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Chapter 2

Limits and Derivatives

Educators


Problem 1

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

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Problem 2

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?

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Problem 3

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

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Problem 4

The point $\mathrm{P}(3,1)$ lies on the curve $y=\sqrt{x-2}$.
(a) If $\mathrm{O}$ is the point $\left(\mathrm{x}_{1} \sqrt{\mathrm{x}-2}\right)$ , use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of $\mathrm{x} :$
$$\begin{array}{llll}{\text { (i) } 2.5} & {\text { (ii) } 2.9} & {\text { (iii) } 2.99} & {\text { (iv) } 2.999} \\ {\text { (v) } 3.5} & {\text { (vi) } 3.1} & {\text { (vii) } 3.01} & {\text { (viii) } 3.001}\end{array}$$
(b) Using the results of part (a), guess the value of the slope of
the tangent line to the curve at $\mathrm{P}(3,1)$ .
(c) Using the slope from part (b), find an equation of the
tangent line to the curve at $\mathrm{P}(3,1) .$
(d) Sketch the curve, two of the secant lines, and the tangent
line.

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Problem 5

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 $t=2$

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Problem 6

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 $t=1$

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Problem 7

The table shows the position of a cyclist.
(a) Find the average velocity for each time period:
$$(i)[1,3] \quad \text { (ii) }[2,3] \quad \text { (iii) }[3,5] \quad \text { (iv) }[3,4]$$
(b) Use the graph of s as a function of to estimate the instan-
taneous velocity when $t=3 .$

Oswaldo J.
Numerade Educator

Problem 8

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.

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Problem 9

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

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