introduction to calculus

791

An Introduction to Calculus: Limits, Derivatives, and Integrals

Windmills have long been used to pump water from wells,grind grain, and saw wood. They are more recently beingused to produce electricity. The propeller radius of thesewindmills range from one to one hundred meters, and thepower output ranges from a hundred watts to a thousandkilowatts. See page 800 in Section 10.1 for some more information and a question and answer about windmills.

10.1 Limits and Motion:The Tangent Problem

10.2 Limits and Motion: The Area Problem

10.3 More on Limits

10.4 NumericalDerivatives and Integrals

C H A P T E R 1 0

5144_Demana_Ch10pp791-838 1/13/06 3:18 PM Page 791

Chapter 10 OverviewBy the beginning of the 17th century, algebra and geometry had developed to the pointwhere physical behavior could be modeled both algebraically and graphically, eachtype of representation providing deeper insights into the other. New discoveries aboutthe solar system had opened up fascinating questions about gravity and its effects onplanetary motion, so that finding the mathematical key to studying motion became thescientific quest of the day. The analytic geometry of Ren Descartes (15961650) putthe final pieces into place, setting the stage for Isaac Newton (16421727) and GottfriedLeibniz (16461716) to stand on the shoulders of giants and see beyond the algebra-ic boundaries that had limited their predecessors. With geometry showing them the way,they created the new form of algebra that would come to be known as the calculus.

In this chapter we will look at the two central problems of motion much as Newton andLeibniz did, connecting them to geometric problems involving tangent lines and areas.We will see how the obvious geometric solutions to both problems led to algebraicdilemmas, and how the algebraic dilemmas led to the discovery of calculus. The lan-guage of limits, which we have used in this book to describe asymptotes, end behavior,and continuity, will serve us well as we make this transition.

792 CHAPTER 10 An Introduction to Calculus: Limits, Derivatives, and Integrals

10.1Limits and Motion: The Tangent Problem

Average Velocityis the change in position divided by the change in time, as in the

following familiar-looking example.

EXAMPLE 1 Computing Average VelocityAn automobile travels 120 miles in 2 hours and 30 minutes. What is its average veloc-ity over the entire 212 hour time interval?SOLUTION The average velocity is the change in position (120 miles) divided bythe change in time (2.5 hours). If we denote position by s and time by t, we have

vave

s

t

122.50

hm

o

iu

lers

s 48 miles per hour.

Notice that the average velocity does not tell us how fast the automobile is traveling atany given moment during the time interval. It could have gone at a constant 48 mph allthe way, or it could have sped up, slowed down, or even stopped momentarily severaltimes during the trip. Scientists like Galileo Galilei (15641642), who studied motionprior to Newton and Leibniz, were looking for formulas that would give velocity as afunction of time, that is, formulas that would give instantaneous values of v for indi-vidual values of t. The step from average to instantaneous sounds simple enough, butthere were complications, as we shall see.

Now try Exercise 1.

Average velocity

BIBLIOGRAPHY

For students: The Hitchhikers Guide toCalculus, Michael Spivak, MathematicalAssociation of America, 1995.How to Ace Calculus, Colin Adams, JoelHass, Abigail Thompson, MathematicalAssociation of America, 1999.Stories about Maxima and Minima,V. M. Tikhomirov (translated by AbeShenitzer), National Council of Teachersof Mathematics, 1990.For teachers: Calculus, Third Edition,Finney, Demana, Waits, Kennedy, Addison-Wesley, 2003.New Directions for Equity in MathematicsEducation, Walter G. Secada, ElizabethFennema, Lisa Byrd Adajian (Eds.) NationalCouncil of Teachers of Mathematics, 1994.

What youll learn about Average Velocity

Instantaneous Velocity

Limits Revisited

The Connection to TangentLines

The Derivative

. . . and whyThe derivative allows us to ana-lyze rates of change, which arefundamental to understandingphysics, economics, engineer-ing, and even history.


Instantaneous VelocityGalileo experimented with gravity by rolling a ball down an inclined plane and record-ing its approximate velocity as a function of elapsed time. Here is what he might haveasked himself when he began his experiments:

SECTION 10.1 Limits and Motion: The Tangent Problem 793

A Velocity Question

A ball rolls a distance of 16 feet in 4 seconds. What is the instantaneous velocityof the ball at a moment of time 3 seconds after it starts to roll?

You might want to visualize the ball being frozen at that moment, and then try to deter-mine its velocity. Well, then the ball would have zero velocity, because it is frozen! Thisapproach seems foolish, since, of course, the ball is moving.

Is this a trick question? On the contrary, it is actually quite profoundit is exactly thequestion that Galileo (among many others) was trying to answer. Notice how easy it isto find the average velocity:

vave

s

t 4

1s

6ec

fo

e

n

e

dts

4 feet per second.

Now, notice how inadequate our algebra becomes when we try to apply the same for-mula to instantaneous velocity:

vave

s

t 0 s

0ec

feo

e

n

tds,

which involves division by 0and is therefore undefined!

So Galileo did the best he could by making t as small as experimentally possible,measuring the small values of s, and then finding the quotients. It only approximatedthe instantaneous velocity, but finding the exact value appeared to be algebraically outof the question, since division by zero was impossible.

Limits RevisitedNewton invented fluxions and Leibniz invented differentials to explain instantaneousrates of change without resorting to zero denominators. Both involved mysterious quanti-ties that could be infinitesimally small without really being zero. (Their 17th-century col-league Bishop Berkeley called them ghosts of departed quantities and dismissed themas nonsense.) Though not well understood by many, the strange quantities modeled thebehavior of moving bodies so effectively that most scientists were willing to accept themon faith until a better explanation could be developed. That development, which tookabout a hundred years, led to our modern understanding of limits.

Since you are already familiar with limit notation, we can show you how this workswith a simple example.

OBJECTIVE

Students will be able to calculate instanta-neous velocities and derivatives using limits.MOTIVATE

Ask students whether or not it makessense to try to find the slope of a parabola.

A DISCLAIMER

Readers who know a little calculus will recognize that the ball in the VelocityQuestion really does have a nonzero instantaneous velocity after 3 seconds (as we will eventually show). The point of the discussion is to show how difficult it is to demonstrate that fact at a singleinstant, since both time and the position of the ball appear not to change.


EXAMPLE 2 Using Limits to Avoid Zero DivisionA ball rolls down a ramp so that its distance s from the top of the ramp after t secondsis exactly t2 feet. What is its instantaneous velocity after 3 seconds?

SOLUTION We might try to answer this question by computing average velocityover smaller and smaller time intervals.

On the interval 3, 3.1:

s

t

33.1.12

332

00.6.11

6.1 feet per second.

On the interval 3, 3.05:

s

t

33.0.0552

332

0.03.

00255

6.05 feet per second.

Continuing this process we would eventually conclude that the must be 6 feet per second.

However, we can see directly what is happening to the quotient by treating it as a limitof the average velocity on the interval 3, t as t approaches 3:

limt3

s

t lim

t3t2

t

33

2

limt3

t

t3

t3 3

Factor the numerator.

limt3

t 3 tt

33

limt3

t 3 Since t 3, tt

33 1

6

Notice that t is not equal to 3 but is approaching 3 as a limit, which allows us to makethe crucial cancellation in the second to last line of Example 2. If t were actually equalto 3, the algebra above would lead to the incorrect conclusion that 00 6. The differ-ence between equaling 3 and approaching 3 as a limit is a subtle one, but it makes allthe difference algebraically.

It is not easy to formulate a rigorous algebraic definition of a limit (which is whyNewton and Leibniz never really did). We have used an intuitive approach to limits sofar in this book and will continue to do so, deferring the rigorous definitions to your cal-culus course. For now, we will use the following informal definition.

Now try Exercise 3.

instantaneous velocity


DEFINITION (INFORMAL) Limit at aWhen we write lim

xyaf x L, we mean that f (x) gets arbitrarily close to L as x

gets arbitrarily close (but not equal) to a.

ARBITRARILY CLOSE

This definition is useless for mathemati-cal proofs until one defines arbitrarilyclose, but if you have a sense of how itapplies to the solution in Example 2above, then you are ready to use limitsto study motion problems.


The Connection to Tangent LinesWhat Galileo discovered by rolling balls down ramps was that the distance traveled wasproportional to the square of the elapsed time. For simplicity, let us suppose that theramp was tilted just enough so that the relation between s, the distance from the top ofthe ramp, and t, the elapsed time, was given (as in Example 2) by

s t2.

Graphing s as a function of t 0 gives the right half of a parabola (Figure 10.1).


Exploration 1 suggests an important general fact: If a, sa and b, sb are two pointson a distance-time graph, then the average velocity over the time interval a, b can bethought of as the slope of the line connecting the two points. In fact, we designate bothquantities with the same symbol: st.

Galileo knew this. He also knew that he wanted to find instantaneous velocity by lettingthe two points become one, resulting in st 00, an algebraic impossibility. Thepicture, however, told a different story geometrically. If, for example, we were to con-nect pairs of points closer and closer to 1, 1, our secant lines would look more andmore like a line that is tangent to the curve at 1, 1 (Figure 10.2).It seemed obvious to Galileo and the other scientists of his time that the slope of the tan-gent line was the long-sought-after answer to the quest for instantaneous velocity. Theycould see it, but how could they compute it without dividing by zero? That was the tan-gent line problem, eventually solved for general functions by Newton and Leibniz inslightly different ways. We will solve it with limits as illustrated in Example 3.

EXPLORATION 1 Seeing Average Velocity

Copy Figure 10.1 on a piece of paper and connect the points 1, 1 and 2, 4with a straight line. (This is called a secant line because it connects two pointson the curve.)

1. Find the slope of the line. 3

2. Find the average velocity of the ball over the time interval 1, 2. 3 ft/sec

3. What is the relationship between the numbers that answer questions 1 and 2? They are the same.

4. In general, how could you represent the average velocity of the ball overthe time interval a, b geometrically?

FIGURE 10.1 The graph of s t2shows the distance s traveled by a ball rolling down a ramp as a functionof the elapsed time t.

0

1234

1 2 3t

s

FIGURE 10.2 A line tangent to thegraph of s t2 at the point 1, 1. Theslope of this line appears to be theinstantaneous velocity at t 1, eventhough st 00 . The geometrysucceeds where the algebra fails!

0

1234

1 2 3t

s

(1, 1)

EXPLORATION EXTENSIONS

Repeat steps 13 using the points (0, 0)and (2, 4).


EXAMPLE 3 Finding the Slope of a Tangent LineUse limits to find the slope of the tangent line to the graph of s t2 at the point 1, 1(Figure 10.2).SOLUTION This will look a lot like the solution to Example 2.

limt1

s

t lim

t1t2

t

11

2

limt1

t

t1

t1 1

Factor the numerator.

limt1

t 1 tt

11

limt1

t 1 Since t 1, tt

11 1

2

If you compare Example 3 to Example 2 it should be apparent that a method for solv-ing the tangent line problem can be used to solve the instantaneous velocity problem,and vice versa. They are geometric and algebraic versions of the same problem!

The DerivativeVelocity, the rate of change of position with respect to time, is only one application ofthe general concept of rate of change. If y f x is any function, we can speak ofhow y changes as x changes.

Now try Exercise 17(a).


Using limits, we can proceed to a definition of the instantaneous rate of change of ywith respect to x at the point where x a. This instantaneous rate of change is calledthe derivative.

THE TANGENT LINE PROBLEM

Although we have focused on Galileoswork with motion problems in order tofollow a coherent story, it was Pierre deFermat (16011665) who first developed amethod of tangents for general curves,recognizing its usefulness for finding relative maxima and minima. Fermat isbest remembered for his work in numbertheory, particularly for Fermats LastTheorem, which states that there are nopositive integers x, y, and z that satisfythe equation xn yn zn if n is an inte-ger greater than 2. Fermat wrote in themargin of a textbook, I have a truly marvelous proof that this margin is toonarrow to contain, but if he had one, heapparently never wrote it down.Although mathematicians tried for over330 years to prove (or disprove) FermatsLast Theorem, nobody succeeded untilAndrew Wiles of Princeton Universityfinally proved it in 1994.

NOTES ON EXAMPLES

To illustrate the local linearity in Example 3, have students graph the function y x2 using a decimal windowon a graphing calculator and zoom inuntil the graph appears linear.

DEFINITION Average Rate of Change

If y f x, then the of y with respect to x on the intervala, b is

yx

f b

b

a

f a.

Geometrically, this is the slope of the through a, f a and b, f b.secant line

average rate of change


A more computationally useful formula for the derivative is obtained by letting x a hand looking at the limit as h approaches 0 (equivalent to letting x approach a).


DEFINITION Derivative at a Point

The , denoted by f a and read f prime ofa is

f a limxa

f x

x

a

f a,

provided the limit exists.

Geometrically, this is the slope of the through a, f a.tangent line

derivative of the function f at x a

The fact that the derivative of a function at a point can be viewed geometrically as theslope of the line tangent to the curve y f x at that point provides us with some insightas to how a derivative might fail to exist. Unless a function has a well-defined slopewhen you zoom in on it at a, the derivative at a will not exist. For example, Figure 10.3shows three cases for which f 0 exists but f 0 does not.

f x x has a graph with f x 3 x has a graph with a verticalno definable slope at x 0. tangent line (no slope) at x 0.

f x has a graph with no definable

slope at x 0.

FIGURE 10.3 Three examples of functions defined at x 0 but not differentiable at x 0.

x 1 for x 01 for x 0

[4.7, 4.7] by [3.1, 3.1](c)

[4.7, 4.7] by [3.1, 3.1](b)

[4.7, 4.7] by [3.1, 3.1](a)

DEFINITION Derivative at a Point (easier for computing)

The , denoted by f a and read f prime ofa is

f a limh0

f a h

h f a,

provided the limit exists.

derivative of the function f at x a

DIFFERENTIABILITY

We say a function is differentiable at aif f a exists, because we can find thelimit of the quotient of differences.


EXAMPLE 4 Finding a Derivative at a PointFind f 4 if f x 2x2 3.SOLUTION

f 4 limh0

f 4 h

h f 4

limh0

limh0

limh0

16h

h2h2

limh0

16 2h hh 1, since h 0.

16

The derivative can also be thought of as a function of x. Its domain consists of all val-ues in the domain of f for which f is differentiable. The function f can be defined byadapting the second definition above.

Now try Exercise 23.

216 8h h2 32h

24 h2 3 2 42 3h


To emphasize the connection with slope yx, Leibniz used the notation dydx forthe derivative. (The dy and dx were his ghosts of departed quantities.) This

has several advantages over the prime notation, as you will learnwhen you study calculus. We will use both notations in our examples and exercises.

EXAMPLE 5 Finding the Derivative of a Function(a) Find f x if f x x2.(b) Find dd

yx if y 1

x.

Leibniz notation

DEFINITION Derivative

If y f x, then the , is the functionf whose value at x is

f x limh0

f x h

h f x,

for all values of x where the limit exists.

derivative of the function f with respect to xALERT

Some students may have trouble seeingthe distinction between the derivative at apoint and the derivative of a function. Thederivative at a point is a number, whilethe derivative of a function is a function.(This function can be evaluated at a givenvalue of x to find the derivative at apoint.)

continued


SOLUTION

(a) f x limh0

f x h

h f x

limh0

x h

h2 x2

limh0

limh0

2xh

h h2

limh0

2x h Since h 0, hh 1.

2x

So f x 2x. (b)

ddyx lim

h0f x h

h f x

limh0

limh0

limh0

xx

hh

1h

limh0

xx

1h

x

12

So ddyx

x

12.


x

x

xx

hh

h

x

1h

1x

h

x2 2xh h2 x2h


FOLLOW-UP

Ask students to explain the distinctionbetween an average velocity and aninstantaneous velocity.ASSIGNMENT GUIDE

Ex. 354, multiples of 3COOPERATIVE LEARNING

Group Activity: Ex. 58NOTES ON EXERCISES

Ex. 1516, 3334, and 5556 are applica-tion problems related to average or instan-taneous velocity.Ex. 4550 provide practice with standard-ized tests.Ex. 5758 require students to use thegraph of a function to create the graph ofits derivative, or vice versa.ONGOING ASSESSMENT

Self-Assessment: Ex. 1, 3, 17, 23, 29Embedded Assessment: Ex. 3538,4344, 5556



CHAPTER OPENER PROBLEM (from page 791)

PROBLEM: For an efficient windmill, the power generated in watts is given bythe equation

P kr2v3

where r is the radius of the propeller in meters, v is the wind velocity in meters persecond, and k is a constant with units of kg/m3. The exact value of k depends onvarious characteristics of the windmill.

Suppose a windmill has a propeller with radius 5 meters and k 0.134 kg/m3.

(a) Find the function Pv which gives power as a function of wind velocity.

(b) Find P7, the rate of change in power generated with respect to wind veloc-ity, when the wind velocity is 7 meters per second.

SOLUTION:

(a) Since r 5 m and k 0.134 kg/m2, we have

P kr2v3 0.13452v3 3.35v3

So, Pv 3.35v3, where v is in meters per second and P is in watts.

(b) P7 limh0P7 h

h P7

limh0

limh0

limh0

limh0

3.35147 21h h2

3.35147

492.45

The rate of change in power generated is about 492 watts per meter/sec.

3.35147h 21h2 h3h

3.3573 147h 21h2 h3 73h

3.357 h3 3.3573h



QUICK REVIEW 10.1 (For help, go to Sections P.1 and P.4.)In Exercises 1 and 2, find the slope of the line determined by thepoints.

1. 2, 3, 5, 1 4/7 2. 3, 1, 3, 3 2/3

In Exercises 36, write an equation for the specified line.

3. Through 2, 3 with slope 32 y (3/2)x 64. Through 1, 6 and 4, 1 y 6 (7/3)(x 1)5. Through 1, 4 and parallel to y 34x 26. Through 1, 4 and perpendicular to y 34x 2

In Exercises 710, simplify the expression assuming h 0.

7. 2 h

h2 4 h 4

8. h 7

9. 2(h1 2)

10. 1(x h

h) 1x

x(x1 h)

12 h 12h

3 h2 3 h 12h

SECTION 10.1 EXERCISES

1. Average Velocity A bicyclist travels 21 miles in 1 hour and 45 minutes. What is her average velocity during the entire 1 34hour time interval? 12 mi per hour

2. Average Velocity An automobile travels 540 kilometers in 4 hours and 30 minutes. What is its average velocity over theentire 4 12 hour time interval? 120 km per hour

In Exercises 36, the position of an object at time t is given by st.Find the instantaneous velocity at the indicated value of t.

3. st 3t 5 at t 4 3

4. st t

21 at t 2 2/9

5. st at2 5 at t 2 4a6. st t 1 at t 1

[Hint: rationalize the numerator.] 1/(22)In Exercises 710, use the graph to estimate the slope of the tangentline, if it exists, to the graph at the given point.

7. x 0 1 8. x 1 1

9. x 2 no tangent 10. x 4 no tangenty

x

2

4

0 1

123

2x

y

y

x4

2

24

2 4

y

x4 2

2

4

2 4

In Exercises 1114, graph the function in a square viewing windowand, without doing any calculations, estimate the derivative of thefunction at the given point by interpreting it as the tangent line slope,if it exists at the point.

11. f x x2 2x 5 at x 3 412. f x 12 x

2 2x 5 at x 2 413. f x x3 6x2 12x 9 at x 0 1214. f x 2 sin x at x 215. A Rock Toss A rock is thrown straight up from level

ground. The distance (in ft) the ball is above the ground (the position function) is f t 3 48t 16t2 at any time t (in sec). Find(a) f 0. 48(b) the initial velocity of the rock. 48 ft/sec

16. Rocket Launch A toy rocket islaunched straight up in the air from level ground. The distance (in ft) the rocket is above the ground (the position function) isf t 170t 16t2 at any time t (in sec). Find(a) f 0. 170(b) the initial velocity of the

rocket. 170 ft/sec


In Exercises 1720, use the limit definition to find

(a) the slope of the graph of the function at the indicated point,(b) an equation of the tangent line at the point.(c) Sketch a graph of the curve near the point without using your

graphing calculator.17. f x 2x2 at x 118. f x 2x x2 at x 219. f x 2x2 7x 3 at x 220. f x

x

12 at x 1

In Exercises 21 and 22, estimate the slope of the tangent line to thegraph of the function, if it exists, at the indicated points.

21. f x x at x 2, 2, and 0. 1; 1; none22. f x tan1 x 1 at x 2, 2, and 0. 0.5; 0.1; 0.5In Exercises 2328, find the derivative, if it exists, of the function atthe specified point.

23. f x 1 x2 at x 2 424. f x 2x 12 x

2 at x 2 425. f x 3x2 2 at x 2 1226. f x x2 3x 1 at x 1 127. f x x 2 at x 2 does not exist28. f x

x

12 at x 1 1

In Exercises 2932, find the derivative of f.29. f x 2 3x 3 30. f x 2 3x2 6x31. f x 3x2 2x 1 32. f x

x

12 (x

12)2

33. Average Speed A lead ball is held at water level and droppedfrom a boat into a lake. The distance the ball falls at 0.1 sec timeintervals is given in Table 10.1.

Table 10.1 Distance Data of the Lead Ball

Time (sec) Distance (ft)0 00.1 0.10.2 0.40.3 0.80.4 1.50.5 2.30.6 3.20.7 4.40.8 5.80.9 7.3

(a) Compute the average speed from 0.5 to 0.6 seconds and from0.8 to 0.9 seconds. 9 ft/sec; 15 ft/sec

(b) Find a quadratic regression model for the distance data andoverlay its graph on a scatter plot of the data.

(c) Use the model in part (b) to estimate the depth of the lake if theball hits the bottom after 2 seconds. 35.9 ft

34. Finding Derivatives from Data A ball is dropped from theroof of a two-story building. The distance in feet above ground ofthe falling ball is given in Table 10.2 where t is in seconds.

(a) Use the data to estimate the average velocity of the ball in theinterval 0.8 t 1. 27.4 ft/sec

(b) Find a quadratic regression model s for the data in Table 10.2and overlay its graph on a scatter plot of the data.

(c) Find the derivative of the regression equation and use it to esti-mate the velocity of the ball at time t 1.

In Exercises 3538, complete the following.

(a) Draw a graph of the function.(b) Find the derivative of the function at the given point if it exists.(c) Writing to Learn If the derivative does not exist at the point,

explain why not.

35. f x at x 236. f x at x 2

37. f x at x 238. f x at x 0

sinx

x if x 0

1 if x 0

x

x

22

if x 21 if x 2

1 x 22 if x 21 x 22 if x 2

4 x if x 2x 3 if x 2

Table 10.2 Distance Data of the Ball

Time (sec) Distance (ft)0.2 30.000.4 28.360.6 25.440.8 21.241.0 15.761.2 9.021.4 0.95



In Exercises 3942, sketch a possible graph for a function that has thestated properties.

39. The domain of f is 0, 5 and the derivative at x 2 is 3.40. The domain of f is 0, 5 and the derivative is 0 at both x 2 and

x 4.41. The domain of f is 0, 5 and the derivative at x 2 is undefined.42. The domain of f is 0, 5, f is nondecreasing on 0, 5, and the

derivative at x 2 is 0.43. Writing to Learn Explain why you can find the derivative

of f x ax b without doing any computations. What is f x? The slope of the line is a; f(x) a

44. Writing to Learn Use the first definition of derivative at a pointto express the derivative of f x x at x 0 as a limit. Thenexplain why the limit does not exist. (A graph of the quotient for xvalues near 0 might help.)

Standardized Test Questions45. True or False When a ball rolls down a ramp, its instantaneous

velocity is always zero. Justify your answer.46. True or False If the derivative of the function f exists at x a,

then the derivative is equal to the slope of the tangent line at x a.Justify your answer.

In Exercises 4750, choose the correct answer. You may use a calculator.

47. Multiple Choice If f x x2 3x 4, find f x. D(A) x2 3 (B) x2 4 (C) 2x 1 (D) 2x 3(E) 2x 3

48. Multiple Choice If f x 5x 3x2, find f x. A(A) 5 6x (B) 5 3x (C) 5x 6 (D) 10x 3(E) 5x 6x2

49. Multiple Choice If f x x3, find the derivative of f at x 2. C(A) 3 (B) 6 (C) 12 (D) 18 (E) Does not exist

50. Multiple Choice If f x x

13, find the derivative of f at

x 1. A

(A) 14 (B) 14 (C)

12 (D)

12 (E) Does not exist

ExplorationsGraph each function in Exercises 5154 and then answer the followingquestions.

(a) Writing to Learn Does the function have a derivative at x 0? Explain.

(b) Does the function appear to have a tangent line at x 0? If so,what is an equation of the tangent line?

51. f x x 52. f x x13 53. f x x13 54. f x tan1 x55. Free Fall A water balloon dropped from a window will fall a

distance of s 16t2 feet during the first t seconds. Find the bal-loons (a) average velocity during the first 3 seconds of falling and(b) instantaneous velocity at t 3.

56. Free Fall on Another Planet It can be established by experi-mentation that heavy objects dropped from rest free fall near thesurface of another planet according to the formula y gt2, where yis the distance in meters the object falls in t seconds after beingdropped. An object falls from the top of a 125 m spaceship whichlanded on the surface. It hits the surface in 5 seconds.(a) Find the value of g. 5 m/sec2

(b) Find the average speed for the fall of the object. 25 m/sec(c) With what speed did the object hit the surface? 50 m/sec

Extending the Ideas57. Graphing the Derivative The graph of f x x2ex is shown

below. Use your knowledge of the geometric interpretation of thederivative to sketch a rough graph of the derivative y f x.

58. Group Activity The graph of y f x is shown below.Determine a possible graph for the function y f x.

[5, 5] by [10, 10]

[0, 10] by [1, 1]




10.2Limits and Motion: The Area Problem

Distance from a Constant VelocityDistance equals rate times time is one of the earliest problem-solving formulas thatwe learn in school mathematics. Given a velocity and a time period, we can use the formula to compute distance traveledas in the following standard example.

EXAMPLE 1 Computing Distance TraveledAn automobile travels at a constant rate of 48 miles per hour for 2 hours and 30 minutes.How far does the automobile travel?

SOLUTION We apply the formula d rt:d 48 mihr2.5 hr 120 miles.

The similarity to Example 1 in Section 10.1 is intentional. In fact, if we represent distance traveled (i.e., the change in position) by s and the time interval by t, theformula becomes

s 48 mph t,

which is equivalent to

s

t 48 mph.

So the two Example 1s are nearly identicalexcept that Example 1 of Section 10.1 didnot make an assumption about constant velocity. What we computed in that instancewas the average velocity over the 2.5-hour interval. This suggests that we could haveactually solved the following, slightly different, problem to open this section.

EXAMPLE 2 Computing Distance TraveledAn automobile travels at an average rate of 48 miles per hour for 2 hours and 30 minutes.How far does the automobile travel?

SOLUTION The distance traveled is s, the time interval has length t, and stis the average velocity.Therefore,

s

s

t t 48 mph2.5 hr 120 miles.

So, given average velocity over a time interval, we can easily find distance traveled. Butsuppose we have a velocity function vt that gives instantaneous velocity as a changingfunction of time. How can we use the instantaneous velocity function to find distance

Now try Exercise 5.

Now try Exercise 1.

OBJECTIVE

Students will be able to calculate definite integrals using areas.MOTIVATE

Ask students to guess the meaning of thefollowing statement: lim

xf (x) 5.

LESSON GUIDE

Day 1: Distance from a Constant Velocity;Distance from a Changing Velocity;Limits at Infinity; The Connection toAreasDay 2: The Definite Integral

What youll learn about Distance from a Constant

Velocity

Distance from a ChangingVelocity

Limits at Infinity

The Connection to Areas

The Definite Integral

. . . and whyLike the tangent line problem,the area problem has manyapplications in every area ofscience, as well as historicaland economic applications.


traveled over a time interval? This was the other intriguing problem about instantaneousvelocity that puzzled the 17th-century scientistsand once again, algebra was inade-quate for solving it, as we shall see.

Distance from a Changing VelocityWhen Galileo began his experiments, heres what he might have asked himself aboutusing a changing velocity to find distance:

SECTION 10.2 Limits and Motion: The Area Problem 805

One might be tempted to offer the following solution:

Velocity times t gives s. But instantaneous velocity occurs at an instant of time, sot 0. That means s 0. So, at any given instant of time, the ball doesnt move.Since any time interval consists of instants of time, the ball never moves at all! (Youmight well ask: Is this another trick question?)As was the case with the Velocity Question in Section 10.1, this foolish-looking exam-ple conceals a very subtle algebraic dilemmaand, far from being a trick question, it isexactly the question that needed to be answered in order to compute the distance trav-eled by an object whose velocity varies as a function of time. The scientists who wereworking on the tangent line problem realized that the distance-traveled problem must berelated to it, but, surprisingly, their geometry led them in another direction. The distancetraveled problem led them not to tangent lines, but to areas.

Limits at InfinityBefore we see the connection to areas, let us revisit another limit concept that will makeinstantaneous velocity easier to handle, just as in the last section. We will again be con-tent with an informal definition.

A Distance Question

Suppose a ball rolls down a ramp and its velocity is always 2t feet per second,where t is the number of seconds after it started to roll. How far does the balltravel during the first 3 seconds?

ZENOS PARADOXES

The Greek philosopher Zeno of Elea(490425 B.C.) was noted for presentingparadoxes similar to the DistanceQuestion. One of the most famous con-cerns the race between Achilles and aslow-but-sure tortoise. Achilles sportinglygives the tortoise a head start, then setsoff to catch him. He must first gethalfway to the tortoise, but by the timehe runs halfway to where the tortoisewas when he started, the tortoise hasmoved ahead. Now Achilles must closehalf of that distance, but by the time hedoes, the tortoise has moved aheadagain. Continuing this argument forever,we see that Achilles can never evencatch the tortoise, let alone pass him, sothe tortoise must win the race.

DEFINITION (INFORMAL) Limit at Infinity

When we write limxy

f x L, we mean that f x gets arbitrarily close to L as x gets arbitrarily large.

EXPLORATION 1 An Infinite Limit

A gallon of water is divided equally and poured into teacups. Find the amountin each teacup and the total amount in all the teacups if there are1. 10 teacups 0.1 gal; 1 gal2. 100 teacups 0.01 gal; 1 gal3. 1 billion teacups 0.000000001 gal; 1 gal4. an infinite number of teacups 0 gal; 1 gal


Find the number of teacups needed if the amount of water in each teacup is (1) 1 cup, (2) 1 tablespoon, (3) an infi-nitely small amount.(Hint: 1 gallon 16 cups; 1 cup 16 tablespoons.)


The preceding Exploration probably went pretty smoothly until you came to the infinitenumber of teacups. At that point you were probably pretty comfortable in saying whatthe total amount would be, and probably a little uncomfortable in saying how muchwould be in each teacup. (Theoretically it would be zero, which is just one reason whythe actual experiment cannot be performed.) In the language of limits, the total amountof water in the infinite number of teacups would look like this:

limny(n 1n ) limnynn 1 gallon

while the total amount in each teacup would look like this:

limny

1n

0 gallons.

Summing up an infinite number of nothings to get something is mysterious enoughwhen we use limits; without limits it seems to be an algebraic impossibility. That is thedilemma that faced the 17th-century scientists who were trying to work with instanta-neous velocity. Once again, it was geometry that showed the way when the algebrafailed.

The Connection to AreasIf we graph the constant velocity v 48 in Example 1 as a function of time t, wenotice that the area of the shaded rectangle is the same as the distance traveled(Figure 10.4). This is no mere coincidence, either, as the area of the rectangle and thedistance traveled over the time interval are both computed by multiplying the sametwo quantities:

48 mph2.5 hr 120 miles.

Now suppose we graph a velocity function that varies continuously as a function of time(Figure 10.5). Would the area of this irregularly-shaped region still give the total dis-tance traveled over the time interval a, b?

Newton and Leibniz (and, actually, many others who had considered this question)were convinced that it obviously would, and that is why they were interested in a cal-culus for finding areas under curves. They imagined the time interval being parti-tioned into many tiny subintervals, each one so small that the velocity over it wouldessentially be constant. Geometrically, this was equivalent to slicing the area into narrow strips, each one of which would be nearly indistinguishable from a narrow rectangle (Figure 10.6).The idea of partitioning irregularly-shaped areas into approximating rectangles was notnew. Indeed, Archimedes had used that very method to approximate the area of a circlewith remarkable accuracy. However, it was an exercise in patience and perseverance, asExample 3 will show.


FIGURE 10.4 For constant velocity,the area of the rectangle is the same as thedistance traveled, since it represents theproduct of the same two quantities:48 mph2.5 hr 120 miles.

FIGURE 10.5 If the velocity varies overthe time interval a, b, does the shadedregion give the distance traveled?

velocity

batime

Velocity (mph)

48

2.5Time (hr)

velocity

batime

FIGURE 10.6 The region is partitionedinto vertical strips. If the strips are narrowenough, they are almost indistinguishablefrom rectangles. The sum of the areas of these rectangles will give the total area and can be interpreted as distance traveled.


EXAMPLE 3 Approximating an Area with Rectangles

Use the six rectangles in Figure 10.7 to approximate the area of the region below thegraph of f x x2 over the interval 0, 3.SOLUTION The base of each approximating rectangle is 12. The height is deter-mined by the function value at the right-hand endpoint of each subinterval. The areasof the six rectangles and the total area are computed in the table below:

Base of Height of Area of Subinterval rectangle rectangle rectangle

0, 12 12 f 12 122 14 1/21/4 0.12512, 1 12 f 1 12 1 121 0.5001, 32 12 f 32 322 94 1294 1.12532, 2 12 f 2 22 4 124 2.0002, 52 12 f 52 522 254 12254 3.12552, 3 12 f 3 32 9 129 4.500

Total Area: 11.375

The six rectangles give a (rather crude) approximation of 11.375 square units for thearea under the curve from 0 to 3.

Figure 10.7 shows that the right rectangular approximation method (RRAM) in Example 4overestimates the true area. If we were to use the function values at the left-hand end-points of the subintervals (LRAM), we would obtain a rectangular approximation(6.875 square units) that underestimates the true area (Figure 10.8). The average of thetwo approximations is 9.125 square units, which is actually a pretty good estimate ofthe true area of 9 square units. If we were to repeat the process with 20 rectangles, theaverage would be 9.01125. This method of converging toward an unknown area byrefining approximations is tedious, but it worksArchimedes used a variation of it2200 years ago to estimate the area of a circle, and in the process demonstrated that theratio of the circumference to the diameter was between 3.140845 and 3.142857.

The calculus step is to move from a finite number of rectangles (yielding an approxi-mate area) to an infinite number of rectangles (yielding an exact area). This brings us tothe definite integral.



FIGURE 10.7 The area under the graph of f (x) x2 is approximated by six rec-tangles, each with base 12. The height ofeach rectangle is the function value at the right-hand endpoint of the subinterval(Example 3).

5

y

10

x2 31

FIGURE 10.8 If we change the rectanglesin Figure 10.7 so that their heights are deter-mined by function values at the left-handendpoints, we get an area approximation(6.875 square units) that underestimates thetrue area.

5

y

10

x2 31


The Definite IntegralIn general, begin with a continuous function y f x over an interval a, b. Dividea, b into n subintervals of length x b an. Choose any value x1 in the firstsubinterval, x2 in the second, and so on. Compute f x1, f x2, f x3, , f xn, multi-ply each value by x, and sum up the products. In sigma notation, the sum of theproducts is

n

i1f xix.

The limit of this sum as n approaches infinity is the solution to the area problem, andhence the solution to the problem of distance traveled. Indeed, it solves a variety ofother problems as well, as you will learn when you study calculus. The limit, if it exists,is called a definite integral.


RIEMANN SUMS

A sum of the form n

i1f(xi)x in which x1

is in the first subinterval, x2 is in the sec-ond, and so on, is called a Riemann sum,in honor of Georg Riemann (18261866),who determined the functions for whichsuch sums had limits as n.

DEFINITE INTEGRAL NOTATION

Notice that the notation for the definiteintegral (another legacy of Leibniz) parallels the sigma notation of the sumfor which it is a limit. The in the limitbecomes a stylized S, for sum. Thex becomes dx (as it did in thederivative), and the f(xi) becomes sim-ply f(x) because we are effectively summing up all the f(x) values along the interval (times an arbitrarily smallchange in x), rendering the subscriptsunnecessary.

DEFINITION Definite Integral

Let f be a function defined on a, b and let n

i1f xix be defined as above. The

definite integral of f over a, b, denoted ba

f x dx, is given by

ba

f xdx limn

n

i1f xix,

provided the limit exists. If the limit exists, we say f is on a, b.integrable

The solution to Example 4 shows that it can be tedious to approximate a definite inte-gral by working out the sum for a large value of n. One of the crowning achievementsof calculus was to demonstrate how the exact value of a definite integral could beobtained without summing up any products at all. You will have to wait until calculus tosee how that is done; meanwhile, you will learn in Section 10.4 how to use a calculatorto take the tedium out of finding definite integrals by summing.

You can also use the area connection to your advantage, as shown in these next twoexamples.

EXAMPLE 4 Computing an Integral

Find 51

2x dx.

SOLUTION This will be the area under the line y 2x over the interval 1, 5. Thegraph in Figure 10.9 shows that this is the area of a trapezoid.

Using the formula A h(b1 2 b2 ), we find that 51

2x dx 4(2(1) 2 2(5) ) 24Now try Exercise 23.

y

x1 5

y = 2x

FIGURE 10.9 The area of the trapezoidequals 5

12x dx. (Example 4)


EXAMPLE 5 Computing an IntegralSuppose a ball rolls down a ramp so that its velocity after t seconds is always 2t feetper second. How far does it fall during the first 3 seconds?

SOLUTION The distance traveled will be the same as the area under the velocitygraph, v t 2t, over the interval 0, 3. The graph is shown in Figure 10.10. Sincethe region is triangular, we can find its area: A 1236 9. The distance traveled in the first 3 seconds, therefore, is s 123 sec6 feetsec 9 feet.



FIGURE 10.10 The area under the velocity graph vt 2t, over the interval 0, 3 is thedistance traveled by the ball in Example 5 during the first 3 seconds.

6

v

t1 2 3

FOLLOW-UP

Ask students to explain why the RRAMapproximation for the area under a curvebecomes more accurate as the number ofrectangles, n, increases.ASSIGNMENT GUIDE

Day 1: Ex. 119 odd, 4750Day 2: Ex. 2145, multiples of 3, 59, 62COOPERATIVE ACTIVITY

Group Activity: Ex. 57, 5964NOTES ON EXERCISES

Ex. 2138 are designed to help studentsunderstand the connection between thedefinite integral and areas.Ex. 4950 require students to estimateareas using data.Ex. 5156 provide practice with standard-ized tests.Ex. 5964 introduce students to some ofthe properties of definite integrals.ONGOING ASSESSMENT

Self-Assessment: Ex. 1, 5, 11, 23, 45Embedded Assessment: Ex. 4344, 58

QUICK REVIEW 10.2 (For help, go to Sections 1.1 and 9.4.)In Exercises 1 and 2, list the elements of the sequence.

1. ak 12 ( 12 k )2 for k 1, 2, 3, 4, . . . , 9, 10

2. ak 14 (2 14 k )2 for k 1, 2, 3, 4, . . . , 9, 10

In Exercises 36, find the sum.

3. 10

k112 k 1

625 4.

n

k1k 1 n(n2

3)

5. 10

k112 k 1

2 5025

6. n

k112 k

2 n(n 1

1)(22n 1)

7. A truck travels at an average speed of 57 mph for 4 hours.How far does it travel? 228 miles

8. A pump working at 5 gal/min pumps for 2 hours. How manygallons are pumped? 600 gal

9. Water flows over a spillway at a steady rate of 200 cubic feetper second. How many cubic feet of water pass over thespillway in 6 hours? 4,320,000 ft3

10. A county has a population density of 560 people per squaremile in an area of 35,000 square miles. What is the popula-tion of the county? 19,600,000 people




In Exercises 14, explain how to represent the problem situation as anarea question and then solve the problem.

1. A train travels at 65 mph for 3 hours. How far does it travel? 195 mi

2. A pump working at 15 gal/min pumps for one-half hour. Howmany gallons are pumped? 450 gal

3. Water flows over a spillway at a steadyrate of 150 cubic feet per second. Howmany cubic feet of water pass over thespillway in one hour? 540,000 ft3

4. A city has a population density of 650people per square mile in an area of 20 square miles. What is thepopulation of the city? 13,000 people

5. An airplane travels at an average velocity of 640 kilometers per hour for 3 hours and 24 minutes. How far does the airplanetravel? 2176 km

6. A train travels at an average velocity of 24 miles per hour for 4 hours and 50 minutes. How far does the train travel? 116 mi

In Exercises 710, estimate the area of the region above the x-axis and under the graph of the function from x 0 to x 5.

7. 8.

9. 10.

x

y

00

1

2

3

4

5

6

1 2 3 4 5x

y

00

1

2

3

4

5

6

1 2 3 4 5

x

y

00

1

2

3

4

5

1 2 3 4 5x

y

00

1

2

3

4

5

1 2 3 4 5

In Exercises 11 and 12, use the 8 rectangles shown to approximate thearea of the region below the graph of f(x) 10 x2 over the interval[1, 3].11. 12.

In Exercises 1316, partition the given interval into the indicated num-ber of subintervals.

13. 0, 2; 4 14. 0, 2; 815. 1, 4; 6 16. 1, 5; 8

In Exercises 1720, complete the following.

(a) Draw the graph of the function for x in the specified interval.Verify that the function is nonnegative in that interval.

(b) On the graph in part (a), draw and shade the approximating rectan-gles for the RRAM using the specified partition. Compute theRRAM area estimate without using a calculator.

(c) Repeat part (b) using the LRAM.(d) Average the RRAM and LRAM approximations from parts (b)

and (c) to find an average estimate of the area. 17. f x x2; 0, 4; 4 subintervals18. f x x2 2; 0, 6; 6 subintervals19. f x 4x x2; 0, 4; 4 subintervals20. f x x3; 0, 3; 3 subintervalsIn Exercises 2128, find the definite integral by computing an area. (Itmay help to look at a graph of the function.)

21. 73

5 dx 20 22. 41

6 dx 30

23. 50

3x dx 37.5 24. 71

0.5x dx 12

25. 41

x 3 dx 16.5 26. 41

3x 2 dx 16.5

27. 22

4 x2 dx 2 28. 60

36 x2 dx 9

5

y

2 311

5

y

2 311

32.5 28.5


It can be shown that the area enclosed between the x-axis and one archof the sine curve is 2. Use this fact in Exercises 2938 to compute thedefinite integral. (It may help to look at a graph of the function.)

29. 0

sin x dx 2 30. 0

sin x 2 dx 2 2

31. 22

sin x 2 dx 2 32. /2/2

cos x dx 2

33. /20

sin x dx 1 34. /20

cos x dx 1

35. 0

2 sin x dx [Hint: All the rectangles are twice as tall.] 4

36. 20

sin ( 2x ) dx [Hint: All the rectangles are twice as wide.]37. 2

0sin x dx 4 38. 3/2

cos x dx 5

In Exercises 3942, find the integral assuming that k is a numberbetween 0 and 4.

39. 40

(kx 3) dx 8k 12 40. k0

(4x 3) dx 2k2 3k

41. 40

(3x k) dx 24 4k 42. 4k

(4x 3) dx

43. Writing to Learn Let gx f x where f has nonnegativefunction values on an interval a, b. Explain why the area abovethe graph of g is the same as the area under the graph of f in thesame interval.

44. Writing to Learn Explain how you can find the area underthe graph of f x 16 x2 from x 0 to x 4 by mental com-putation only.

45. Falling Ball Suppose a ball is dropped from a tower and itsvelocity after t seconds is always 32t feet per second. How far doesthe ball fall during the first 2 seconds? 64 ft

46. Accelerating Automobile Suppose an automobile acceleratesso that its velocity after t seconds is always 6t feet per second.How far does the car travel in the first 7 seconds? 147 ft

47. Rock Toss A rock is thrown straight up from level ground. Thevelocity of the rock at any time t (sec) is vt 48 32t ftsec.(a) Graph the velocity function.(b) At what time does the rock reach its maximum height? (c) Find how far the rock has traveled at its maximum height.

48. Rocket Launch A toy rocket is launched straight up from levelground. Its velocity function is f t 170 32t feet per second,where t is the number of seconds after launch.(a) Graph the velocity function.(b) At what time does the rocket reach its maximum height? (c) Find how far the rocket has traveled at its maximum

height. 451.6 ft

49. Finding Distance Traveled as Area A ball is pushed off theroof of a three-story building. Table 10.3 gives the velocity (in feetper second) of the falling ball at 0.2-second intervals until it hitsthe ground 1.4 seconds later.

(a) Draw a scatter plot of the data. (b) Find the approximate building height using RRAM areas as in

Example 4. Use the fact that if the velocity function is alwaysnegative the distance traveled will be the same as if the absolutevalue of the velocity values were used. 33.86 ft

50. Work Work is defined as force times distance. A full water barrelweighing 1250 pounds has a significant leak and must be lifted 35 feet. Table 10.4 displays the weight of the barrel measured after each 5 feet of movement. Find the approximate work in foot-pounds done in lifting the barrel 35 feet. 31,500 ft-pounds

Standardized Test Questions51. True or False When estimating the area under a curve using

LRAM, the accuracy typically improves as the number n of subin-tervals is increased.

52. True or False The statement limx

f x L means thatf x gets arbitrarily large as x gets arbitrarily close to L.

Table 10.4 Weight of a Leaking Water Barrel

Distance (ft) Weight (lb)0 12505 1150

10 105015 95020 85025 75030 650

Table 10.3 Velocity Data of the Ball

Time Velocity

0.2 5.050.4 11.430.6 17.460.8 24.211.0 30.621.2 37.061.4 43.47



It can be shown that the area of the region enclosed by the curve y x, the x-axis, and the line x 9 is 18. Use this fact inExercises 5356 to choose the correct answer. Do not use a calculator.

53. Multiple Choice 0

92x dx A

(A) 36 (B) 27 (C) 18 (D) 9 (E) 6


9

x 5 dx E(A) 14 (B) 23 (C) 33 (D) 45 (E) 63


14

x 5 dx C(A) 9 (B) 13 (C) 18 (D) 23 (E) 28


33x dx D

(A) 54 (B) 18 (C) 9 (D) 6 (E) 3

Explorations57. Group Activity You may have erroneously assumed that the

function f had to be positive in the definition of the definite integral. It is a fact that 2

0sin x dx 0. Use the definition of

the definite integral to explain why this is so. What does this implyabout 1

0x 1 dx?

58. Area Under a Discontinuous Function Let

f x .(a) Draw a graph of f. Determine its domain and range.(b) Writing to Learn How would you define the area under f

from x 0 to x 4? Does it make a difference if the functionhas no value at x 2?

Extending the IdeasGroup Activity From what you know about definite integrals,decide whether each of the following statements is true or false forintegrable functions (in general). Work with your classmates to justifyyour answers.

59. ba

f x dx ba

gx dx ba

f x gx dx true

60. ba

8 f x dx 8 ba

f x dx true

61. ba

f x gx dx ba

f x dx ba

gx dx false

62. ca

f x dx bc

f x dx ba

f x dx for a c b true

63. ba

f x ab

f x false

64. aa

f x dx 0 true

1 if x 2x if x 2



SECTION 10.3 More on Limits 813

10.3More on Limits

A Little HistoryProgress in mathematics occurs gradually and without much fanfare in the earlystages. The fanfare occurs much later, after the discoveries and innovations have beencleaned up and put into perspective. Calculus is certainly a case in point. Most of theideas in this chapter pre-date Newton and Leibniz. Others were solving calculusproblems as far back as Archimedes of Syracuse (ca. 287212 B.C.), long before cal-culus was discovered. What Newton and Leibniz did was to develop the rules ofthe game, so that derivatives and integrals could be computed algebraically. Mostimportantly, they developed what has come to be called the Fundamental Theorem ofCalculus, which explains the connection between the tangent line problem and thearea problem.

But the methods of Newton and Leibniz depended on mysterious infinitesimal quan-tities that were small enough to vanish and yet were not zero. Jean Le Rond dAlembert(17171783) was a strong proponent of replacing infinitesimals with limits (the strat-egy that would eventually work), but these concepts were not well understood untilKarl Weierstrass (18151897) and his student Eduard Heine (18211881) introducedthe formal, unassailable definitions that are used in our higher mathematics coursestoday. By that time, Newton and Leibniz had been dead for over 150 years.

Defining a Limit InformallyThere is nothing difficult about the following limit statements:

limx3

(2x 1) 5 limx

x2 3 limn

1n

0

That is why we have used limit notation throughout this book. Particularly when elec-tronic graphers are available, analyzing the limiting behavior of functions alge-braically, numerically, and graphically can tell us much of what we need to knowabout the functions.

What is difficult is to come up with an air-tight definition of what a limit really is. If ithad been easy, it would not have taken 150 years. The subtleties of the epsilon-deltadefinition of Weierstrass and Heine are as beautiful as they are profound, but they arenot the stuff of a precalculus course. Therefore, even as we look more closely at limitsand their properties in this section, we will continue to refer to our informal defini-tion of limit (essentially that of dAlembert). We repeat it here for ready reference:

OBJECTIVE

Students will be able to use the propertiesof limits and evaluate one-sided limits,two-sided limits, and limits involvinginfinity.MOTIVATE

Discuss whether limx2

int (x) exists, whereint x represents the greatest integer that isless than or equal to x.LESSON GUIDE

Day 1: A Little History; Defining a LimitInformally; Properties of Limits; Limitsof Continuous Functions; One-Sided andTwo-Sided LimitsDay 2: Limits Involving InfinityTEACHING NOTE

For reference, the formal epsilon-deltadefinition of a limit is given below. Notethat f(x) gets arbitrarily close to Lmeans f(x) is within units of L, and x gets arbitrarily close to a means x is within units of a. f(x) has a limit Las x approaches a (that is, lim

xaf(x) L)

if and only if for any 0 there exists anumber 0 such that, whenever x a , we have f(x) L .

DEFINITION (INFORMAL) Limit at aWhen we write lim

xyaf x L, we mean that f x gets arbitrarily close to L as x

gets arbitrarily close (but not equal) to a.

What youll learn about A Little History

Defining a Limit Informally

Properties of Limits

Limits of ContinuousFunctions

One-Sided and Two-SidedLimits

Limits Involving Infinity

. . . and whyLimits are essential concepts inthe development of calculus.



EXPLORATION 1 Whats the Limit?

As a class, discuss the following two limit statements until you really under-stand why they are true. Look at them every way you can. Use your calculators.Do you see how the above definition verifies that they are true? In particular,can you defend your position against the challenges that follow the statements?(This exploration is intended to be free-wheeling and philosophical. You cantprove these statements without a stronger definition.)1. lim

x27x 14.000000000000000001

Challenges: Isnt 7x getting arbitrarily close to that number as x approaches 2? How can you tell that 14 is the limit and 14.000000000000000001

is not?

2. limx0

x2

x

2x 2

Challenges: How can the limit be 2 when the quotient isnt even defined at 0? Wont there be an asymptote at x 0? The denominator equals 0

there. How can you tell that 2 is the limit and 1.99999999999999999999

is not?

Solve Graphically

The graph in Figure 10.11a suggeststhat the limit exists and is about 3.

FIGURE 10.11a A graph of f x x3 1x 1.

Solve Numerically

The table also gives compelling evidence that the limit is 3.

FIGURE 10.11b A table of valuesfor f x x3 1x 1.

Solve Algebraically

limx1

x

x

3

11

limx1

limx1

x2 x 1

1 1 1

3

x 1x2 x 1

x 1X

Y1 = (X31)/(X1)

.997

.998

.99911.0011.0021.003

2.9912.9942.997ERROR3.0033.0063.009

Y1

(b)[4, 4] by [2, 8](a)

X=1.0212766 Y=3.0642825

1

EXAMPLE 1 Finding a Limit

Find limx1

x

x

3

11

.

As convincing as the graphical and numerical evidence is, the best evidence is algebraic. The limit is 3.


Discuss the following limit statement anddecide whether it is true:

limx0

x sinx

1 0.



Properties of LimitsWhen limits exist, there is nothing unusual about the way they interact algebraicallywith each other. You could easily predict that the following properties would hold.These are all theorems that one could prove with a rigorous definition of limit, but wemust state them without proof here.


Properties of Limits

If limxc

f x and limxc

gx both exist, then

1. Sum Rule limxc

f x gx limxc

f x limxc

gx

2. Difference Rule limxc

f x gx limxc

f x limxc

gx

3. Product Rule limxc

f x gx limxc

f x limxc

gx

4. Constant Multiple limxc

k gx k limxc

gxRule

5. Quotient Rule limxc

gf x

x

l

l

i

i

mx

mx

c

c

g

f

x

x

,

provided limxc

gx 0

6. Power Rule limxc

f xn limxc

f xn for na positive integer

7. Root Rule limxc

n f x n limxc

f x for n 2 a positive integer, provided n lim

xcf x

and limxc

n f x are real numbers.

EXAMPLE 2 Using the Limit Properties

You will learn in Example 10 that limx0

sin

x

x 1. Use this fact, along with the limit

properties, to find the following limits:

(a) limx0

x

x

sin x (b) lim

x01

x

c2os2 x (c) lim

x03

s3

inx

x

SOLUTION

(a) limx0

x

x

sin x lim

x0 ( xx sinx x ) lim

x0x

x lim

x0sin

x

x Sum Rule

1 1

2continued


(b) limx0

1

x

c2os2 x lim

x0si

x

n2

2 x Pythagorean identity

limx0 (sinx x )(sinx x )

limx0 (sinx x ) limx0 (sinx x ) Product Rule

1 1

1

(c) limx0

3

s3inx

x lim

x0 3 sinx x 3 limx0sinx x Root Rule 3 1

1

Limits of Continuous FunctionsRecall from Section 1.2 that a function is continuous at a if lim

xaf x f a. This means

that the limit (at a) of a function can be found by plugging in a provided the function iscontinuous at a. (The condition of continuity is essential when employing this strategy. Forexample, plugging in 0 does not work on any of the limits in Example 2.)

EXAMPLE 3 Finding Limits by SubstitutionFind the limits.

(a) limx0

ex

c

os

t2a

x

n x (b) lim

n16log2

nn

SOLUTION

You might not recognize these functions as being continuous, but you can use the limitproperties to write the limits in terms of limits of basic functions.

(a) limx0

ex

c

os

t2a

x

n x Quotient Rule

Difference and Power Rules

e0

c

os

ta0n

20

Limits of continuous functions

1

10

1

limx0

ex limx0

tan x

limx0

cos x2

limx0

ex tan xlim

x0cos2 x



continued


(b) limn16

log2

nn

Quotient Rule

log2

1166 Limits of continuous functions

44

1

Example 3 hints at some important properties of continuous functions that follow fromthe properties of limits. If f and g are both continuous at x a, then so are f g, f g,fg, and fg (with the assumption that g(a) does not create a zero denominator in the quo-tient). Also, the nth power and nth root of a function that is continuous at a will also becontinuous at a (with the assumption that f (a) is real).

One-sided and Two-sided LimitsWe can see that the limit of the function in Figure 10.11 is 3 whether x approaches 1 fromthe left or right. Sometimes the values of a function f can approach different values as x approaches a number c from opposite sides. When this happens, the limit of f as x approaches c from the left is the of f at c and the limit of f as xapproaches c from the right is the of f at c. Here is the notation we use:left-hand: lim

xycf x The limit of f as x approaches c from the left.

right-hand: limxyc

f x The limit of f as x approaches c from the right.

EXAMPLE 4 Finding Left- and Right-Hand Limits

Find limxy2

f x and limxy2

f x where f x .SOLUTION Figure 10.12 suggests that the left- and right-hand limits of f exist butare not equal. Using algebra we find:

limxy2

f x limxy2

x2 4x 1 Definition of f

22 4 2 1

3

limxy2

f x limxy2

2x 3 Definition of f

2 2 3

1

You can use trace or tables to support the above results.

Now try Exercise 27, parts (a) and (b).

x2 4x 1 if x 22x 3 if x 2

right-hand limitleft-hand limit


limn16

nlimn16

log2 n


FIGURE 10.12 A graph of thepiecewise-defined function.

f x (Example 4)

x2 4x 1 x 22x 3 x 2

[2, 8] by [3, 7]

NOTES ON EXAMPLES

The functions in Example 3 are in factcontinuous at the limit points, so wecould have found the limits using directsubstitution instead of using the methodshown.


The limit limxyc

f x is sometimes called the of f at c to distinguish itfrom the one-sided left-hand and right-hand limits of f at c. The following theoremindicates how these limits are related.

two-sided limit


The limit of the function f of Example 4 as x approaches 2 does not exist, so f is dis-continuous at x 2. However, discontinuous functions can have a limit at a point ofdiscontinuity. The function f of Example 1 is discontinuous at x 1 because f 1 doesnot exist, but it has the limit 3 as x approaches 1. Example 5 illustrates another way afunction can have a limit and still be discontinuous.

EXAMPLE 5 Finding a Limit at a Point of Discontinuity

Let

f x Find lim

xy3f x and prove that f is discontinuous at x 3.

SOLUTION Figure 10.13 suggests that the limit of f as x approaches 3 exists. Usingalgebra we find

limxy3

x

x

2

39

limxy3

x

x

3

x3 3

limxy3

x 3 We can assume x 3.

6.

Because f 3 2 limxy3

f x, f is discontinuous at x 3.

EXAMPLE 6 Finding One-Sided and Two-Sided Limits

Let f x intx (the greatest integer function). Find:(a) lim

xy3intx (b) lim

xy3intx (c) lim

xy3intx


x

x

2

39

if x 3

2 if x 3.

THEOREM One-sided and Two-sided Limits

A function f x has a limit as x approaches c if and only if the left-hand and right-hand limits at c exist and are equal. That is,

limxyc

f x L limxyc

f x L and limxyc

f x L.

FIGURE 10.13 A graph of the functionin Example 5.

[4.7, 4.7] by [5, 10]

continued


SOLUTION Recall that int x is equal to the greatest integer less than or equal to x.For example, int3 3. From the definition of f and its graph in Figure 10.14 we cansee that

(a) limxy3

intx 2

(b) limxy3

intx 3

(c) limxy3

intx does not exist.

Limits Involving InfinityThe informal definition that we have for a limit refers to lim

xyaf x L where both a and

L are real numbers. In Section 10.2 we adapted the definition to apply to limits of the form lim

xyf x L so that we could use this notation in describing definite integrals.

This is one type of limit at infinity. Notice that the limit itself L is a finite real num-ber, assuming the limit exists, but that the values of x are approaching infinity.



Notice that limits, whether at a or at infinity, are always finite real numbers; otherwise,the limits do not exist. For example, it is correct to write

limx0

x

12 does not exist,

since it approaches no real number L. In this case, however, it is also convenient to write

limx0

x

12

,

which gives us a little more information about why the limit fails to exist. (It increaseswithout bound.) Similarly, it is convenient to write

limx0

ln x

,

since ln x decreases without bound as x approaches 0 from the right. In this context, thesymbols

and

are sometimes called .infinite limits

INFINITE LIMITS ARE NOT LIMITS

It is important to realize that an infinitelimit is not a limit, despite what thename might imply. It describes a specialcase of a limit that does not exist. Recallthat a sawhorse is not a horse and a bad-minton bird is not a bird.

ARCHIMEDES (CA. 287212 B.C.)

The Greek mathematician Archimedesfound the area of a circle using a methodinvolving infinite limits. See Exercise 89for a modern version of his method.

NOTES ON EXAMPLES

Examples 5 and 6 demonstrate that it isnot necessarily true that lim

xcf (x) f (c).

Make certain that students understand therole of continuity in evaluating limits.

FIGURE 10.14 The graph of f(x) int(x). (Example 6)

[5, 5] by [5, 5]

DEFINITION Limits at Infinity

When we write limxy

f x L, we mean that f x gets arbitrarily close to L as xgets arbitrarily large. We say that f has a limit L as x approaches .When we write lim

xf x L, we mean that f x gets arbitrarily close to L as x

gets arbitrarily large. We say that f has a limit L as x approaches .


EXAMPLE 7 Investigating Limits as xyLet f x sin xx. Find lim

xyf x and lim

xyf x.

SOLUTION The graph of f in Figure 10.15 suggests thatlimxy

sin

x

x lim

xysin

x

x 0.

In Section 1.3, we used limits to describe the unbounded behavior of the functionf x x3 as xy :

limxy

x3 and limxy

x3

The behavior of the function gx ex as x y can be described by the followingtwo limits:

limxy

ex and limxy

ex 0

The function gx ex has unbounded behavior as x y and has a finite limit asxy .

EXAMPLE 8 Using Tables to Investigate Limits as xy

Let f x xex. Find limxy

f x and limxy

f x.

SOLUTION The tables in Figure 10.16 suggest that

limxy

xex 0 and limxy

xex .

FIGURE 10.16 The table in (a) suggests that the values of f (x) xex approach 0 asxy and the table in (b) suggests that the values of f (x) xex approach as xy .(Example 8)

The graph of f in Figure 10.17 supports these results.Now try Exercise 49.

X

Y2 = Xe^(X)

0102030405060

02.2E59.7E93E149E183E237E27

Y2

(b)

X

Y2 = Xe^(X)

0102030405060

04.5E44.1E83E122E161E205E25

Y2

(a)



FIGURE 10.15 The graph of f (x) (sin x)x. (Example 7)

[20, 20] by [2, 2]

FIGURE 10.17 The graph of thefunction f (x) xex. (Example 8)

[5, 5] by [5, 5]

FOLLOW-UP

Ask students to discuss the followingstatement and determine for what valuesof k and c it is true: lim

xckx kc.

ASSIGNMENT GUIDE

Day 1: Ex. 339, multiples of 3Day 2: Ex. 4287, multiples of 3COOPERATIVE ACTIVITY

Group Activity: Ex. 3536, 8487NOTES ON EXERCISES

Ex. 160 are basic exercises involvingleft-hand, right-hand, and two-sided limits.Ex. 7378 provide practice with standard-ized tests.Ex. 8487 give students an opportunity tocreate functions with specified properties.ONGOING ASSESSMENT

Self-Assessment: Ex. 11, 19, 23, 27, 37,41, 47, 49, 51, 55, 63, 65, 67Embedded Assessment: Ex. 3740


In Section 2.6 we used the graph of f x 1x 2 to state that

limxy2

x

12 and limxy2 x

12 .

Either one of these unbounded limits allows us to conclude that the vertical line x 2is a vertical asymptote of the graph of f (Figure 10.18).

EXAMPLE 9 Investigating Unbounded LimitsFind lim

xy21x 22.

SOLUTION The graph of f x 1x 22 in Figure 10.19 suggests that

limxy2

x

122 and limxy2 x

122 .

This means that the limit of f as x approaches 2 does not exist. The table of values inFigure 10.20 agrees with this conclusion. The graph of f has a vertical asymptote atx 2.

Not all zeros of denominators correspond to vertical asymptotes as illustrated inExamples 5 and 7.

EXAMPLE 10 Investigating a Limit at x 0Find lim

xy0sin xx.

SOLUTION The graph of f x sin xx in Figure 10.15 suggests this limit exists.The table of values in Figure 10.21 suggest that

limxy0

sin

x

x 1.

FIGURE 10.21 A table of values for f (x) (sin x)x. (Example 10)Now try Exercise 63.

X

Y1 = sin(X)/X

.03.02.010.01.02.03

.99985

.99993

.99998ERROR.99998.99993.99985

Y1



EXPLORATION 2 Investigating a Logistic Function

Let f x 1 50

23x.

1. Use tables and graphs to find limxy

f x and limxy

f x. 50; 02. Identify any horizontal asymptotes. y 50; y 0

3. How is the numerator of the fraction for f related to part 2?

FIGURE 10.18 The graph of f (x) 1(x 2) with its vertical asymptoteoverlaid.

[4.7, 4.7] by [5, 5]

FIGURE 10.19 The graph of f (x) 1/(x 2)2 in Example 9.

[4, 6] by [2, 10]


For what value of x is f(x) equal to 25?

FIGURE 10.20 A table of values for

f (x) (x

12)2

. (Example 9)

X

Y1 = 1/(X2)2

1.91.991.99922.0012.012.1

100100001E6ERROR1E610000100

Y1



QUICK REVIEW 10.3 (For help, go to Sections 1.2 and 1.3.)In Exercises 1 and 2, find (a) f 2, (b) f 0, and (c) f 2.

1. f x (22x

x

41)2 2. f x

sinx

x

In Exercises 3 and 4, find the (a) vertical asymptotes and (b) hori-zontal asymptotes of the graph of f, if any.

3. f x 2x

x2

2

43

4. f x 2 x3

x

1x2

In Exercises 5 and 6, the end behavior asymptote of the function f is one of the following. Which one is it?

(a) y 2x2 (b) y 2x2 (c) y x3 (d) y x3

5. f x 2x3

3 3x

x

2 1 (b) 6. f x x

4 2x

x

2

3x 1

(c)

In Exercises 7 and 8, find (a) the points of continuity and (b) the points of discontinuity of the function.

7. f x x 2 8. gx

Exercises 9 and 10 refer to the piecewise-defined function

f x .9. Draw the graph of f .

10. Find the points of continuity and the points of discontinuity of f .

3x 1 if x 14 x2 if x 1

2x 1x2 4


In Exercises 110, find the limit by direct substitution if it exists.

1. limxy1

x x 12 4 2. limxy3

x 112 4096

3. limxy2

x3 2x 3 7 4. limxy2

x3 x 5 1

5. limxy2

x 5 7 6. limxy2

x 42 3 3.30

7. limxy0

ex sin x 0 8. limxy

ln (sin 2x) 09. lim

xyax2 2 a2 2 10. lim

xyax

x

2

2

11 a

a2

2

11

In Exercises 1118, (a) explain why you cannot use substitution tofind the limit and (b) find the limit algebraically if it exists.

11. limxy3

x2

x27

x

912

12. limxy3

x2

x2

2

x

915

13. limxy1

x

x

3

11

14. limxy2x3 2

x

x

2

2x 2

15. limxy2

x

x

2

24

16. limxy2

x

x

2

24

17. limxy0

x 3 18. limxy0

x

x

22

In Exercises 1922, use the fact that limxy0

sin

x

x 1, along with

the limit properties, to find the following limits.

19. limxy0

2xs

2in

x

x 1 20. lim

xy0sin

x

3x 3

21. limxy0

sin

x

2 x 0 22. lim

xy0x

2s

x

in x 1

In Exercises 2326, find the limits.

23. limxy0

loe

gx

4

x

x2 2 24. limxy0

35s

s

iin

n

x

x

4c

c

o

o

s

s

x

x 4

25. limxy2

ls

n

in2

2x

x

ln 26. lim

xy27

lox

g

3 x

9 2

In Exercises 2730, use the given graph to find the limits or to explainwhy the limits do not exist.

27. (a) limxy2

f x 3(b) lim

xy2f x 1

(c) limxy2

f x none

28. (a) limxy3

f x 2(b) lim

xy3f x 4

(c) limxy3

f x

29. (a) limxy3

f x 4(b) lim

xy3f x 4

(c) limxy3

f x 4

0 1

123

2 3x

y

0 2

213

41 3x

y

0 2

243

41 3x

y


30. (a) limxy1

f x 1(b) lim

xy1f x 3

(c) limxy1

f x

In Exercises 31 and 32, the graph of a function y f (x) is given.Which of the statements about the function are true and which arefalse?

31. (a) limxy1

f x 1 true(b) lim

xy0f x 0 true

(c) limxy0

f x 1 false(d) lim

xy0f x lim

xy0f x

(e) limxy0

f x exists false (f) limxy0

f x 0 false(g) lim

xy0f x 1 false (h) lim

xy1f x 1 true

( i ) limxy1

f x 0 false (j) limxy2

f x 2 true

32. (a) limxy1

f x 1 true(b) lim

xy2f x does not exist

(c) limxy2

f x 2 false(d) lim

xy1f x 2 true

(e) limxy1

f x 1(f) lim

xy1f x does not exist

(g) limxy0

f x limxy0

f x false(h) lim

xycf x exists for every c in 1, 1. false (not x 0)

( i ) limxyc

f x exists for every c in 1, 3. trueIn Exercises 33 and 34, use a graph of f to find (a) lim

xy0f x,

(b) limxy0

f x, and (c) limxy0

f x if they exist.33. f x 1 x1x 34. f x 1 x12x35. Group Activity Assume that lim

xy4f x 1 and lim

xy4gx 4.

Find the limit.

(a) limxy4

gx 2 6 (b) limxy4

xf x 4

(c) limxy4

g2x 16 (d) limxy4

f xgx

1 2

36. Group Activity Assume that limxya

f x 2 and limxya

gx 3. Find the limit.

(a) limxya

f x gx 1 (b) limxya

f x gx 6

(c) limxya

3gx 1 8 (d) limxya

23

f xgx

In Exercises 3740, complete the following for the given piecewise-defined function f .(a) Draw the graph of f .(b) Determine lim

xyaf x and lim

xyaf x.

(c) Writing to Learn Does limxya

f x exist? If it does, give its value.If it does not exist, give an explanation.

37. a 2, f x 38. a 1, f x 39. a 0, f x 40. a 3, f x In Exercises 4146, find the limit.

41. limxy2

int (x) 2 42. limxy2

int (x) 143. lim

xy0.0001int (x) 0 44. lim

xy52int 2x 4

45. limxy3

1 46. limxy0

25x

x

52

In Exercises 4754, find (a) limxy

y and (b) limxy

y.

47. y 1co

s x

x 0; 0 48. y

x

x

sin x 1; 1

49. y 1 2x ; 1 50. y 1 x

2x 0;

51. y x sin x 52. y ex sin x

53. y ex sin x 54. y ex cos x

In Exercises 5560, use graphs and tables to find the limit and identifyany vertical asymptotes.

55. limxy3

x

13 ; x 3 56. limxy3 x

x

3 ; x 3

57. limxy2

x

12 ; x 2 58. limxy2 x

x

2 ; x 2

59. limxy5

x

152 ; x 5 60. limxy2 x2

1 4

In Exercises 6164, determine the limit algebraically if possible.Support your answer graphically.

61. limxy0

1 x

x

3 1 3 62. lim

xy013

x

x 13

19

63. limxy0

tan

x

x 1 64. lim

xy2x

x2

44 none

x 3x 3

1 x2 if x 38 x if x 3

x 3 if x 0x2 2x if x 0

2 x if x 1x 1 if x 1

2 x if x 21 if x 2

x2 4 if x 2


0 2

24

11 3

x

y

1 2

3

1 3x

y

1 2

23

1 3x

y


In Exercises 6572, find the limit.

65. limxy0

x

x2

66. limxy0

x

x

2

0

67. limxy0 [x sin ( 1x )] 0 68. limxy27 cos ( 1x )

69. limxy1

x

x

2

11

undefined 70. limxy

lln

n

x

x

2 2

71. limxy

lln

n

x

x2

12

72. limxy

3x 0

Standardized Test Questions

73. True or False If f x , thenlimxy3

f x is undefined. Justify your answer.74. True or False If f x and gx are two functions and

limxy0

f x does not exist, then limxy0

f x gx cannot exist. Justify your answer.

Multiple Choice In Exercises 7578, match the function y f x with the table. Do not use a calculator.

75. y x2

x

2x3 3

B 76. y x2

x

2x3 3

A

77. y x2

x

2x3 9

C 78. y x

x

3

237

D

X

X = 2.7

3.73.83.944.14.24.3

Y12.72.82.933.13.23.3

(E)

X

X=2.7

24.3925.2426.11ERROR27.9128.8429.79

Y1

(D)

2.72.82.933.13.23.3

X

X=2.7

23.733.863.9ERROR55.925.815.7

Y1

(C)

2.72.82.933.13.23.3

X

X=2.7

3.73.83.9ERROR4.14.24.3

Y1

(B)

2.72.82.933.13.23.3

X

X=2.7

52.382.2172.1ERROR188.198.268.3

Y1

(A)

2.72.82.933.13.23.3

x 2 if x 38 x if x 3

ExplorationsIn Exercises 7982, complete the following for the given piecewise-defined function f .(a) Draw the graph of f .(b) At what points c in the domain of f does lim

xycf x exist?

(c) At what points c does only the left-hand limit exist?(d) At what points c does only the right-hand limit exist?

79. f x 80. f x

81. f x 82. f x 83. Rabbit Population The population of rabbits over a 2-year

period in a certain county is given in Table 10.5.

(a) Draw a scatter plot of the data in Table 10.5.

(b) Find a logistic regression model forthe data. Find the limit of that modelas time approaches infinity.

(c) What can you conclude about thelimit of the rabbit population growthin the county?

(d) Provide a reasonable explanation for the population growth limit.

Table 10.5 Rabbit Population

Beginning of Number Month (in thousands)

0 102 124 146 168 22

10 3012 3514 3916 4418 4820 5022 51

x2 if 2 x 0 or 0 x 21 if x 02x if x 2 or x 2

1 x2 if 1 x 0x if 0 x 12 if x 1

sin x if x 0csc x if 0 x

cos x if x 0cos x if 0 x



Group Activity In Exercises 8487, sketch a graph of a functiony f(x) that satisfies the stated conditions. Include any asymptotes.84. lim

xy0f x , lim

xyf x , lim

xyf x 2

85. limxy4

f x , limxy

f x , limxy

f x 286. lim

xyf x 2, lim

xy2f x ,

limxy2

f x , limxy

f x 87. lim

xy1f x , lim

xy2f x , lim

xy2f x , lim

xyf x

Extending the Ideas 88. Properties of Limits Find the limits of f, g, and fg as x

approaches c.

(a) f x x

22 , g x x

2, c 0 ; 0; 2

(b) f x 1x , g x 3

x, c 0 ; 0; none

(c) f x x 3

1 , g x x 12, c 1 ; 0; 0

(d) f x x

114, g x x 1

2, c 1 ; 0;

(e) Writing to Learn Suppose that limxyc

f x andlimxyc

g x 0. Based on your results in parts (a)(d), whatcan you say about lim

xya f x g x? nothing

89. Limits and the Area of a Circle Consider an n-sided regularpolygon made up of n congruent isosceles triangles, each withheight h and base b. The figure shows an 8-sided regular polygon.

bh

(a) Show that the area of an 8-sided regular polygon is A 4hband the area of the n-sided regular polygon is A 12nhb.

(b) Show that the base b of the n-sided regular polygon is b 2h tan 180n.

(c) Show that the area A of the n-sided regular polygon is A nh2 tan 180n.

(d) Let h 1. Construct a table of values for n and A for n 4, 8,16, 100, 500, 1000, 5000, 10000, 100000. Does A have a limitas ny ?

(e) Repeat part (d) with h 3.(f) Give a convincing argument that lim

nyA h2, the area of a

circle of radius h.90. Continuous Extension of a Function Let

f x (a) Sketch several possible graphs for f.(b) Find a value for a so that the function is continuous at x 2. 1

In Exercises 9193, (a) graph the function, (b) verify that the functionhas one removable discontinuity, and (c) give a formula for a continu-ous extension of the function. [Hint: See Exercise 90.]

91. y 2x

x

24

92. y 5x

5x

93. y x

x

3

11

91. (b) y 2x

x

24

2(

x

x

22)

2

(c) y 292. (b) y x5

5x

x

x

55

1

(c) y 193. (b) y x

x

3

11

x2 x 1(c) y x2 x 1

(x 1)(x2 x 1)

x 1

x2 3x 3 if x 2a if x 2.




10.4Numerical Derivatives and Integrals

Derivatives on a CalculatorAs computers and sophisticated calculators have become indispensable tools formodern engineers and mathematicians (and, ultimately, for modern students of math-ematics), numerical techniques of differentiation and integration have re-emerged asprimary methods of problem-solving. This is no small irony, as it was precisely toavoid the tedious computations inherent in such methods that calculus was inventedin the first place. Although nothing can diminish the magnitude of calculus as a sig-nificant human achievement, and although nobody can get far in m

introduction to calculus

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