Chapter 22, Net Change in Amount and Area: Introducing the Definite Integral Video Solutions, Calculus: An Integrated Approach to Functions and Their Rates of Change

Problem 1

It s u season and a health clinic has set up a u shot program for its patients. The clinic is open on Saturday from 8:00 A.M. to 4:00 P.M. (16:00) giving u shots on a rst-come, rst-serve basis. The clinic has the capacity to serve 30 patients per hour. The function $r(t)$, whose graph is given below, gives the rate at which people are arriving at the clinic for shots.
Explain your answers to the questions below by relating them to points, lengths, or areas on the graph. ${ }^{5}$
(a) At what time does a line start forming?
(b) At approximately what time is the length of the line increasing most rapidly?
(c) At approximately what time is the line the longest? (The answers to (b) and (c) are different. Explain why.)
(d) When the line is longest, approximately how many people are in line?
(e) Approximate the longest amount of time a person could wait for a shot. (Assume that doors close to new arrivals at 4:00 P.M. but everyone who has arrived by $4: 00$
P.M. is served.)
(f) Approximately how long is the line at 3:00 P.M.?
(g) Approximate the number of people who came to the clinic for a u shot this day.

Sajay Krishnan Paruthiyil

Numerade Educator

01:39

Problem 2

Maple syrup is being poured at a decreasing rate out of a tank. By taking readings from the valve on the tank, we have the following information on the rate at which the syrup is leaving the tank.$$
\begin{array}{lccccc}
t \text { (seconds) } & 0 & 2 & 4 & 6 & 8 \\
\left.\hline \text { rate (in cm }^{3} / \mathrm{sec}\right) & 10 & 9 & 7 & 4 & 2
\end{array}
$$
(a) Find a good upper bound for the amount of maple syrup that has been poured out between time $t=0$ and $t=8$.
(b) Find a good lower bound for this same amount.

Audrey Fong

Numerade Educator

08:49

Problem 3

An industrial chemist is making a mixture in a large container. A certain chemical, B, is being introduced into the mixture at an ever-increasing rate. Some of the rates have been registered below. Time $t=0$ marks the rst introduction of this chemical into the mixture.
$$
\begin{array}{lcccccc}
\text { time (in minutes) } & 0 & 3 & 7 & 10 & 13 & 15 \\
\hline \text { rate (in grams/min.) } & 10 & 12 & 20 & 23 & 25 & 29
\end{array}
$$
Determine reasonable upper and lower bounds for the number of grams of chemical B in the mixture at time $t=13$.

Luis Amaro

Numerade Educator

03:03

Problem 4

(a) By partitioning the interval $[0, \pi / 2]$ into four equal pieces and using the areas of inscribed and circumscribed rectangles as appropriate, nd upper and lower bounds for the area between the graph of $\sin x$ and the $x$ -axis for $x$ in the interval $[0, \pi / 2] .$ Draw a picture illustrating what you have done. (Note: You will have to use your calculator to get some of the values of $\sin x$ and to get a numerical answer.)
(b) Using the work you did in part (a), nd upper and lower bounds for the area under the graph of $\sin x$ between $x=0$ and $x=\pi$. Explain what you have done using a picture.
(c) Using the work you did in part (b), give upper and lower bounds for the area under the graph of $\cos x$ between $x=-\pi / 2$ and $\pi / 2$.

Elliott Walker

Numerade Educator

01:57

Problem 5

Suppose velocity (in miles per hour) is given by $v(t)=3 t$, where $t$ is measured in hours. We are interested in the distance traveled from $t=0$ to $t=k$, where $k$ is a constant.
(a) By solving the differential equation $d s / d t=3 t$ and using the initial condition $s(0)=s_{0}$, nd the distance function $s(t) .$ Using $s(t)$, nd
i. $s(0)$.
ii. $s(k)$.
iii. the distance traveled between $t=0$ and $t=k$.
(b) Find the area under the graph of $v(t)$ from $t=0$ to $t=k$. Verify that your answers to part (a) iii and (b) are the same.

Colin O'Haire

Numerade Educator

01:57

Problem 6

Suppose velocity (in miles per hour) is given by $v(t)=m t+\mathrm{c}$, where $m$ and $c$ are positive constants.
(a) Using your knowledge of the area of a trapezoid, nd the area under the graph of $v(t)$ on the interval $[a, b]$, where $a$ and $b$ are positive constants.
(b) By solving the differential equation $d s / d t=m t+c$ and using the initial condition $s(0)=s_{0}$, nd the distance function $s(t) .$ Using $s(t)$ nd
i. $s(a)$.
ii. $s(b)$.
iii. the distance traveled between $t=a$ and $t=b$. Verify that your answers to parts (a) and (b) iii are the same.

Colin O'Haire

Numerade Educator

01:00

Problem 7

Below is the graph of the velocity of a bee traveling in a straight line from a clover to a hive. Find the following.
(a) the distance traveled between $t=1$ and $t=3$.
(b) the distance traveled between $t=0$ and $t=8$.
(c) the distance traveled between $t=3$ and $t=5$.

Kratika Bhadauria

Numerade Educator

01:09

Problem 8

(a) The velocity of an object at time $t$ is given by $v(t)=t^{2} \mathrm{ft} / \mathrm{sec} .$ Partition the time interval $[0,3]$ into 3 equal pieces each of length 1 second. Find upper and lower bounds for the distance the object traveled between time $t=0$ and $t=3$.
(b) Illustrate your work in part (a) by graphing $v(t)$ and using areas of inscribed and circumscribed rectangles. Draw two pictures, one illustrating the upper bound and the other the lower bound.
(c) Repeat part (a), but this time partition the interval into 6 equal pieces, each of length $1 / 2$. Make a sketch indicating the areas you have found.
(d) What is the difference between $R_{n}$ and $L_{n}$ if the interval is partitioned into 50 equal pieces? 100 equal pieces?
(e) Into how many equal pieces must we partition $[0,3]$ to be sure that the difference between the right- and left-hand sums is less than or equal to $0.01$ ?

Carson Merrill

Numerade Educator

01:00

Problem 9

Suppose $v(t)$ gives the velocity of a trekker on the time interval $[0,3]$ and suppose that $v(t)$ is positive and decreasing over this interval. If we use a left-hand sum to approximate the distance she has covered over this time interval, will the approximation give a lower bound or an upper bound?

Stephen Hobbs

Numerade Educator