Chapter 2

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume \(V\) of water remaining in the tank (in gallons) after t minutes. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t(\min ) & {5} & {10} & {15} & {20} & {25} & {30} \\ \hline V(g a l) & {694} & {444} & {250} & {111} & {28} & {0} \\\ \hline\end{array}$$ $$\begin{array}{l}{\text { (a) If } P \text { is the point }(15,250) \text { on the graph of } V \text { , find the slopes }} \\ {\text { of the secant lines PQ when } Q \text { is the point on the graph }} \\ {\text { with } t=5,10,20,25, \text { and } 30 \text { . }}\end{array} $$ $$\begin{array}{l}{\text { (b) Estimate the slope of the tangent line at } P \text { by averaging the }} \\ {\text { slopes of two secant lines. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Use a graph of the function to estimate the slope of the }} \\ {\text { tangent line at } P . \text { (This slope represents the rate at which the }} \\ {\text { water is flowing from the tank after } 15 \text { minutes.) }}\end{array}$$

Given that $$\lim _{x \rightarrow 2} f(x)=4 \quad \lim _{x \rightarrow 2} g(x)=-2 \quad \lim _{x \rightarrow 2} h(x)=0$$ find the limits that exist. If the limit does not explain why. (a) $$\lim _{x \rightarrow 2}[f(x)+5 g(x)]$$ (b) $$\lim _{x \rightarrow 2}[g(x)]^{3}$$ (c) $$\lim _{x \rightarrow 2} \sqrt{f(x)}$$ (d) $$\lim _{x \rightarrow 2} \frac{3 f(x)}{g(x)}$$ (e) $$\lim _{x \rightarrow 2} \frac{g(x)}{h(x)}$$ (f) $$\lim _{x \rightarrow 2} \frac{g(x) h(x)}{f(x)}$$

Explain in your own words what is meant by the equation $$\lim _{x \rightarrow 2} f(x)=5$$ Is it possible for this statement to be true and yet \(f(2)=3 ?\) Explain.

Write an equation that expresses the fact that a function f is continuous at the number 4 .

$$ \begin{array}{l}{\text { Explain in your own words the meaning of each of the }} \\ {\text { following. }} \\ {\text { (a) } \lim _{x \rightarrow \infty} f(x)=5 \quad \text { (b) } \lim _{x \rightarrow-\infty} f(x)=3}\end{array} $$

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. $$\begin{array}{|c|c|c|c|c|c|}\hline t(\min ) & {36} & {38} & {40} & {42} & {44} \\ \hline \text { Heartbeats } & {2530} & {2661} & {2806} & {2948} & {3080} \\ \hline\end{array}$$ The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of \(t\) . $$\begin{array}{ll}{\text { (a) } t=36} & {\text { and } t=42 \quad \text { (b) } t=38 \text { and } t=42} \\ {\text { (c) } t=40 \text { and } t=42} & {\text { (d) } t=42 \text { and } t=44}\end{array}$$ What are your conclusions?

Graph the curve \(y=e^{x}\) in the viewing rectangles \([-1,1]\) by $$[0,2],[-0.5,0.5]\( by \)[0.5,1.5],$$ and $$[-0.1,0.1]\( by \)[0.9,1.1]$$ . What do you notice about the curve as you zoom in toward the point \((0,1) ?\)

Explain what it means to say that $$\lim _{x \rightarrow 1^{-}} f(x)=3 \quad \text { and } \quad \lim _{x \rightarrow 1^{+}} f(x)=7$$ In this situation is it possible that \(\lim _{x \rightarrow 1}\) \(\mathrm{f}(\mathrm{x})\) exists? Explain.

If \(\mathrm{f}\) is continuous on \((-\infty, \infty),\) what can you say about its graph?

$$ \begin{array}{l}{\text { (a) Can the graph of } y=f(x) \text { intersect a vertical asymptote? }} \\ {\text { Can it intersect a horizontal asymptote? Illustrate by }} \\ {\text { sketching graphs. }} \\ {\text { (b) How many horizontal asymptotes can the graph of } y=f(x)} \\ {\text { have? Sketch graphs to illustrate the possibilities. }}\end{array} $$

The point \(\mathrm{P}\left(1, \frac{1}{2}\right)\) lies on the curve \(y=x /(1+x)\). $$\begin{array}{l}{\text { (a) If } \mathrm{O} \text { is the point }(\mathrm{x}, \mathrm{x} /(1+\mathrm{x})), \text { use your calculator to find }} \\ {\text { the slope of the secant line PQ (correct to six decimal places)}}\end{array}$$ $$\begin{array}{lll}{\text { (i) } 0.5} & {\text { (ii) } 0.9} & {\text { (iii) } 0.99} & {\text { (iv) } 0.999} \\ {\text { (v) } 1.5} & {\text { (vi) } 1.1} & {\text { (vii) } 1.01} & {\text { (vili) } 1.001}\end{array}$$ $$\begin{array}{l}{\text { (b) Using the results of part (a), guess the value of the slope of }} \\ {\text { the tangent line to the curve at } P\left(1, \frac{1}{2}\right)}\end{array}$$ $$\begin{array}{l}{\text { (c) Using the slope from part (b) , find an equation of the }} \\ {\text { tangent line to the curve at } P\left(1, \frac{1}{2}\right)}\end{array}$$

Explain the meaning of each of the following. $${ (a) }\lim _{x \rightarrow-3} f(x)=\infty \quad \text { (b) } \lim _{x \rightarrow 4^{+}} f(x)=-\infty$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow-2}\left(3 x^{4}+2 x^{2}-x+1\right)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 2} \frac{2 x^{2}+1}{x^{2}+6 x-4}$$

If a ball is thrown into the air with a velocity of 40 \(\mathrm{ft} / \mathrm{s}\) , its height in feet t seconds later is given by \(\mathrm{y}=40 \mathrm{t}-16 \mathrm{t}^{2}\) . (a) Find the average velocity for the time period beginning when \(t=2\) and lasting $$\begin{array}{ll}{\text { (i) } 0.5 \text { second }} & {\text { (il) } 0.1 \text { second }} \\ {\text { (iii) } 0.05 \text { second }} & {\text { (iv) } 0.01 \text { second }}\end{array}$$ (b) Estimate the instantaneous velocity when \(t=2\)

Use a graph to find a number \(\delta\) such that $$\quad if \quad\left|x-\frac{\pi}{4}\right|<\delta \quad then \quad|\tan x-1|<0.2$$

\(5-8\) Find an equation of the tangent line to the curve at the given point. $$y=\frac{x-1}{x-2}, \quad(3,2)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 8}(1+\sqrt[3]{x})\left(2-6 x^{2}+x^{3}\right)$$

Sketch the graph of a function that is continuous everywhere except at \(x=3\) and is continuous from the left at \(3 .\)

If a rock is thrown upward on the planet Mars with a velocity of \(10 \mathrm{m} / \mathrm{s},\) its height in meters t seconds later is given by \(\mathrm{y}=10 \mathrm{t}-1.86 \mathrm{t}^{2}\) (a) Find the average velocity over the given time intervals: $$\begin{array}{ll}{\text { (i) }[1,2]} & {\text { (ii) }[1,1.5]} \\ {\text { (iv) }[1,1.01]} & {\text { (v) }[1,1.001] {\text { (iii) }[1,1.1] }}\end{array}$$ (b) Estimate the instantaneous velocity when \(t=1\)

\(5-8\) Find an equation of the tangent line to the curve at the given point. $$y=2 x^{3}-5 x,(-1,3)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{t \rightarrow-1}\left(t^{2}+1\right)^{3}(t+3)^{5}$$

\(5-10\) Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. $$ \begin{array}{l}{\lim _{x \rightarrow 0^{+}} f(x)=\infty, \quad \lim _{x \rightarrow 0^{-}} f(x)=-\infty, \quad \lim _{x \rightarrow \infty} f(x)=1} \\\ {\lim _{x \rightarrow-\infty} f(x)=1}\end{array} $$

Sketch the graph of a function that has a jump discontinuity at \(x=2\) and a removable discontinuity at \(x=4,\) but is continuous elsewhere.

For the limit $$\lim _{x \rightarrow 1}\left(4+x-3 x^{3}\right)=2$$ illustrate Definition 2 by finding values of \(\delta\) that correspond to \(\varepsilon=1\) and \(\varepsilon=0.1 .\)

\(5-8\) Find an equation of the tangent line to the curve at the given point. $$y=\sqrt{x}, \quad(1,1)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 1}\left(\frac{1+3 x}{1+4 x^{2}+3 x^{4}}\right)^{3}$$

A parking lot charges \(\$ 3\) for the first hour (or part of an hour) and \(\$ 2\) for each succeeding hour (or part), up to a daily maximum of \(\$ 10\) . (a) Sketch a graph of the cost of parking at this lot as a function of the time parked there. (b) Discuss the discontinuities of this function and their significance to someone who parks in the lot.

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion \(s=2 \sin \pi t+3 \cos \pi t,\) where \(t\) is measured in seconds. (a) Find the average velocity during each time period: $$\begin{array}{ll}{\text { (i) }[1,2]} & {\text { (ii) }[1,1.1]} \\ {\text { (iii) }} & {[1,1.01]} & {[\text { iv) }[1,1.001]}\end{array}$$ (b) Estimate the instantaneous velocity of the particle when t= 1.

For the limit $$\lim _{x \rightarrow 0} \frac{\mathrm{e}^{x}-1}{x}=1$$ illustrate Definition 2 by finding values of \(\delta\) that correspond to \(\varepsilon=0.5\) and \(\varepsilon=0.1\)

\(5-8\) Find an equation of the tangent line to the curve at the given point. $$y=\frac{2 x}{(x+1)^{2}}, \quad(0,0)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{u \rightarrow-2} \sqrt{u^{4}+3 u+6}$$

\(5-10\) Sketch the graph of an example of a function f that satisfies all of the given conditions. $$ \lim _{x \rightarrow-2} f(x)=\infty, \quad \lim _{x \rightarrow-\infty} f(x)=3, \quad \lim _{x \rightarrow \infty} f(x)=-3 $$

Explain why each function is continuous or discontinuous. (a) The temperature at a specific location as a function of time (b) The temperature at a specific time as a function of the distance due west from New York City (c) The altitude above sea level as a function of the distance due west from New York City (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time

The point \(P(1,0)\) lies on the curve \(y=\sin (10 \pi / x)\). $$\begin{array}{l}{\text { (a) If } Q \text { is the point }(x, \sin (10 \pi / x)) \text { , find the slope of the secant }} \\ {\text { line } P Q \text { (correct to four decimal places) for } x=2,1.5,1.4 \text { , }} \\\ {1.3,1.2,1.1,0.5,0.6,0.7,0.8, \text { and } 0.9 . \text { Do the slopes }} \\\ {\text { appear to be approaching a limit? }}\end{array}$$ $$\begin{array}{l}{\text { (b) Use a graph of the curve to explain why the slopes of the }} \\ {\text { secant lines in part (a) are not close to the slope of the tan- }} \\ {\text { gent line at } P .}\end{array}$$ $$\begin{array}{l}{\text { (c) By choosing appropriate secant lines, estimate the slope of }} \\ {\text { the tangent line at } \mathrm{P} .}\end{array}$$

(a) Find the slope of the tangent to the curve $$y=3+4 x^{2}-2 x^{3}$$ at the point where \(x=\) a. (b) Find equations of the tangent lines at the points \((1,5)\) and \((2,3) .\) (c) Graph the curve and both tangents on a common screen.

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 4^{-}} \sqrt{16-x^{2}}$$

\(5-10\) Sketch the graph of an example of a function f that satisfies all of the given conditions. $$ \begin{array}{l}{f(0)=3, \quad \lim _{x \rightarrow 0^{-}} f(x)=4, \quad \lim _{x \rightarrow 0^{+}} f(x)=2} \\ {\lim _{x \rightarrow-\infty} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{-}} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{+}} f(x)=\infty} \\ {\lim _{x \rightarrow \infty} f(x)=3}\end{array} $$

If \(f\) and \(g\) are continuous functions with \(f(3)=5\) and $$\lim _{x \rightarrow 3}[2 f(x)-g(x)]=4,$$ find \(g(3)\)

(a) What is wrong with the following equation? $$\frac{x^{2}+x-6}{x-2}=x+3$$ (b) In view of part (a), explain why the equation $$\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}=\lim _{x \rightarrow 2}(x+3)$$ is correct.

\(5-10\) Sketch the graph of an example of a function f that satisfies all of the given conditions. $$ \lim _{x \rightarrow 3} f(x)=-\infty, \quad \lim _{x \rightarrow \infty} f(x)=2, \quad f(0)=0, \quad f $$

\(10-12\) Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. \(f(x)=x^{2}+\sqrt{7-x}, \quad a=4\)

J Use the graph of the function \(f(x)=1 /\left(1+e^{1 / x}\right)\) to state the value of each limit, if it exists. If it does not exist, explain why. $${ (a) }\lim _{x \rightarrow 0^{-}} f(x) \quad \text { (b) } \lim _{x \rightarrow 0^{+}} f(x) \quad \text { (c) } \lim _{x \rightarrow 0} f(x)$$

Evaluate the limit, if it exists. $$\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}$$

$$ \begin{array}{c}{\text { Guess the value of the limit }} \\ {\quad \lim _{x \rightarrow \infty} \frac{x^{2}}{2^{x}}}\end{array} $$ $$ \begin{array}{l}{\text { by evaluating the function } f(x)=x^{2} / 2^{x} \text { for } x=0,1,2,3,} \\ {4,5,6,7,8,9,10,20,50, \text { and } 100 . \text { Then use a graph of } f} \\ {\text { to support your guess. }}\end{array} $$

\(10-12\) Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. \(f(x)=\left(x+2 x^{3}\right)^{4}, \quad a=-1\)

Sketch the graph of the following function and use it to determine the values of a for which \(\lim _{x \rightarrow a}\) \(f(x)\) exists: $$f(x)=\left\\{\begin{array}{ll}{2-x} & {\text { if } x<-1} \\ {x} & {\text { if }-1 \leq x<1} \\ {(x-1)^{2}} & {\text { if } x \geqslant 1}\end{array}\right.$$

Evaluate the limit, if it exists. $$\lim _{x \rightarrow-4} \frac{x^{2}+5 x+4}{x^{2}+3 x-4}$$

$$ \begin{aligned} \text { (a) Use a graph of } & \\ & f(x)=\left(1-\frac{2}{x}\right)^{x} \end{aligned} $$ $$ \begin{array}{l}{\text { to estimate the value of } \lim _{x \rightarrow \infty} f(x) \text { correct to two }} \\ {\text { decimal places. }} \\\ {\text { (b) Use a table of values of } f(x) \text { to estimate the limit to }} \\ {\text { four decimal places. }}\end{array} $$

\(10-12\) Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. \(h(t)=\frac{2 t-3 t^{2}}{1+t^{3}}, \quad a=1\)