Chapter 1, Functions and Sequences Video Solutions, Biocalculus Calculus for the Life Sciences

Section 1

Four Ways to Represent a Function

00:32

Problem 1

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Amrita B.

Numerade Educator

01:31

Problem 2

If
$$f(x)=\frac{x^{2}-x}{x-1} \quad \text { and } \quad g(x)=x$$
is it true that $f=g ?$

Nandini S.

Numerade Educator

01:51

The graph of a function $f$ is given.
(a) State the value of $f(1) .$
(b) Estimate the value of $f(-1)$
(c) For what values of $x$ is $f(x)=1 ?$
(d) Estimate the value of $x$ such that $f(x)=0$ .
(e) State the domain and range of $f$ .
(f) On what interval is $f$ increasing?

Alexander C.

Numerade Educator

04:18

Problem 4

The graphs of $f$ and $g$ are given.
(a) State the values of $f(-4)$ and $g(3) .$
(b) For what values of $x$ is $f(x)=g(x) ?$
(c) Estimate the solution of the equation $f(x)=-1$
(d) On what interval is $f$ decreasing?
(e) State the domain and range of $f .$
(f) State the domain and range of $g .$

Nandini S.

Numerade Educator

01:08

Problem 5

Determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

Tyler M.

Numerade Educator

01:22

Problem 6

Determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

Nandini S.

Numerade Educator

05:15

Problem 7

Determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

Khoi V.

Numerade Educator

01:04

Problem 8

Determine whether the curve is the graph of a function of $x$. If it is, state the domain and range of the function.

Nandini S.

Numerade Educator

05:58

Problem 9

Global temperature Shown is a graph of the global average temperature $T$ during the 20 th century.
(a) What was the global average temperature in 1950$?$
(b) In what year was the average temperature $14.2^{\circ}$ ?
(c) When was the temperature smallest? Largest?
(d) Estimate the range of $T .$

Vanna T.

Numerade Educator

01:32

Problem 10

Tree ring width Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000.
(a) What is the range of the ring width function?
(b) What does the graph tend to say about the temperature of the earth? Does the graph reflect the volcanic eruptions of the mid-19th century?

Nandini S.

Numerade Educator

03:22

Problem 11

Esophageal pH A healthy esophagus has a pH of about 7.0. When acid reflux occurs, stomach acid (which has pH ranging from 1.0 to 3.0) flows backward from the stomach into the esophagus. When the pH of the esophagus is less than 4.0, the episode is called "clinical acid reflux" and can cause ulcers and damage the lining of the esophagus. The graph shows esophageal pH for a sleeping patient with acid
reflux. During what time interval is the patient considered to have an episode of clinical acid reflux?

Oswaldo J.

Numerade Educator

01:38

Problem 12

Tadpole weights The figure shows the average body weights of tadpoles raised in different densities. The
function f shows body weights when the density is 10 tadpolesyL. For functions t and h the densities are 80 and 160 tadpolesyL, respectively. What do these graphs tell you about the effect of crowding?

Nandini S.

Numerade Educator

02:46

Problem 13

Species richness Tropical regions receive more rainfall and intense sunlight and have longer growing seasons than regions farther from the equator. As a result, they enjoy greater species richness, that is, greater numbers of species. The graph shows how species richness varies with latitude for ants.

(a) How many species would you expect to find at $30^{\circ} \mathrm{S}$ ? At $20^{\circ} \mathrm{N} ?$
(b) If you found about 100 ant species at a certain location, at roughly what latitude would you be?
(c) What symmetry property does this function possess?

Vanna T.

Numerade Educator

05:25

Problem 14

In this section we discussed examples of ordinary, every- day functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.

Nandini S.

Numerade Educator

01:07

Problem 15

The graph shown gives the weight of a certain person as a function of age. Describe in words how this person's weight varies over time. What do you think happened when this person was 30 years old?

Amrita B.

Numerade Educator

01:49

Problem 16

Ground reaction force in walking The graph shows the horizontal force exerted by the ground on a person during walking. Positive values are forces in the forward direction and negative values are forces in the backward direction. Give an explanation for the shape of the graph of the force function, including the points where it crosses the axis.

Nandini S.

Numerade Educator

00:29

Problem 17

You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.

Amrita B.

Numerade Educator

01:40

Problem 18

Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?

Meghan M.

Numerade Educator

01:04

Problem 19

Bacteria count Shown is a typical graph of the number N of bacteria grown in a batch culture as a function of time t. Describe what you think is happening during each of the four phases.

Amrita B.

Numerade Educator

01:21

Problem 20

Sketch a rough graph of the number of hours of daylight as a function of the time of year.

Nandini S.

Numerade Educator

03:52

Problem 21

Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

Charles C.

Numerade Educator

01:53

Problem 22

You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.

Nandini S.

Numerade Educator

00:31

Problem 23

Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.

Amrita B.

Numerade Educator

01:08

Problem 24

Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.

Nandini S.

Numerade Educator

04:05

Problem 25

Bird count The table shows the number of house finches, in thousands, observed in the Christmas bird count in California.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1980} & {1985} & {1990} & {1995} & {2000} & {2005} & {2010} \\ \hline \text { Count } & {74} & {92} & {88} & {107} & {70} & {61} & {78} \\ \hline\end{array}$$

(a) Use the data to sketch a rough graph of the bird count as a function of time.
(b) Use your graph to estimate the count in 1997.

Vanna T.

Numerade Educator

01:33

Problem 26

Blood alcohol concentration Researchers measured the blood alcohol concentration $(\mathrm{BAC})$ of eight adult male subjects after rapid consumption of 30 $\mathrm{mL}$ of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in $m g / m L )$ of the eight men.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline \text { (hours) } & {0.0} & {0.2} & {0.5} & {0.75} & {1.0} & {1.25} & {1.5} \\ \hline \mathrm{BAC} & {0} & {0.25} & {0.41} & {0.40} & {0.33} & {0.24} \\ \hline\end{array}$$
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline \text { (hours) } & {1.75} & {2.0} & {2.25} & {2.5} & {3.0} & {3.5} & {4.0} \\ \hline \mathrm{BAC} & {0.22} & {0.18} & {0.15} & {0.12} & {0.07} & {0.03} & {0.01} \\ \hline\end{array}$$

(a) Use the readings to sketch the graph of the BAC as a function of $t$ .
(b) Use your graph to describe how the concentration of alcohol varies with time.

Nandini S.

Numerade Educator

12:14

Problem 27

If $f(x)=3 x^{2}-x+2$, find $f(2), f(-2), f(a), f(-a)$, $f(a+1), 2 f(a), f(2 a), f\left(a^{2}\right),[f(a)]^{2}$, and $f(a+h)$.

Jeremy S.

Numerade Educator

02:45

Problem 28

A spherical balloon with radius $r$ inches has volume $V(r)=\frac{4}{3} \pi r^{3} .$ Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r+1$ inches.

Nandini S.

Numerade Educator

01:06

Problem 29

Evaluate the difference quotient for the given function. Simplify your answer.
$f(x)=4+3 x-x^{2}, \quad \frac{f(3+h)-f(3)}{h}$

Amrita B.

Numerade Educator

02:59

Problem 30

Evaluate the difference quotient for the given function. Simplify your answer.
$f(x)=x^{3}, \quad \frac{f(a+h)-f(a)}{h}$

Nandini S.

Numerade Educator

00:48

Problem 31

Evaluate the difference quotient for the given function. Simplify your answer.
$f(x)=\frac{1}{x}, \quad \frac{f(x)-f(a)}{x-a}$

Amrita B.

Numerade Educator

02:07

Problem 32

Evaluate the difference quotient for the given function. Simplify your answer.
$f(x)=\frac{x+3}{x+1}, \quad \frac{f(x)-f(1)}{x-1}$

Nandini S.

Numerade Educator

00:39

Problem 33

Find the domain of the function.
$f(x)=\frac{x+4}{x^{2}-9}$

Amrita B.

Numerade Educator

00:57

Problem 34

Find the domain of the function.
$f(x)=\frac{2 x^{3}-5}{x^{2}+x-6}$

Nandini S.

Numerade Educator

00:34

Problem 35

Find the domain of the function.
$f(t)=\sqrt[3]{2 t-1}$

Amrita B.

Numerade Educator

01:36

Problem 36

Find the domain of the function.
$g(t)=\sqrt{3-t}-\sqrt{2+t}$

Nandini S.

Numerade Educator

00:48

Problem 37

Find the domain of the function.
$h(x)=\frac{1}{\sqrt[4]{x^{2}-5 x}}$

Amrita B.

Numerade Educator

02:06

Problem 38

Find the domain of the function.
$f(u)=\frac{u+1}{1+\frac{1}{u+1}}$

Nandini S.

Numerade Educator

02:10

Problem 39

Find the domain of the function.
$F(p)=\sqrt{2-\sqrt{p}}$

Rajesh R.

Numerade Educator

03:39

Problem 40

Find the domain and range and sketch the graph of the
function $h(x)=\sqrt{4-x^{2}}$

Nandini S.

Numerade Educator

00:36

Problem 41

Find the domain and sketch the graph of the function.
$f(x)=2-0.4 x$

Amrita B.

Numerade Educator

01:37

Problem 42

Find the domain and sketch the graph of the function.
$F(x)=x^{2}-2 x+1$

Nandini S.

Numerade Educator

00:34

Problem 43

Find the domain and sketch the graph of the function.
$f(t)=2 t+t^{2}$

Amrita B.

Numerade Educator

01:48

Problem 44

Find the domain and sketch the graph of the function.
$H(t)=\frac{4-t^{2}}{2-t}$

Nandini S.

Numerade Educator

00:29

Problem 45

Find the domain and sketch the graph of the function.
$g(x)=\sqrt{x-5}$

Amrita B.

Numerade Educator

01:43

Problem 46

Find the domain and sketch the graph of the function.
$F(x)=|2 x+1|$

Nandini S.

Numerade Educator

00:53

Problem 47

Find the domain and sketch the graph of the function.
$G(x)=\frac{3 x+|x|}{x}$

Amrita B.

Numerade Educator

04:00

Problem 48

Find the domain and sketch the graph of the function.
$g(x)=|x|-x$

Nandini S.

Numerade Educator

00:35

Problem 49

Find the domain and sketch the graph of the function.
$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {1-x} & {\text { if } x \geqslant 0}\end{array}\right.$

Amrita B.

Numerade Educator

02:43

Problem 50

Find the domain and sketch the graph of the function.
$f(x)=\left\{\begin{array}{ll}{3-\frac{1}{2} x} & {\text { if } x \leq 2} \\ {2 x-5} & {\text { if } x>2}\end{array}\right.$

Nandini S.

Numerade Educator

00:31

Problem 51

Find the domain and sketch the graph of the function.
$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x \leqslant-1} \\ {x^{2}} & {\text { if } x>-1}\end{array}\right.$

Amrita B.

Numerade Educator

01:44

Problem 52

Find the domain and sketch the graph of the function.
$f(x)=\left\{\begin{array}{ll}{x+9} & {\text { if } x<-3} \\ {-2 x} & {\text { if }|x| \leq 3} \\ {-6} & {\text { if } x>3}\end{array}\right.$

Nandini S.

Numerade Educator

00:59

Problem 53

Find a formula for the described function and state its domain.

A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.

Amrita B.

Numerade Educator

02:13

Problem 54

Find a formula for the described function and state its domain.

A rectangle has area 16$\mathrm{m}^{2}$. Express the perimeter of the rectangle as a function of the length of one of its sides.

Nandini S.

Numerade Educator

00:56

Problem 55

Find a formula for the described function and state its domain.

Express the area of an equilateral triangle as a function of the length of a side.

Amrita B.

Numerade Educator

01:46

Problem 56

Find a formula for the described function and state its domain.

Express the surface area of a cube as a function of its volume.

Nandini S.

Numerade Educator

00:58

Problem 57

Find a formula for the described function and state its domain.

An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base.

Amrita B.

Numerade Educator

03:59

Problem 58

A cell phone plan has a basic charge of $\$ 35$ a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost $C$ as a function of the number $x$ of minutes used and graph $C$ as a function of $x$ for 0$\leqslant x \leqslant 600$ .

Nandini S.

Numerade Educator

01:50

Problem 59

A hotel chain charges $\$ 75$ each night for the first two night and $\$ 50$ for each additional night's stay. Express the total cost $T$ as a function of the number of nights $x$ that a guest stays.

Amrita B.

Numerade Educator

01:42

Problem 60

The function in Example 11 is called a step function because its graph looks like stairs. Give two other examples of step functions that arise in everyday life.

Nandini S.

Numerade Educator

00:41

Problem 61

Temperature chart The figure shows the temperature of a patient infected with the malaria species P. malariae. Estimate the period of the temperature function.

Amrita B.

Numerade Educator

00:47

Problem 62

Malarial fever A temperature chart is shown for a patient with a fever induced by the malaria species P. falciparum. What do you think is happening?

Nandini S.

Numerade Educator

00:20

Problem 63

Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

Amrita B.

Numerade Educator

01:26

Problem 64

Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

Nandini S.

Numerade Educator

00:36

Problem 65

(a) If the point s5, 3d is on the graph of an even function, what other point must also be on the graph?
(b) If the point s5, 3d is on the graph of an odd function, what other point must also be on the graph?

Amrita B.

Numerade Educator

01:10

Problem 66

A function f has domain f25, 5g and a portion of its graph is shown.
(a) Complete the graph of f if it is known that f is even.
(b) Complete the graph of f if it is known that f is odd.

Nandini S.

Numerade Educator

00:33

Problem 67

Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$f(x)=\frac{x}{x^{2}+1}$

Amrita B.

Numerade Educator

00:49

Problem 68

Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$f(x)=\frac{x^{2}}{x^{4}+1}$

Nandini S.

Numerade Educator

00:30

Problem 69

Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$f(x)=\frac{x}{x+1}$

Amrita B.

Numerade Educator

01:04

Problem 70

Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$f(x)=x|x|$

Nandini S.

Numerade Educator

00:29

Problem 71

Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$f(x)=1+3 x^{2}-x^{4}$

Amrita B.

Numerade Educator

01:26

Problem 72

Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$f(x)=1+3 x^{3}-x^{5}$

Nandini S.

Numerade Educator

00:50

Problem 73

If $f$ and $g$ are both even functions, is $f+g$ even? If $f$ and $g$ are both odd functions, is $f+g$ odd? What if $f$ is even and $g$ is odd? Justify your answers.

Amrita B.

Numerade Educator

01:32

Problem 74

If $f$ and $g$ are both even functions, is the product $f g$ even? If $f$ and $g$ are both odd functions, is $f g$ odd? What if $f$ is even and $g$ is odd? Justify your answers.

Nandini S.

Numerade Educator