). Evaluate \(\lim _{t \rightarrow \infty} C(t)\) and interpret your result.

Prove that \(\frac{d}{d x} \ln |x|=\frac{1}{x}(x \neq 0)\) for the case \(x<0 .\)

Joan is looking straight out a window of an apartment building at a height of \(32 \mathrm{ft}\) from the ground. A boy throws a tennis ball straight up by the side of the building where the window is located. Suppose the height of the ball (measured in feet) from the ground at time \(t\) is \(h(t)=4+64 t-16 t^{2} .\) a. Show that \(h(0)=4\) and \(h(2)=68\). b. Use the intermediate value theorem to conclude that the ball must cross Joan's line of sight at least once. c. At what time(s) does the ball cross Joan's line of sight? Interpret your results.

The total worldwide box-office receipts for a long-running blockbuster movie are approximated by the function $$ T(x)=\frac{120 x^{2}}{x^{2}+4} $$ where \(T(x)\) is measured in millions of dollars and \(x\) is the number of months since the movie's release. a. What are the total box-office receipts after the first month? The second month? The third month? b. What will the movie gross in the long run (when \(x\) is very large)?

If \(f\) is differentiable and \(c\) is a constant, then $$ \frac{d}{d x}[f(c x)]=c f^{\prime}(c x) $$

The percentage of a certain brand of computer chips that will fail after \(t\) yr of use is estimated to be $$ P(t)=100\left(1-e^{-0.1 t}\right) $$ a. What percentage of this brand of computer chips are expected to be usable after 3 yr? b. Evaluate \(\lim _{t \rightarrow \infty} P(t)\). Did you expect this result?

If \(f\) is differentiable, then $$ \frac{d}{d x} \sqrt{f(x)}=\frac{f^{\prime}(x)}{2 \sqrt{f(x)}} $$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(\lim _{x \rightarrow 0} f(x)=3\), then \(f(0)=3\).

The rate of production \(R\) in photosynthesis is related to the light intensity \(I\) by the function $$ R(I)=\frac{a I}{b+I^{2}} $$ where \(a\) and \(b\) are positive constants. a. Taking \(a=b=1\), compute \(R(I)\) for \(I=0,1,2,3,4\), and 5 . b. Evaluate \(\lim _{I \rightarrow \infty} R(I)\). c. Use the results of parts (a) and (b) to sketch the graph of \(R\). Interpret your results.

In Section \(9.4\), we proved that $$ \frac{d}{d x}\left(x^{n}\right)=n x^{n-1} $$ for the special case when \(n=2\). Use the chain rule to show that $$ \frac{d}{d x}\left(x^{1 / n}\right)=\frac{1}{n} x^{1 / n-1} $$ for any nonzero integer \(n\), assuming that \(f(x)=x^{1 / n}\) is differentiable. Hint: Let \(f(x)=x^{1 / n}\) so that \([f(x)]^{n}=x\). Differentiate both sides with respect to \(x\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(\lim _{x \rightarrow 0} f(x)=4\) and \(\lim _{x \rightarrow 0} g(x)=0\), then \(\lim _{x \rightarrow 0} f(x) g(x)=0\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(\lim _{x \rightarrow 2} f(x)=3\) and \(\lim _{x \rightarrow 2} g(x)=0\), then \(\lim _{x \rightarrow 2}[f(x)] /[g(x)]\) does not exist.

Certain proteins, known as enzymes, serve as catalysts for chemical reactions in living things. In 1913 Leonor Michaelis and L. M. Menten discovered the following formula giving the initial speed \(V\) (in moles/liter/second) at which the reaction begins in terms of the amount of substrate \(x\) (the substance heing acted upon, measured in moles/liters) present: $$ V=\frac{a x}{x+b} $$ where \(a\) and \(b\) are positive constants. Evaluate $$ \lim _{x \rightarrow \infty} \frac{a x}{x+b} $$ and interpret your result.

Suppose \(f\) is continuous on \([a, b]\) and \(f(a)

Show by means of an example that \(\lim _{x \rightarrow a}[f(x) g(x)]\) may exist even though neither \(\lim _{x \rightarrow a} f(x)\) nor \(\lim _{x \rightarrow a} g(x)\) exists. Does this example contradict Theorem \(1 ?\)

Let \(f(x)=\frac{x^{2}}{x^{2}+1} .\) a. Show that \(f\) is continuous for all values of \(x\). b. Show that \(f(x)\) is nonnegative for all values of \(x\). c. Show that \(f\) has a zero at \(x=0 .\) Does this contradict Theorem 5 ?

Let \(f(x)=x-\sqrt{1-x^{2}}\). a. Show that \(f\) is continuous for all values of \(x\) in the interval \([-1,1]\). b. Show that \(f\) has at least one zero in \([-1,1]\). c. Find the zeros of \(f\) in \([-1,1]\) by solving the equation \(f(x)=0 .\)

a. Prove that a polynomial function \(y=P(x)\) is continuous at every number \(x\). Follow these steps: (i) Use Properties 2 and 3 of continuous functions to establish that the function \(g(x)=x^{n}\), where \(n\) is a positive integer, is continuous everywhere. (ii) Use Properties 1 and 5 to show that \(f(x)=c x^{n}\), where \(c\) is a constant and \(n\) is a positive integer, is continuous everywhere. (iii) Use Property 4 to complete the proof of the result. b. Prove that a rational function \(R(x)=p(x) / q(x)\) is continuous at every point \(x\), where \(q(x) \neq 0\). Hint: Use the result of part (a) and Property 6 .