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

Daniel J.
Numerade Educator

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.
$$
\begin{array}{|l|c|c|c|c|c|}
\hline t \text { (min) } & 36 & 38 & 40 & 42 & 44 \\
\hline \text { Heartbeats } & 2530 & 2661 & 2806 & 2948 & 3080 \\
\hline
\end{array}
$$
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 $.
(a) $ t = 36 $ and $ t = 42 $
(b) $ t = 38 $ and $ t = 42 $
(c) $ t = 40 $ and $ t = 42 $
(d) $ t = 42 $ and $ t = 44 $
What are your conclusions?

Daniel J.
Numerade Educator

Problem 3

The point $ P(2, -1) $ lies on the curve $ y = 1/(1-x) $.

(a) If $ Q $ is the point $ (x, 1/(1-x)) $, use your calculator to find the slope of the secant line $ PQ $ (correct to six decimal places) for the following values of $ x $:
(i) $ 1.5 $ (ii) $ 1.9 $ (iii) $ 1.99 $ (iv) $ 1.999 $
(v) $ 2.5 $ (vi) $ 2.1 $ (vii) $ 2.01 $ (viii) $ 2.001 $

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $ P(2, -1) $.

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $ P(2, -1) $.

Daniel J.
Numerade Educator

Problem 4

The point $ P(0.5, 0) $ lies on the curve $ y = \cos \pi x $.

(a) If $ Q $ is the point $ (x, \cos \pi x) $, use your calculator to find the slope of the secant line $ PQ $ (correct to six decimal places) for the following values of $ x $:

(i) $ 0 $ (ii) $ 0.4 $ (iii) $ 0.49 $
(iv) $ 0.499 $ (v) $ 1 $ (vi) $ 0.6 $
(vii) $ 0. 51 $ (viii) $ 0.501 $

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $ P(0.5, 0) $.

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $ P(0.5, 0) $.

(d) Sketch the curve, two of the secant lines, and the tangent line.

Daniel J.
Numerade Educator

Problem 5

If a ball is thrown into the air with a velocity of $ 40 ft/s $, its height in feet $ t $ seconds later is given by $ y = 40t - 16t^2 $.

(a) Find the average velocity for the time period beginning when $ t = 2 $ and lasting
(i) 0.5 seconds (ii) 0.1 seconds
(iii) 0.05 seconds (iv) 0.01 seconds
(b) Estimate the instantaneous velocity when $ t = 2 $.

Daniel J.
Numerade Educator

Problem 6

If a rock is thrown upward on the planet Mars with a velocity of $ 10 m/s $, its height in meters $ t $ seconds after is given by $ y = 10t - 1.86t^2 $.

(a) Find the average velocity over the given time intervals:
(i) $ [1, 2] $ (ii) $ [1, 1.5] $
(iii) $ [1, 1.1] $ (iv) $ [1, 1.01] $
(v) $ [1, 1.001] $
(b) Estimate the instantaneous velocity when $ t = 1 $.

Daniel J.
Numerade Educator

Problem 7

The table shows the position of a motorcyclist after accelerating from rest.

(a) Find the average velocity for each time period:
(i) $ [2, 4] $ (ii) $ [3, 4] $ (iii) $ [4, 5] $ (iv) $ [4, 6] $
(b) Use the graph of $ s $ as a function of $ t $ to estimate the instantaneous velocity when $ t = 3 $.

Daniel 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:
(i) $ [1, 2] $ (ii) $ [1, 1.1] $
(iii) $ [1, 1.01] $ (iv) $ [1, 1.001] $
(b) Estimate the instantaneous velocity of the particle when $ t =1 $.

Daniel J.
Numerade Educator

Problem 9

The point $ P(1, 0) $ lies on the curve $ y = \sin (10\pi /x) $.

(a) If $ Q $ is the point $ (x, \sin (10\pi /x)) $, find the slope of the secant line $ PQ $ (correct to four decimal places) for $ x $ = 2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9 .
Do the slopes appear to be approaching a limit?
(b) Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at $ P $.
(c) By choosing appropriate secant lines, estimate the slope of the tangent line at $ P $.

Daniel J.
Numerade Educator