Calculus

Essential Calculus (Available Titles CengageNOW) | 1st Edition | ISBN: 9780495014423 | Authors: James Stewart
New Search

Essential Calculus (Available Titles CengageNOW) 1st Edition solutions

Get Unlimited Answers

Author: James Stewart
Publisher: Brooks Cole
ISBN: 9780495014423
Buy the book

Select Chapter:

We have answers for this textbook’s questions
Check out the solutions to this books Chapter Problems

Chapter 1 Problems

Chapter: 1 Problem: 1.6.1

Essential Calculus (Available Titles CengageNOW) 1
For the function whose graph is given, state the following. (a) (b) (c) (d) (e) (f ) The equations of the asymptotes lim xl fx lim xl lim fx xl1 f x lim xl1 lim f x xl2 f x f 1

Chapter: 1 Problem: 1.2.1

Essential Calculus (Available Titles CengageNOW) 1
(a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that and sketch several members of the family. (c) Which function belongs to both families?

Chapter: 1 Problem: 1

Essential Calculus (Available Titles CengageNOW) 1
(a) What is a function? What are its domain and range? f A t B (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?

Chapter: 1 Problem: 1.3.1

Essential Calculus (Available Titles CengageNOW) 1
If a ball is thrown into the air with a velocity of 40 fts, its t2 tt height in feet seconds later is given by . (a) Find the average velocity for the time period beginning when and lasting (i) 0.5 second (ii) 0.1 second (iii) 0.05 second (iv) 0.01 second (b) Estimate the instantaneous velocity when

Chapter: 1 Problem: 1.4.1

Essential Calculus (Available Titles CengageNOW) 1
Given that find the limits that exist. If the limit does not exist, explain why. (a) (b) (c) (d) (e) (f) (g) (h) limxla 2 fx hx f x lim xla f x tx lim xla tx fx lim xla f x hx lim xla 1 f x lim x l a s 3 hx lim xla f x 2 lim xla fx hx lim xla lim hx 8 xla lim tx 0 xla fx 3 1.4 EXERCISES Note that . s

Chapter: 1 Problem: 1.5.1

Essential Calculus (Available Titles CengageNOW) 1
Write an equation that expresses the fact that a function is continuous at the number 4.

Chapter: 1 Problem: 1.1.1

Essential Calculus (Available Titles CengageNOW) 1
The graph of a function is given. (a) State the value of . (b) Estimate the value of . (c) For what values of is ? (d) Estimate the values of such that . (e) State the domain and range of . (f ) On what interval is increasing?

Chapter: 1 Problem: 1.6.2

Essential Calculus (Available Titles CengageNOW) 1
For the function t whose graph is given, state the following. (a) (b) (c) (d) (e) (f ) T

Chapter: 1 Problem: 1.2.2

Essential Calculus (Available Titles CengageNOW) 1
What do all members of the family of linear functions have in common? Sketch several members of the family.

Chapter: 1 Problem: 2

Essential Calculus (Available Titles CengageNOW) 1
Discuss four ways of representing a function. Illustrate your discussion with examples.

Chapter: 1 Problem: 1.3.2

Essential Calculus (Available Titles CengageNOW) 1
If an arrow is shot upward on the moon with a velocity of 58 ms, its height in meters seconds later is given by (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 .

Chapter: 1 Problem: 1.4.2

Essential Calculus (Available Titles CengageNOW) 1
The graphs of and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why. (a) (b) (c) (d) (e) (f ) limxl1 lim s3 f x xl2 x 3 f x lim xl1 fx tx lim xl0 f xtx lim xl1 lim f x tx xl2 f x tx 1 x

Chapter: 1 Problem: 1.5.2

Essential Calculus (Available Titles CengageNOW) 1
If is continuous on , what can you say about its graph?

Chapter: 1 Problem: 1.1.2

Essential Calculus (Available Titles CengageNOW) 1
The graphs of and t are given. (a) State the values of and . (b) For what values of is ? (c) Estimate the solution of the equation . (d) On what interval is decreasing? (e) State the domain and range of (f ) State the domain and range of .

Chapter: 1 Problem: 1.6.3

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Chapter: 1 Problem: 1.2.3

Essential Calculus (Available Titles CengageNOW) 1
What do all members of the family of linear functions have in common? Sketch several members of the family.

Chapter: 1 Problem: 3

Essential Calculus (Available Titles CengageNOW) 1
(a) What is an even function? How can you tell if a function is even by looking at its graph? (b) What is an odd function? How can you tell if a function is odd by looking at its graph?

Chapter: 1 Problem: 1.3.3

Essential Calculus (Available Titles CengageNOW) 1
Use the given graph of to state the value of each quantity, if it exists. If it does not exist, explain why. (a) (b) (c) (d) (e)

Chapter: 1 Problem: 1.4.3

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Chapter: 1 Problem: 1.5.3

Essential Calculus (Available Titles CengageNOW) 1
a) From the graph of , state the numbers at which is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether is continuous from the right, or from the left, or neither.

Chapter: 1 Problem: 1.1.3

Essential Calculus (Available Titles CengageNOW) 1
Determine whether the curve is the graph of a function of . If it is, state the domain and range of the function.

Chapter: 1 Problem: 1.6.4

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Chapter: 1 Problem: 1.2.4

Essential Calculus (Available Titles CengageNOW) 1
Find expressions for the quadratic functions whose graphs are shown.

Chapter: 1 Problem: 4

Essential Calculus (Available Titles CengageNOW) 1
What is a mathematical model?

Chapter: 1 Problem: 1.3.4

Essential Calculus (Available Titles CengageNOW) 1
For the function whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) (b) (c) (d) (e)

Chapter: 1 Problem: 1.4.4

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Chapter: 1 Problem: 1.5.4

Essential Calculus (Available Titles CengageNOW) 1
From the graph of , state the intervals on which is continuous.

Chapter: 1 Problem: 1.1.4

Essential Calculus (Available Titles CengageNOW) 1
Determine whether the curve is the graph of a function of . If it is, state the domain and range of the function.

Chapter: 1 Problem: 1.6.5

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Chapter: 1 Problem: 1.2.5

Essential Calculus (Available Titles CengageNOW) 1
Find an expression for a cubic function if and f1 f 0 f 2 0 f f1 6 y

Chapter: 1 Problem: 5

Essential Calculus (Available Titles CengageNOW) 1
Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f ) Rational function

Chapter: 1 Problem: 1.3.5

Essential Calculus (Available Titles CengageNOW) 1
For the function whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) (b) (c) (d) (e) (f ) (g) (h)

Chapter: 1 Problem: 1.4.5

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Chapter: 1 Problem: 1.5.5

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of a function that is continuous everywhere except at x 3 and is continuous from the left at 3. 6

Chapter: 1 Problem: 1.1.5

Essential Calculus (Available Titles CengageNOW) 1
Determine whether the curve is the graph of a function of . If it is, state the domain and range of the function.

Chapter: 1 Problem: 1.6.6

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Chapter: 1 Problem: 1.2.6

Essential Calculus (Available Titles CengageNOW) 1
Some scientists believe that the average surface temperature of the world has been rising steadily. They have modeled the temperature by the linear function , where is temperature in and represents years since 1900. (a) What do the slope and -intercept represent? (b) Use the equation to predict the a

Chapter: 1 Problem: 6

Essential Calculus (Available Titles CengageNOW) 1
Sketch by hand, on the same axes, the graphs of the following functions. (a) (b) (c) (d)

Chapter: 1 Problem: 1.3.6

Essential Calculus (Available Titles CengageNOW) 1
ketch the graph of the following function and use it to determine the values of for which exists: f x  2 x x x 1 2 if x  1 if 1 x  1 if x  1 a limxla

Chapter: 1 Problem: 1.4.6

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Chapter: 1 Problem: 1.5.6

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of a function that has a jump discontinuity at and a removable discontinuity at , but is continuous elsewhere.

Chapter: 1 Problem: 1.1.6

Essential Calculus (Available Titles CengageNOW) 1
Determine whether the curve is the graph of a function of . If it is, state the domain and range of the function.

Chapter: 1 Problem: 1.6.7

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Chapter: 1 Problem: 1.2.7

Essential Calculus (Available Titles CengageNOW) 1
If the recommended adult dosage for a drug is (in mg), then to determine the appropriate dosage for a child of age , pharmacists use the equation . Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of . What does it represent? (b) What is the dosage for a newborn?

Chapter: 1 Problem: 7

Essential Calculus (Available Titles CengageNOW) 1
Sketch by hand, on the same axes, the graphs of the following functions. (a) (b) (c) (d)

Chapter: 1 Problem: 1.3.7

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. lim f1 2 xl1 lim fx 2 xl1 f x 2 f f x  2

Chapter: 1 Problem: 1.4.7

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Chapter: 1 Problem: 1.5.7

Essential Calculus (Available Titles CengageNOW) 1
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 significan

Chapter: 1 Problem: 1.1.7

Essential Calculus (Available Titles CengageNOW) 1
The graph shown gives the weight of a certain person as a function of age. Describe in words how this persons weight varies over time. What do you think happened when this person was 30 years old?

Chapter: 1 Problem: 1.6.8

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Chapter: 1 Problem: 1.2.8

Essential Calculus (Available Titles CengageNOW) 1
The manager of a weekend flea market knows from past experience that if he charges dollars for a rental space at the flea market, then the number of spaces he can rent is given by the equation . (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of s

Chapter: 1 Problem: 8

Essential Calculus (Available Titles CengageNOW) 1
Suppose that has domain and has domain . (a) What is the domain of ? (b) What is the domain of ? (c) What is the domain of

Chapter: 1 Problem: 1.3.8

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. xl2 lim fx 0 xl0 lim fx 1 xl0 f x 1 lim f1 2 xl1

Chapter: 1 Problem: 1.4.8

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Chapter: 1 Problem: 1.5.8

Essential Calculus (Available Titles CengageNOW) 1
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

Chapter: 1 Problem: 1.1.8

Essential Calculus (Available Titles CengageNOW) 1
The graph shown gives a salesmans distance from his home as a function of time on a certain day. Describe in words what the graph indicates about his travels on this day.

Chapter: 1 Problem: 1.6.9

Essential Calculus (Available Titles CengageNOW) 1
Guess the value of the limit by evaluating the function for and . Then use a graph of to support your guess.

Chapter: 1 Problem: 1.2.9

Essential Calculus (Available Titles CengageNOW) 1
The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function . (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent?

Chapter: 1 Problem: 9

Essential Calculus (Available Titles CengageNOW) 1
How is the composite function defined? What is its domain?

Chapter: 1 Problem: 1.3.9

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. lim f x 2 xl3 lim fx 2 xl3 f x 4 lim f2 1 f

Chapter: 1 Problem: 1.4.9

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Chapter: 1 Problem: 1.5.9

Essential Calculus (Available Titles CengageNOW) 1
If and are continuous functions with and limx l 3 2 f x tx 4 t3 f t f 3

Chapter: 1 Problem: 1.1.9

Essential Calculus (Available Titles CengageNOW) 1
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.

Chapter: 1 Problem: 1.6.10

Essential Calculus (Available Titles CengageNOW) 1
Determine and (a) by evaluating for values of that approach 1 from the left and from the right, (b) by reasoning as in Example 1, and ; (c) from a graph of . f

Chapter: 1 Problem: 1.2.10

Essential Calculus (Available Titles CengageNOW) 1
Jason leaves Detroit at 2:00 PM and drives at a constant speed west along I-96. He passes Ann Arbor, 40 mi from Detroit, at 2:50 PM. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represe

Chapter: 1 Problem: 10

Essential Calculus (Available Titles CengageNOW) 1
Suppose the graph of is given. Write an equation for each of the graphs that are obtained from the graph of as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the x-axis. (f ) Reflect about the y-axis. (g)

Chapter: 1 Problem: 1.3.10

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the given conditions. xl4 lim f x 3 xl4 lim fx 3 xl1 f x 3 f 3 3 f2 1

Chapter: 1 Problem: 1.4.10

Essential Calculus (Available Titles CengageNOW) 1
(a) What is wrong with the following equation? (b) In view of part (a), explain why the equation x 2 x 6 x 2 x 3 lim lim xl2 x 2 x 6 x 2 lim xl2 x 3 x 2 x is correct.

Chapter: 1 Problem: 1.5.10

Essential Calculus (Available Titles CengageNOW) 1
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number . fx x a 4 2 s7 x a lim

Chapter: 1 Problem: 1.1.10

Essential Calculus (Available Titles CengageNOW) 1
Sketch a rough graph of the number of hours of daylight as a function of the time of year.

Chapter: 1 Problem: 1.6.11

Essential Calculus (Available Titles CengageNOW) 1
Use a graph to estimate all the vertical and horizontal asymptotes of the curve y x 3 x 3 2x 1 2.

Chapter: 1 Problem: 1.2.11

Essential Calculus (Available Titles CengageNOW) 1
Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at and 173 chirps per minute at . (a) Find a linear equation that models the temperature T as a fu

Chapter: 1 Problem: 11

Essential Calculus (Available Titles CengageNOW) 1
Explain what each of the following means and illustrate with a sketch. (a) (b) (c) (d) (e) limxl fx L lim xla lim fx xla f x L lim xla lim f x L xla f x L f

Chapter: 1 Problem: 1.3.11

Essential Calculus (Available Titles CengageNOW) 1
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). lim xl2 x 2 2x x 2 x 2 f x 2.5, 2.1, 2.05, 2.01, 2.005, 2.001, l 1.9, 1.95, 1.99, 1.995, 1.999

Chapter: 1 Problem: 1.4.11

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim xl2 x 2 x 6 x 2 li

Chapter: 1 Problem: 1.5.11

Essential Calculus (Available Titles CengageNOW) 1
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number . f x x 2x a 1 3 4 11. f

Chapter: 1 Problem: 1.1.11

Essential Calculus (Available Titles CengageNOW) 1
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

Chapter: 1 Problem: 1.6.12

Essential Calculus (Available Titles CengageNOW) 1
(a) Use a graph of to estimate the value of correct to two decimal places. (b) Use a table of values of to estimate the limit to four decimal places.

Chapter: 1 Problem: 1.2.12

Essential Calculus (Available Titles CengageNOW) 1
The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and

Chapter: 1 Problem: 12

Essential Calculus (Available Titles CengageNOW) 1
Describe several ways in which a limit can fail to exist. Illustrate with sketches.

Chapter: 1 Problem: 1.3.12

Essential Calculus (Available Titles CengageNOW) 1
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). lim xl 1 x 2 2x x 2 x 2 1.9 x 0, 0.5, 0.9, 0.95, 0.99, 0.999, lim x 2, 1.5, 1.1, 1.01, 1.001 x 0,

Chapter: 1 Problem: 1.4.12

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. im xl4 x 2 5x 4 x2 3x 4 lim

Chapter: 1 Problem: 1.5.12

Essential Calculus (Available Titles CengageNOW) 1
Use the definition of continuity and the properties of limits to show that the function is continuous on the interval

Chapter: 1 Problem: 1.1.12

Essential Calculus (Available Titles CengageNOW) 1
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.

Chapter: 1 Problem: 1.6.13

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl3 x 2 x 3 fx

Chapter: 1 Problem: 1.2.13

Essential Calculus (Available Titles CengageNOW) 1
At the surface of the ocean, the water pressure is the same as the air pressure above the water, . Below the surface, the water pressure increases by for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure ?

Chapter: 1 Problem: 13

Essential Calculus (Available Titles CengageNOW) 1
State the following Limit Laws. (a) Sum Law (b) Difference Law (c) Constant Multiple Law (d) Product Law (e) Quotient Law (f ) Power Law (g) Root Law

Chapter: 1 Problem: 1.3.13

Essential Calculus (Available Titles CengageNOW) 1
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). lim xl0 sin x x tan x x 1,,,, , 0.5 0.2 0.1 0.05 0.01

Chapter: 1 Problem: 1.4.13

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim xl2 x 2 x 6 x 2 li

Chapter: 1 Problem: 1.5.13

Essential Calculus (Available Titles CengageNOW) 1
Explain why the function is discontinuous at . Sketch the graph of the function. f x 1 x 1 2 a

Chapter: 1 Problem: 1.1.13

Essential Calculus (Available Titles CengageNOW) 1
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.

Chapter: 1 Problem: 1.6.14

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl5 6 x 5 l

Chapter: 1 Problem: 1.2.14

Essential Calculus (Available Titles CengageNOW) 1
The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost as a function of the distance driven assuming that a linear relationship gives a suitable model.

Chapter: 1 Problem: 14

Essential Calculus (Available Titles CengageNOW) 1
What does the Squeeze Theorem say?

Chapter: 1 Problem: 1.3.14

Essential Calculus (Available Titles CengageNOW) 1
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). x 1,,,, , 0.5 0.2 0.1 0.05 0.01 15, 15.5, 15.9, 15.95, 15.99

Chapter: 1 Problem: 1.4.14

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim xl 4 x 2 4x x 2 3x 4 li

Chapter: 1 Problem: 1.5.14

Essential Calculus (Available Titles CengageNOW) 1
Explain why the function is discontinuous at . Sketch the graph of the function. f x  1 x 1 2 if x 1 if x 1 f x

Chapter: 1 Problem: 1.1.14

Essential Calculus (Available Titles CengageNOW) 1
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.

Chapter: 1 Problem: 1.6.15

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim cot x xl1 2 x x 1 2 15.

Chapter: 1 Problem: 1.2.15

Essential Calculus (Available Titles CengageNOW) 1
Suppose the graph of is given. Write equations for the 19. f graphs that are obtained from the graph of as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the -axis. (f ) Reflect about the -axis. (g) Stret

Chapter: 1 Problem: 15

Essential Calculus (Available Titles CengageNOW) 1
(a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval ? What can you say about the graph of such a function?

Chapter: 1 Problem: 1.3.15

Essential Calculus (Available Titles CengageNOW) 1
Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. lim xl0 sx 4 2 x 1

Chapter: 1 Problem: 1.4.15

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim tl3 t 2 9 2t 2 7t 3 15.

Chapter: 1 Problem: 1.5.15

Essential Calculus (Available Titles CengageNOW) 1
Explain why the function is discontinuous at . Sketch the graph of the function. f x  1 x 2 1x if x  1 if x  1 f x

Chapter: 1 Problem: 1.1.15

Essential Calculus (Available Titles CengageNOW) 1
A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.

Chapter: 1 Problem: 1.1.16

Essential Calculus (Available Titles CengageNOW) 1
A jet takes off from an airport and lands an hour later at another airport, 400 miles away. If represents the time in minutes since the plane has left the terminal, let be the horizontal distance traveled and be the altitude of the plane. (a) Sketch a possible graph of . (b) Sketch a possible graph o

Chapter: 1 Problem: 1.6.16

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl lim cot x x

Chapter: 1 Problem: 1.2.16

Essential Calculus (Available Titles CengageNOW) 1
Explain how the following graphs are obtained from the graph of . (a) (b) (c) (d) (e) (f)

Chapter: 1 Problem: 16

Essential Calculus (Available Titles CengageNOW) 1
What does the Intermediate Value Theorem say?

Chapter: 1 Problem: 1.3.16

Essential Calculus (Available Titles CengageNOW) 1
Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. lim xl0 tan 3x tan 5x

Chapter: 1 Problem: 1.4.16

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. im xl1 x 2 4x x 2 3x 4 lim

Chapter: 1 Problem: 1.5.16

Essential Calculus (Available Titles CengageNOW) 1
Explain why the function is discontinuous at . Sketch the graph of the function. f x  x 2 x x 2 1 1 if x 1 if x 1 16.

Chapter: 1 Problem: 1.1.17

Essential Calculus (Available Titles CengageNOW) 1
If , find , , , ,[ fa] fa h 2 fa , 2 f a 1 2f a f2a fx 3x f2 f 2

Chapter: 1 Problem: 1.6.17

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim x l2 sec x lim

Chapter: 1 Problem: 1.2.17

Essential Calculus (Available Titles CengageNOW) 1
The graph of is given. Match each equation with its graph and give reasons for your choices. (a) (b) (c) (d) (e)

Chapter: 1 Problem: 17

Essential Calculus (Available Titles CengageNOW) 1
(a) What does it mean to say that the line is a vertical asymptote of the curve ? Draw curves to illustrate the various possibilities. (b) What does it mean to say that the line is a horizontal asymptote of the curve ? Draw curves to illustrate the various possibilities.

Chapter: 1 Problem: 1.3.17

Essential Calculus (Available Titles CengageNOW) 1
Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. lim xl1 x6 1 x10 1 l

Chapter: 1 Problem: 1.4.17

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim hl0 4 h 2 16 h lim

Chapter: 1 Problem: 1.5.17

Essential Calculus (Available Titles CengageNOW) 1
Explain, using Theorems 4, 5, 6, and 8, why the func- c t , . tion is continuous at every number in its domain. State the domain. 17 cannot be copied

Chapter: 1 Problem: 1.1.18

Essential Calculus (Available Titles CengageNOW) 1
A spherical balloon with radius inches has volume . Find a function that represents the amount of air required to inflate the balloon from a radius of inches to a radius of inches.

Chapter: 1 Problem: 1.6.18

Essential Calculus (Available Titles CengageNOW) 1
Find the limit im xl 3x 5 x 4 l

Chapter: 1 Problem: 1.2.18

Essential Calculus (Available Titles CengageNOW) 1
The graph of is given. Draw the graphs of the following functions. (a) (b) (c) (d) x y 0 1 1 y 1 2 y 2f x f x 3 y f x 4 y f x 4 f @ ! $ % f

Chapter: 1 Problem: 18

Essential Calculus (Available Titles CengageNOW) 1
If f is continuous on and and then there exists a number r such that a  r  1 f r  1, 1 f1 4 f 1 3, limxl2 f4x

Chapter: 1 Problem: 1.3.18

Essential Calculus (Available Titles CengageNOW) 1
Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. lim xl0 9x 5x x l

Chapter: 1 Problem: 1.4.18

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim hl0 s1 h 1 h l

Chapter: 1 Problem: 1.5.18

Essential Calculus (Available Titles CengageNOW) 1
Explain, using Theorems 4, 5, 6, and 8, why the func- c t , . tion is continuous at every number in its domain. State the domain. 17 cannot be copied

Chapter: 1 Problem: 1.1.19

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the difference quotient for the given function. Simplify your answer. fx 4 3x x 2 19. f 3 h f 3 h fx 4

Chapter: 1 Problem: 1.6.19

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl x 3 5x 2x 3 x 2 4 19

Chapter: 1 Problem: 1.2.19

Essential Calculus (Available Titles CengageNOW) 1
The graph of is given. Use it to graph the following functions. (a) (b) (c) (d)

Chapter: 1 Problem: 19

Essential Calculus (Available Titles CengageNOW) 1
Let be a function such that . Then there exists a number such that if , then  f x 6   1 0 

Chapter: 1 Problem: 1.3.19

Essential Calculus (Available Titles CengageNOW) 1
(a) By graphing the function and zooming in toward the point where the graph crosses the y-axis, estimate the value of . (b) Check your answer in part (a) by evaluating for values of x that approach 0.

Chapter: 1 Problem: 1.4.19

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim xl2 x 2 x 3 8 19

Chapter: 1 Problem: 1.5.19

Essential Calculus (Available Titles CengageNOW) 1
Explain, using Theorems 4, 5, 6, and 8, why the func- c t , . tion is continuous at every number in its domain. State the domain. 17 cannot be copied

Chapter: 1 Problem: 1.1.20

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the difference quotient for the given function. Simplify your answer. h f x x 3 f 3 fa h fa h f x

Chapter: 1 Problem: 1.6.20

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim tl t 2 2 t 3 t 2 1 lim

Chapter: 1 Problem: 1.2.20

Essential Calculus (Available Titles CengageNOW) 1
(a) How is the graph of related to the graph of ? Use your answer and Figure 18(a) to sketch the graph of . (b) How is the graph of related to the graph of ? Use your answer and Figure 17(a) to sketch the graph of

Chapter: 1 Problem: 20

Essential Calculus (Available Titles CengageNOW) 1
If for all and exists, then limx l 0 f x  1.

Chapter: 1 Problem: 1.3.20

Essential Calculus (Available Titles CengageNOW) 1
(a) Estimate the value of by graphing the function . State your answer correct to two decimal places. (b) Check your answer in part (a) by evaluating for values of x that approach 0.

Chapter: 1 Problem: 1.4.20

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. im xl1 x 2 2x 1 x 4 1 lim

Chapter: 1 Problem: 1.5.20

Essential Calculus (Available Titles CengageNOW) 1
Explain, using Theorems 4, 5, 6, and 8, why the func- c t , . tion is continuous at every number in its domain. State the domain. 17 cannot be copied

Chapter: 1 Problem: 1.1.21

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the difference quotient for the given function. Simplify your answer. x a f x 1 x ff x fa x a

Chapter: 1 Problem: 1.6.21

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim ul 4u4 5 u2 22u2 1 lim t

Chapter: 1 Problem: 1.2.21

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y x 3 y

Chapter: 1 Problem: 21

Essential Calculus (Available Titles CengageNOW) 1
The graph of is given. (a) Find each limit, or explain why it does not exist. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (b) State the equations of the horizontal asymptotes. (c) State the equations of the vertical asymptotes. (d) At what numbers is discontinuous? Explain.

Chapter: 1 Problem: 1.3.21

Essential Calculus (Available Titles CengageNOW) 1
(a) Evaluate the function for 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of (b) Evaluate for 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.

Chapter: 1 Problem: 1.4.21

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim xl7 sx 2 3 x 7 li

Chapter: 1 Problem: 1.5.21

Essential Calculus (Available Titles CengageNOW) 1
Explain, using Theorems 4, 5, 6, and 8, why the func- c t , . tion is continuous at every number in its domain. State the domain. 17 cannot be copied

Chapter: 1 Problem: 1.1.22

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the difference quotient for the given function. Simplify your answer. f x x 3 x 1 f x fx f 1 x 1 f x x

Chapter: 1 Problem: 1.6.22

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl x 2 s9x 2 1 l

Chapter: 1 Problem: 1.2.22

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y 1 x 2 y

Chapter: 1 Problem: 22

Essential Calculus (Available Titles CengageNOW) 1
Sketch the graph of an example of a function that satisfies all of the following conditions: ,, , , , f is continuous from the right at 3 lim xl3 lim fx 2 xl3 fx lim xl3 lim fx xl lim f x 0 xl f x 2 f 0 x y 1 1 f

Chapter: 1 Problem: 1.3.22

Essential Calculus (Available Titles CengageNOW) 1
(a) Evaluate for , 0.5, 0.1, 0.05, 0.01, and 0.005. (b) Guess the value of . (c) Evaluate for successively smaller values of until you finally reach values for . Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (In Section 3.7 a method for

Chapter: 1 Problem: 1.4.22

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim hl 0 3 h 1 31 h lim

Chapter: 1 Problem: 1.5.22

Essential Calculus (Available Titles CengageNOW) 1
Explain, using Theorems 4, 5, 6, and 8, why the func- c t , . tion is continuous at every number in its domain. State the domain. 17 cannot be copied

Chapter: 1 Problem: 1.1.23

Essential Calculus (Available Titles CengageNOW) 1
Find the domain of the function. f x x 3x 1 fx

Chapter: 1 Problem: 1.6.23

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl (s9x 23. 2 x 3x)

Chapter: 1 Problem: 1.2.23

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y x 1 2 4

Chapter: 1 Problem: 23

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. im t l 0 t 3 tan3 2t

Chapter: 1 Problem: 1.3.23

Essential Calculus (Available Titles CengageNOW) 1
Use the given graph of to find a number such that x  x 2    0.5   0.2 fx

Chapter: 1 Problem: 1.4.23

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim  xl4 1 4 1 x 4 x li

Chapter: 1 Problem: 1.5.23

Essential Calculus (Available Titles CengageNOW) 1
Locate the discontinuities of the function and illustrate by graphing. y y tan sx 1 1 sin x 21

Chapter: 1 Problem: 1.1.24

Essential Calculus (Available Titles CengageNOW) 1
Find the domain of the function. f x 5x 4 x 2 3x 2 f x

Chapter: 1 Problem: 1.6.24

Essential Calculus (Available Titles CengageNOW) 1
Find the limit im xl (sx 2 ax sx2 bx ) lim

Chapter: 1 Problem: 1.2.24

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y x y x 1 2 4x 3 2 y 1

Chapter: 1 Problem: 24

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. im t l 0 t 3 tan3 2t

Chapter: 1 Problem: 1.3.24

Essential Calculus (Available Titles CengageNOW) 1
Use the given graph of to find a number such that  x 1    2 f

Chapter: 1 Problem: 1.4.24

Essential Calculus (Available Titles CengageNOW) 1
Evaluate the limit, if it exists. lim tl 0  1 t 1 t 2 t l

Chapter: 1 Problem: 1.5.24

Essential Calculus (Available Titles CengageNOW) 1
Locate the discontinuities of the function and illustrate by graphing. y tan sx 1

Chapter: 1 Problem: 1.1.25

Essential Calculus (Available Titles CengageNOW) 1
Find the domain of the function. ft st s 3 t f x

Chapter: 1 Problem: 1.6.25

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl cos x

Chapter: 1 Problem: 1.2.25

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y 1 2 cos x y

Chapter: 1 Problem: 25

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. "Find the limit. im t l 0 t 3 tan3 2t "

Chapter: 1 Problem: 1.3.25

Essential Calculus (Available Titles CengageNOW) 1
Use a graph to find a number such that if then  x 2     s4x 1 3   0.5 

Chapter: 1 Problem: 1.4.25

Essential Calculus (Available Titles CengageNOW) 1
(a) Estimate the value of by graphing the function . (b) Make a table of values of for x close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct. f x fx x(s1 3x 1) lim xl0 x s1 3x 1 lim tl

Chapter: 1 Problem: 1.5.25

Essential Calculus (Available Titles CengageNOW) 1
Use continuity to evaluate the limit. im 26. sinx sin x xl4 5 sx s5 x y y

Chapter: 1 Problem: 1.1.26

Essential Calculus (Available Titles CengageNOW) 1
Find the domain of the function. tu su s4 u ft

Chapter: 1 Problem: 1.6.26

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl sin2 x

Chapter: 1 Problem: 1.2.26

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y 4 sin 3x

Chapter: 1 Problem: 26

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. lim xl1 x 2 9 x 2 2x 3 lim

Chapter: 1 Problem: 1.3.26

Essential Calculus (Available Titles CengageNOW) 1
Use a graph to find a number such that if then  x 2 |  0.1  6

Chapter: 1 Problem: 1.4.26

Essential Calculus (Available Titles CengageNOW) 1
missing problem

Chapter: 1 Problem: 1.5.26

Essential Calculus (Available Titles CengageNOW) 1
Use continuity to evaluate the limit. lim xl lim 26. sinx sin x xl4

Chapter: 1 Problem: 1.1.27

Essential Calculus (Available Titles CengageNOW) 1
Find the domain of the function. hx 1 s 4 x 2 5x 27.

Chapter: 1 Problem: 1.6.27

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim x l (x sx )

Chapter: 1 Problem: 1.2.27

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y sin x2 y

Chapter: 1 Problem: 27

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. lim hl0 h 1 3 1 h lim

Chapter: 1 Problem: 1.3.27

Essential Calculus (Available Titles CengageNOW) 1
A machinist is required to manufacture a circular metal disk with area . (a) What radius produces such a disk? (b) If the machinist is allowed an error tolerance of in the area of the disk, how close to the ideal radius in part (a) must the machinist control the radius? (c) In terms of the definition

Chapter: 1 Problem: 1.4.27

Essential Calculus (Available Titles CengageNOW) 1
missing problem

Chapter: 1 Problem: 1.5.27

Essential Calculus (Available Titles CengageNOW) 1
Show that is continuous on . fx  x 2 if x  1 sx if x  1 f

Chapter: 1 Problem: 1.1.28

Essential Calculus (Available Titles CengageNOW) 1
Find the domain and range and sketch the graph of the function hx s4 x 2 hx

Chapter: 1 Problem: 1.6.28

Essential Calculus (Available Titles CengageNOW) 1
Find the limit im xl x 3 2x 3 5 2x2 lim

Chapter: 1 Problem: 1.2.28

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y 1 x 42

Chapter: 1 Problem: 28

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. lim tl2 t 2 4 t 3 8 l

Chapter: 1 Problem: 1.3.28

Essential Calculus (Available Titles CengageNOW) 1
A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given by Tw 0.1w2 2

Chapter: 1 Problem: 1.4.28

Essential Calculus (Available Titles CengageNOW) 1
missing problem

Chapter: 1 Problem: 1.5.28

Essential Calculus (Available Titles CengageNOW) 1
Show that is continuous on . f x  sin x if x  4 cos x if x  4 fx

Chapter: 1 Problem: 1.1.29

Essential Calculus (Available Titles CengageNOW) 1
Find the domain and sketch the graph of the function f x 5 x

Chapter: 1 Problem: 1.6.29

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim xl x 4 x 5 lim

Chapter: 1 Problem: 1.2.29

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y sx 3 4

Chapter: 1 Problem: 29

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. lim rl9 sr r 9 4 li

Chapter: 1 Problem: 1.3.29

Essential Calculus (Available Titles CengageNOW) 1
Prove the statement using the definition of limit and illustrate with a diagram like Figure 15. lim 2 x 3) 2 xl1 2x 3 5 , 

Chapter: 1 Problem: 1.4.29

Essential Calculus (Available Titles CengageNOW) 1
lim xl0 sx 1 sin2 2x 0 lim xl0 x 4 cos 2 x 0. 2x tx x x limxl1 tx 4 x 2 2 limxl4 f x 4x 9 fx x x

Chapter: 1 Problem: 1.5.29

Essential Calculus (Available Titles CengageNOW) 1
Find the numbers at which the function is discontinuous. At which of these points is continuous from the right, from the left, or neither? Sketch the graph of .

Chapter: 1 Problem: 1.1.30

Essential Calculus (Available Titles CengageNOW) 1
Find the domain and sketch the graph of the function Fx 1 2 f x 5 x 3 hx s4

Chapter: 1 Problem: 1.6.30

Essential Calculus (Available Titles CengageNOW) 1
Find the limit lim x l x 2 x 4 lim xl

Chapter: 1 Problem: 1.2.30

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y x 2 y sx 3 4 3 y 1 x

Chapter: 1 Problem: 30

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. lim vl 4 4 v  4 v  lim

Chapter: 1 Problem: 1.3.30

Essential Calculus (Available Titles CengageNOW) 1
Prove the statement using the definition of limit and illustrate with a diagram like Figure 15. lim xl2 ( 1 lim 2 x 3) 2 xl1

Chapter: 1 Problem: 1.4.30

Essential Calculus (Available Titles CengageNOW) 1
missing problem

Chapter: 1 Problem: 1.5.30

Essential Calculus (Available Titles CengageNOW) 1
The gravitational force exerted by the Earth on a unit mass at a distance r from the center of the planet is where M is the mass of the Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?

Chapter: 1 Problem: 1.1.31

Essential Calculus (Available Titles CengageNOW) 1
Find the domain and sketch the graph of the function f t t 2 6t Fx

Chapter: 1 Problem: 1.6.31

Essential Calculus (Available Titles CengageNOW) 1
Find the limit im xl x x 3 x 5 1 x 2 x 4 lim

Chapter: 1 Problem: 1.2.31

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y x 1 1 2 x 2 8x y

Chapter: 1 Problem: 31

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. lim s l16 4 ss s 16 l

Chapter: 1 Problem: 1.3.31

Essential Calculus (Available Titles CengageNOW) 1
Prove the statement using the definition of limit and illustrate with a diagram like Figure 15. lim 7 3x 5 xl3 31. 1 4x 13 lim xl2

Chapter: 1 Problem: 1.4.31

Essential Calculus (Available Titles CengageNOW) 1
missing problem

Chapter: 1 Problem: 1.5.31

Essential Calculus (Available Titles CengageNOW) 1
For what value of the constant is the function continuous on ? fx  cx 2 2x x 3 cx if x  2 if x  2 ,

Chapter: 1 Problem: 1.1.32

Essential Calculus (Available Titles CengageNOW) 1
Find the domain and sketch the graph of the function Ht 4 t 2 2 t f t

Chapter: 1 Problem: 1.6.32

Essential Calculus (Available Titles CengageNOW) 1
(a) Graph the function How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits and (b) By calculating values of , give numerical estimates of the limits in part (a). f x lim xl s2x 2 1 3x 5 lim xl s2x 2 1 3x 5 fx s2x 2 1 3x 5 lim xl (c) Calculate

Chapter: 1 Problem: 1.2.32

Essential Calculus (Available Titles CengageNOW) 1
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations y 1 s 3 y x 1 1 2

Chapter: 1 Problem: 32

Essential Calculus (Available Titles CengageNOW) 1
Find the limit. lim v l2 v 2 2v 8 v 4 16 li

Chapter: 1 Problem: 1.3.32

Essential Calculus (Available Titles CengageNOW) 1
Prove the statement using the definition of limit and illustrate with a diagram like Figure 15. im xl4 lim 7 3x 5 xl3

Chapter: 1 Problem: 1.4.32

Essential Calculus (Available Titles CengageNOW) 1
missing problem

Chapter: 1 Problem: 1.5.32

Essential Calculus (Available Titles CengageNOW) 1
Find the constant that makes continuous on tx  x 2 c 2 cx 20 if x  4 if x  4 172