Solved: (a) Evaluate for , 0.5, 0.1, 0.05, 0.01, and

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?

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 .

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

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

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

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

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?

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.

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.

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

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

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.

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.

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.

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.

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 of . (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity.

If , find , , , ,[ fa] fa h 2 fa , 2 f a 1 2f a f2a fx 3x f2 f 2

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.

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

Evaluate the difference quotient for the given function. Simplify your answer. h f x x 3 f 3 fa h fa h f x

Evaluate the difference quotient for the given function. Simplify your answer. x a f x 1 x ff x fa x a

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

Find the domain of the function. f x x 3x 1 fx

Find the domain of the function. f x 5x 4 x 2 3x 2 f x

Find the domain of the function. ft st s 3 t f x

Find the domain of the function. tu su s4 u ft

Find the domain of the function. hx 1 s 4 x 2 5x 27.

Find the domain and range and sketch the graph of the function hx s4 x 2 hx

Find the domain and sketch the graph of the function f x 5 x

Find the domain and sketch the graph of the function Fx 1 2 f x 5 x 3 hx s4

Find the domain and sketch the graph of the function f t t 2 6t Fx

Find the domain and sketch the graph of the function Ht 4 t 2 2 t f t

Find the domain and sketch the graph of the function tx sx 5 Ht

Find the domain and sketch the graph of the function Fx  2x 1  tx s

Find the domain and sketch the graph of the function G 2 x 3x  x  x 35.

Find the domain and sketch the graph of the function tx  x  x G 2 x

Find the domain and sketch the graph of the function f x  x 2 1 x if x  0 if x  0 tx

Find the domain and sketch the graph of the function f x  3 1 2 x 2x 5 if x 2if x  2 f x

Find the domain and sketch the graph of the function f x  x 2 x 2 if x 1 if x  1 39. f

Find the domain and sketch the graph of the function f x  1 3x 2 7 2x if x 1 if  x   1 if x  1 f x 

Find an expression for the function whose graph is the given curve. The line segment joining the points a 2, 1 4, 6 10

Find an expression for the function whose graph is the given curve. The line segment joining the points 3, 2 6, 3 2, 1 4

Find an expression for the function whose graph is the given curve. The bottom half of the parabola x y 1 2 0 3, 2

Find an expression for the function whose graph is the given curve. The top half of the circle x 1 2 2 y 2 1 x

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

A rectangle has area 16 m . Express the perimeter of the rectangle as a function of the length of one of its sides.

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

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

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

A taxi company charges two dollars for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost (in dollars) of a ride as a function of the distance traveled (in miles) for , and sketch the graph of this function.

In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of $14,000? On $26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I.

The functions in Example 6 and Exercises 50 and 51(a) are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.

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

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

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

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

Determine whether is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f x x x 2 1 f 0

Determine whether is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. fx x 2 x 4 1 f x

Determine whether is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f x x x 1 fx

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

Determine whether is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f x 1 3x 2 x 4 f x

Determine whether is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually f x 1 3x 3 x 5 f x

(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?

What do all members of the family of linear functions have in common? Sketch several members of the family.

What do all members of the family of linear functions have in common? Sketch several members of the family.

Find expressions for the quadratic functions whose graphs are shown.

Find an expression for a cubic function if and f1 f 0 f 2 0 f f1 6 y

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 average global surface temperature in 2100.

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?

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 spaces rented cant be negative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent?

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?

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 represent?

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 function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.

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 what does it represent? (c) What is the y-intercept of the graph and what does it represent?

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 ?

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. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the y-intercept represent? (e) Why does a linear function give a suitable model in this situation?

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) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3.

Explain how the following graphs are obtained from the graph of . (a) (b) (c) (d) (e) (f)

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

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

The graph of is given. Use it to graph the following functions. (a) (b) (c) (d)

(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

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

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

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

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

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

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

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

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

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

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

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

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

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  2 x 1 y

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 4 tanx  4 y  2 x

Find , , , and and state their domains. tx 3x f x x 2 1 3 2x 2 f t f

Find , , , and and state their domains. fx s1 x tx s1 x tx 3x f

Find the functions (a) , (b) , (c) , and (d) and their domains. f x x tx 2x 1 2 1 t t f

Find the functions (a) , (b) , (c) , and (d) and their domains. fx 1 x tx 1x 3 f x x

Find the functions (a) , (b) , (c) , and (d) and their domains. f x sin x tx 1 sx fx 1

Find the functions (a) , (b) , (c) , and (d) and their domains. tx 5x f x 1 3x 2 3x 2 f x sin

Find the functions (a) , (b) , (c) , and (d) and their domains. x 2 f x x 1 x tx x 1 x 2

Find the functions (a) , (b) , (c) , and (d) and their domains. tx x fx s2x 3 2 1 tx x

Find (cannot be copied)

Find (cannot be copied)

Express the function in the form f t. f Fx x Fx sin(sx ) 2 110 f t.

Express the function in the form f t. f Fx sin(sx ) 2

Express the function in the form f t. f ut scos t 48.

Express the function in the form f t. f ut tan t 1 tan t ut

Express the function in the form f t h. ut H  x 1 3x 2 f

Express the function in the form f t h. ut Hx s 8 2  x H  x 1

Express the function in the form f t h. ut Hx sec4 (sx ) Hx

Use the table to evaluate each expression. (a) (b) (c) (d) (e) (f )

Use the given graphs of and to evaluate each expression, or explain why it is undefined. (a) (b) (c) (d) (e) (f )

A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of . (a) Express the radius of the balloon as a function of the time (in seconds). (b) If is the volume of the balloon as a function of the radius, find and interpret it.

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of . (a) Express the radius of this circle as a function of the time (in seconds). (b) If is the area of this circle as a function of the radius, find and interpret it.

An airplane is flying at a speed of at an altitude of one mile and passes directly over a radar station at time . (a) Express the horizontal distance (in miles) that the plane has flown as a function of . (b) Express the distance between the plane and the radar station as a function of . (c) Use composition to express as a function o t.

The Heaviside function H is defined by It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage in a circuit if the switch is turned on at time and 120 volts are applied instantaneously to the circuit. Write a formula for in terms of . (c) Sketch the graph of the voltage in a circuit if the switch is turned on at time seconds and 240 volts are applied instantaneously to the circuit. Write a formula for in terms of . (Note that starting at corresponds to a translation.)

The Heaviside function defined in Exercise 57 can also be used to define the ramp function , which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function . (b) Sketch the graph of the voltage in a circuit if the switch is turned on at time and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for in terms of for . (c) Sketch the graph of the voltage in a circuit if the switch is turned on at time seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for in terms of for .

Let and be linear functions with equations and . Is also a linear function? If so, what is the slope of its graph?

If you invest dollars at 4% interest compounded annually, then the amount of the investment after one year is . Find , , and . What do these compositions represent? Find a formula for the composition of copies of .

(a) If and , find a function such that . (Think about what operations you would have to perform on the formula for to end up with the formula for .) (b) If and , find a function such that t f t h hx

If and , find a function such that .

(a) Suppose and are even functions. What can you say about and ? (b) What if and are both odd?

Suppose is even and is odd. What can you say about ?

Suppose t is an even function and let . Is h always an even function?

Suppose t is an odd function and let . Is h always an odd function? What if is odd? What if is even?

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 2.

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 .

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)

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)

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)

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

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

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

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

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

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

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,

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

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

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

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

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

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

(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.

(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.

(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.

(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 evaluating the limit will be explained.) ; (d) Graph the function h in the viewing rectangle by . Then zoom in toward the point where the graph crosses the y-axis to estimate the limit of as x approaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part (c).

Use the given graph of to find a number such that x  x 2    0.5   0.2 fx

Use the given graph of to find a number such that  x 1    2 f

Use a graph to find a number such that if then  x 2     s4x 1 3   0.5 

Use a graph to find a number such that if then  x 2 |  0.1  6

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 of , what is ? What is ? What is ? What is ? What value of is given? What is the corresponding value of ?

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 2.155w 20 where is the temperature in degrees Celsius and is the 39. 2 0 power input in watts. (a) How much power is needed to maintain the temperature at ? (b) If the temperature is allowed to vary from by up to , what range of wattage is allowed for the input power? (c) In terms of the definition of , what is ? What is ? What is ? What is ? What value of is given? What is the corresponding value of ? 2

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 , 

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

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

Prove the statement using the definition of limit and illustrate with a diagram like Figure 15. im xl4 lim 7 3x 5 xl3

Prove the statement using the definition of limit. lim x l3 x 5 3 5

Prove the statement using the definition of limit. im x l 6  x 4 3 9 2 l

Prove the statement using the definition of limit. lim 7 x l5 4 3x 5  7 lim

Prove the statement using the definition of limit. lim x l3 x 2 x 12 x 3 lim 7 x l5

Prove the statement using the definition of limit. im c c x l a x a l

Prove the statement using the definition of limit. lim x l a lim c c x

Prove the statement using the definition of limit. lim 3 0 x l 0 x w

Prove the statement using the definition of limit. lim x l 0 x lim 3 0

Prove the statement using the definition of limit. lim 9 x 0 x l 0  x  0 li

Prove the statement using the definition of limit. lim xl9 s 4 lim 9 x 0 x l

Prove the statement using the definition of limit.  x 2 9   x 3  x 3  lim x l Show that if , then . If you let be the smaller of the numbers 1 and , show that this works.]

Prove the statement using the definition of limit.  lim x 3   1 x l3 x x 2 x 4 8    lim x 3   1 x l3 x  x 4  

(a) For the limit , use a graph to find a value of that corresponds to (b) By using a computer algebra system to solve the cubic equation , find the largest possible value of that works for any given . (c) Put in your answer to part (b) and compare with your answer to part (a).

If is the Heaviside function defined in Example 6, prove, using Definition 4, that does not exist. [Hint: Use an indirect proof as follows. Suppose that the limit is . Take in the definition of a limit and try to arrive at a contradiction.]

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 . sin 7x 7 si

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

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). cannot be copied

(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.

Evaluate the limit, if it exists. lim xl2 x 2 x 6 x 2 li

Evaluate the limit, if it exists. im xl4 x 2 5x 4 x2 3x 4 lim

Evaluate the limit, if it exists. lim xl2 x 2 x 6 x 2 li

Evaluate the limit, if it exists. lim xl 4 x 2 4x x 2 3x 4 li

Evaluate the limit, if it exists. lim tl3 t 2 9 2t 2 7t 3 15.

Evaluate the limit, if it exists. im xl1 x 2 4x x 2 3x 4 lim

Evaluate the limit, if it exists. lim hl0 4 h 2 16 h lim

Evaluate the limit, if it exists. lim hl0 s1 h 1 h l

Evaluate the limit, if it exists. lim xl2 x 2 x 3 8 19

Evaluate the limit, if it exists. im xl1 x 2 2x 1 x 4 1 lim

Evaluate the limit, if it exists. lim xl7 sx 2 3 x 7 li

Evaluate the limit, if it exists. lim hl 0 3 h 1 31 h lim

Evaluate the limit, if it exists. lim  xl4 1 4 1 x 4 x li

Evaluate the limit, if it exists. lim tl 0  1 t 1 t 2 t l

(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

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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

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x 1  t lim xl1 lim tx xl1 lim tx xl1 tx lim xl0 lim tx xl1 lim tx xl1 tx tx  x 1 x 2 x 1 if x 1 if 1  x  1 if x  1 lim xl0  1 x 1  x  lim  xl0 1 x 1  x   lim xl6 2x 12  x 6  lim xl3 33. (2x  x 3 ) lim (b) Does exist? (c) Sketch the graph of F

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(a) If the symbol denotes the greatest integer function defined in Example 8, evaluate (i) (ii) (iii) (b) If n is an integer, evaluate (i) (ii) (c) For what values of does exist?

Let . (a) Sketch the graph of (b) If is an integer, evaluate (i) (ii) (c) For what values of does exist?

If , show that exists but is not equal to .

In the theory of relativity, the Lorentz contraction formula expresses the length L of an object as a function of its velocity with respect to an observer, where is the length of the object at rest and c is the speed of light. Find and interpret the result. Why is a left-hand limit necessary?

Find the limit. lim xl0 sin 3x x

Find the limit. lim xl0 sin 4x sin 6x

Find the limit. im tl0 tan 6t sin 2t

Find the limit. lim tl0 sin2 3t t 2

Find the limit. im x cot x l0 sin   tan 

Find the limit. lim xl0 lim x cot x

If is a polynomial, show that .

If r is a rational function, use Exercise 49 to show that limxla rx ra for every number a in the domain of r. p li

To prove that sine has the Direct Substitution Property we need to show that for every real number a. If we let , then and . So an equivalent statement is that Use (1) to show that this is true

Prove that cosine has the Direct Substitution Property.

Show by means of an example that may exist even though neither nor exists.

Show by means of an example that may exist even though neither nor exists.

Is there a number a such that exists? If so, find the value of a and the value of the limit.

The figure shows a fixed circle with equation and a shrinking circle with radius and center the origin. P is the point , Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the -axis. What happens to R as shrinks, that is, as ?

Write an equation that expresses the fact that a function is continuous at the number 4.

If is continuous on , what can you say about its graph?

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.

From the graph of , state the intervals on which is continuous.

Sketch the graph of a function that is continuous everywhere except at x 3 and is continuous from the left at 3. 6

Sketch the graph of a function that has a jump discontinuity at and a removable discontinuity at , but is continuous elsewhere.

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 significance to someone who parks in the lot.

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 York City (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time

If and are continuous functions with and limx l 3 2 f x tx 4 t3 f t f 3

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

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

Use the definition of continuity and the properties of limits to show that the function is continuous on the interval

Explain why the function is discontinuous at . Sketch the graph of the function. f x 1 x 1 2 a

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

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

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.

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

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

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

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

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

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

Locate the discontinuities of the function and illustrate by graphing. y y tan sx 1 1 sin x 21

Locate the discontinuities of the function and illustrate by graphing. y tan sx 1

Use continuity to evaluate the limit. im 26. sinx sin x xl4 5 sx s5 x y y

Use continuity to evaluate the limit. lim xl lim 26. sinx sin x xl4

Show that is continuous on . fx  x 2 if x  1 sx if x  1 f

Show that is continuous on . f x  sin x if x  4 cos x if x  4 fx

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 .

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?

For what value of the constant is the function continuous on ? fx  cx 2 2x x 3 cx if x  2 if x  2 ,

Find the constant that makes continuous on tx  x 2 c 2 cx 20 if x  4 if x  4 172

Which of the following functions has a removable discontinuity at ? If the discontinuity is removable, find a function that agrees with for and is continuous at . a. f x a 2 x 2 2x 8 x 2 t b. fx a 7 x 7  x c. f x a 4 x 3 64 x 4 d. fx a 9 3 s

Suppose that a function is continuous on [0, 1] except at 0.25 and that and . Let N 2. Sketch two possible graphs of , one showing that might not satisfy the conclusion of the Intermediate Value Theorem and one showing that might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesnt satisfy the hypothesis).

If , show that there is a number such that .

Use the Intermediate Value Theorem to prove that there is a positive number such that . (This proves the existence of the number .)

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. x 1, 2 x

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. s 0, 1 3

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. cos x x 0, 1 tan

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. tan x 2x 0, 1.4 s

(a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. cos x x 2

(a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. x 5 x cos x x 2 2x 3 0 3 co

(a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. x 5 x 2 4 0 x

(a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. sx 5 1 x 3 x 5

Is there a number that is exactly 1 more than its cube?

(a) Show that the absolute value function is continuous everywhere. (b) Prove that if is a continuous function on an interval, then so is . (c) Is the converse of the statement in part (b) also true? In other words, if is continuous, does it follow that is continuous? If so, prove it. If not, find a counterexample.

A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving at 7:00 PM. The following morning, he starts at 7:00 AM at the top and takes the same path back, arriving at the monastery at 7:00 PM. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.

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

For the function t whose graph is given, state the following. (a) (b) (c) (d) (e) (f ) T

Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Sketch the graph of an example of a function that satisfies all of the given conditions. cannot be copied

Guess the value of the limit by evaluating the function for and . Then use a graph of to support your guess.

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

Use a graph to estimate all the vertical and horizontal asymptotes of the curve y x 3 x 3 2x 1 2.

(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.

Find the limit lim xl3 x 2 x 3 fx

Find the limit lim xl5 6 x 5 l

Find the limit lim cot x xl1 2 x x 1 2 15.

Find the limit lim xl lim cot x x

Find the limit lim x l2 sec x lim

Find the limit im xl 3x 5 x 4 l

Find the limit lim xl x 3 5x 2x 3 x 2 4 19

Find the limit lim tl t 2 2 t 3 t 2 1 lim

Find the limit lim ul  4u4 5 u2 22u2 1 lim t

Find the limit lim xl x 2 s9x 2 1 l

Find the limit lim xl (s9x 23. 2 x 3x)

Find the limit im xl (sx 2 ax sx2 bx ) lim

Find the limit lim xl cos x

Find the limit lim xl sin2 x

Find the limit lim x l  (x sx )

Find the limit im xl x 3 2x 3 5 2x2 lim

Find the limit lim xl x 4 x 5 lim

Find the limit lim x l  x 2 x 4 lim xl

Find the limit im xl x x 3 x 5 1 x 2 x 4 lim

(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 the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check your calculation for the second limit.]

Find the horizontal and vertical asymptotes of each curve. Check your work by graphing the curve and estimating the asymptotes. cannot be copied

Find the horizontal and vertical asymptotes of each curve. Check your work by graphing the curve and estimating the asymptotes. cannot be copied

(a) Estimate the value of by graphing the function . (b) Use a table of values of to guess the value of the limit. (c) Prove that your guess is correct.

(a) Use a graph of to estimate the value of to one decimal place. (b) Use a table of values of to estimate the limit to four decimal places. (c) Find the exact value of the limit.

Estimate the horizontal asymptote of the function by graphing for . Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

Find a formula for a function that has vertical asymptotes and and horizontal asymptote .

Find a formula for a function that satisfies the following conditions: fx 0 f lim f 2 0 xl0 lim xl3 lim fx  xl3 f x  lim f 2

By the end behavior of a function we mean the behavior of its values as and as . (a) Describe and compare the end behavior of the functions by graphing both functions in the viewing rectangles 2, 2 by and by . 2, 2 10, 10 10,000, 10,000 Qx 3x 5 Px 3x 5 5x 3 2x x l x l (b) Two functions are said to have the same end behavior if their ratio approaches 1 as . Show that P and Q have the same end behavior.

Let and be polynomials. Find if the degree of is (a) less than the degree of and (b) greater than the degree of .

Make a rough sketch of the curve ( an integer) for the following five cases: (i) (ii) , odd (iii) , even (iv) , odd (v) , even Then use these sketches to find the following limits. (a) (b) (c) (d)

Find if, for all , 4x 1 x  fx  4x 2 3x x 2 43.

In the theory of relativity, the mass of a particle with velocity is where is the mass of the particle at rest and is the speed of light. What happens as ?

(a) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 Lmin. Show that the concentration of salt minutes later (in grams per liter) is (b) What happens to the concentration as ?

(a) Show that . ; (b) By graphing the function in part (a) and the line on a common screen, find a number such that when What if 1.9 is replaced by 1.99? x  N 4x 2 5x 2x 2 1  1.9

How close to do we have to take so that x 3 4  10,000 3

Prove, using Definition 6, that im xl3 1 x 3 4  1

Prove that . im xl1 5 x 1 49. 3

For the limit illustrate Definition 7 by finding values of that correspond to and .

Use a graph to find a number such that  whenever x  N

For the limit illustrate Definition 8 by finding a value of that corresponds to .

(a) How large do we have to take so that ? (b) Taking in (5), we have the statement Prove this directly using Definition 7.

Prove, using Definition 8, that .

Prove that and if these limits exist. lim x l fx lim tl0 f1t lim x l fx lim tl0 f 1t lim x l

(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?

Discuss four ways of representing a function. Illustrate your discussion with examples.

(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?

What is a mathematical model?

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

Sketch by hand, on the same axes, the graphs of the following functions. (a) (b) (c) (d)

Sketch by hand, on the same axes, the graphs of the following functions. (a) (b) (c) (d)

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

How is the composite function defined? What is its domain?

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) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. (i) Stretch horizontally by a factor of 2. ( j) Shrink horizontally by a factor of 2.

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

Describe several ways in which a limit can fail to exist. Illustrate with sketches.

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

What does the Squeeze Theorem say?

(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?

What does the Intermediate Value Theorem say?

(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.

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

Let be a function such that . Then there exists a number such that if , then  f x 6   1 0 

If for all and exists, then limx l 0 f x  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.

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

Find the limit. im t l 0 t 3 tan3 2t

Find the limit. im t l 0 t 3 tan3 2t

Find the limit. "Find the limit. im t l 0 t 3 tan3 2t "

Find the limit. lim xl1 x 2 9 x 2 2x 3 lim

Find the limit. lim hl0 h 1 3 1 h lim

Find the limit. lim tl2 t 2 4 t 3 8 l

Find the limit. lim rl9 sr r 9 4 li

Find the limit. lim vl 4 4 v  4 v  lim

Find the limit. lim s l16 4 ss s 16 l

Find the limit. lim v l2 v 2 2v 8 v 4 16 li

Find the limit. lim x l  1 2x x 2 1 x 2x 2 lim

Find the limit. lim xl 1 2x 2 x 4 5 x 3x 4 lim

Find the limit. lim xl (sx 2 4x 1 x) l

Find the limit. lim xl1  1 x 1 1 x 2 3x 2  lim

Find the limit. lim xl0 cot 2x csc x

Find the limit. im t l 0 t 3 tan3 2t

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered. y cos2 x x 2

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered. y sx 2 x 1 sx 2 x y c

If for , find