Análisis globalizado de los datos recopilados

11. Resultados

11.2 Análisis globalizado de los datos recopilados

T1. State the definitions of continuity at a point and on a closed interval.

T2. a. For function f in Figure 2-7h, find • lim f(x)

Figure 2-7i

For Problems T11–T13, let f(x) =

T11. Show that f is discontinuous at x = 2 if k = 1.

Sketch the graph.

T12. Find the value of k that makes f continuous at x = 2.

T13. On your grapher, plot the graph using the value of k from Problem T11. Use Boolean variables to restrict the domains of the two branches. Sketch the graph.

T14. Temperature Versus Depth Problem: During the day the soil at Earth’s surface warms up. Heat from the surface penetrates to greater depths. But before the temperature lower down reaches the surface temperature, night comes and Earth’s surface cools. Figure 2-7j shows what the temperature, T, in degrees Celsius, might look like as a function of depth, x, in feet.

Figure 2-7j

a. From the graph, what does the limit of T seem to be as x approaches infinity? How deep do you think you would have to be so that the temperature varies no more than 1 degree from this limit? What feature does the graph have when T equals this limit?

b. The equation of the function T in Figure 2-7i is T(x) = 20 + 8(0.97x_{) cos 0.5x. Use this}

equation to calculate a positive number x = c such that T does not vary more than

0.1 unit from the limit whenever x c. c. Just for fun, see if you can figure out the

approximate time of day for which the graph in Figure 2-7j applies.

T15. Glacier Problem: To determine how far a glacier has traveled in a given time interval, naturalists drive a metal stake into the surface of the glacier. From a point not on the glacier, they measure the distance, d(t), in centimeters, from its original position that the stake has moved in time t, in days. Every ten days they record this distance, getting the values shown in the table. t (days) d (t) (cm) 0 0 10 6 20 14 30 24 40 36 50 50

a. Show that the equation d(t) = 0.01t2_{+ 0.5t}

fits all the data points in the table. Use the most time-efficient way you can think of to do this problem.

b. Use the equation to find the average rate the glacier is moving during the interval t = 20

to t = 20.1.

c. Write an equation for the average rate from 20 days to t days. Perform the appropriate algebra, then find the limit of the average

to be speeding up or slowing down as time goes on? How do you reach this conclusion?

T16. Calvin and Phoebe’s Acceleration Problem: Calvin and Phoebe are running side by side along the jogging trail. At time t = 0, each one

starts to speed up. Their speeds are given by the following, where p(t) and c(t) are in ft/s and t is in seconds.

c(t) = 16 – 6(2-t₎ For Calvin. p(t) = 10 + For Phoebe.

Show that each is going the same speed when t = 0. What are the limits of their speeds as t approaches infinity? Surprising?!

T17. Let f(x) =

What value of k makes f continuous at x = 2?

What feature will the graph of f have at this point?

T18. Let h(x) = x3_{. Show that the number 7 is}

between h(1) and h(2). Since h is continuous on the interval [1, 2], what theorem allows you to conclude that there is a real number

between 1 and 2?

T19. What did you learn as a result of taking this test that you did not know before?

rate as t approaches 20. What is the instantaneous rate the glacier is moving at t = 20? What mathematical name is given to this rate?

Properties of Limits

Finding the average velocity of a moving vehicle requires dividing

the distance it travels by the time it takes to go that distance. You

can calculate the instantaneous velocity by taking the limit of the

average velocity as the time interval approaches zero. You can also

use limits to find exact values of definite integrals. In this chapter

you’ll study limits, the foundation for the other three concepts

of calculus.

Mathematical Overview

Informally, the limit of a function

f

x

approaches

c

is the

y

-value that

f

(

x

) stays close to when

x

is kept close enough to

c

but not equal to

c

. In Chapter 2, you will formalize the concept of

limit by studying it in four ways.

Graphically Numerically x f(x) 3.01 3.262015 3.001 3.251200... 3.0001 3.250120... 3.00001 3.250012... ·· · ···

Algebraically 0 < |x – c| < | f(x) – L| < , f(x) is within units of L whenever x is within units of c, the definition of limit.

Verbally

I have learned that a limit is a y-value that f(x) can be kept

arbitrarily close to just by keeping x close enough to c but not equal

to c. Limits involving infinity are related to vertical and horizontal

asymptotes. Limits are used to find exact values of derivatives.

The icon at the top of each

even-numbered page of this chapter

shows that

f

(

x

) is close to

L

when

x

2-1 Numerical Approach to the Definition

Exploratory Problem Set 2-1

1. Figure 2-1a shows the function

Figure 2-1a

a. Show that f(2) takes the indeterminate form 0/0. Explain why there is no value for f(2). b. The number y = 3 is the limit of f(x) as x

approaches 2. Make a table of values of f(x) for each 0.001 unit of x from 1.997 to 2.003. Is it true that f(x) stays close to 3 when x is kept close to 2 but not equal to 2?

c. How close to 2 would you have to keep x for f(x) to stay within 0.0001 unit of 3? Within 0.00001 unit of 3? How could you keep f(x) arbitrarily close to 3 just by keeping x close enough to 2 but not equal to 2?

d. The missing point at x = 2 is called a removable discontinuity. Why do you suppose this name is used?

2. Let g(x) = (x – 3) sin

Plot the graph of g using a window that includes y = 2 with x = 3 as a grid point. Then zoom in on the point (3, 2) by a factor of 10 in both the x- and y-directions. Sketch the resulting graph. Does g(x) seem to be approaching a limit as x approaches 3? If so, what does the limit equal? If not, explain why not.

3. Let h(x) = sin

Plot the graph of h using a window with a y-range of about 0 to 3 with x = 3 as a grid

point. Then zoom in on the point (3, 2) by a factor of 10 in the x-direction. Leave the y-scale the same. Sketch the resulting graph. Does h(x) appear to approach a limit as x approaches 3? If so, what does the limit equal? If not, explain why not.

OBJECTIVE Find the limit of f(x) as x approaches c if f(c) is undefined.

In Section 1-2 you learned that L is the limit of f(x) as x approaches c if and only if you can keep f(x) arbitrarily close to L by keeping x close enough to c but not equal to c. In this chapter you’ll acquire a deeper understanding of the meaning and properties of limits.

2-2 Graphical and Algebraic Approaches

Here is the formal definition of limit. You should commit this definition to memory so that you can say it orally and write it correctly without having to look at the text. As you progress through the chapter, the various parts of the definition will become clearer to you.

Notes:

• f(x) is pronounced “the limit of f(x) as x approaches c.”

• “For any number > 0” can also be read as “for any positive number .” The same is true for > 0.

• The optional words “no matter how small” help you focus on keeping f(x) close to L.

• The restriction “but x c” is needed because the value of f(c) may be undefined or different from the limit.

EXAMPLE 1 State whether the function graphed in Figure 2-2a has a limit at the given

x-value, and explain why or why not. If there is a limit, give its value. a. x = 1 b. x = 2 c. x = 3

OBJECTIVE Given a function f , state whether f(x) has a limit L as x approaches c, and if so, explain how close you can keep x to c for f(x) to stay within a given number units of L.

DEFINITION: Limit

L = f(x) if and only if

for any number > 0, no matter how small there is a number > 0 such that

if x is within units of c, but x c, then f(x) is within units of L.

In Figure 2-1a, you saw a function for which f(x) stays close to 3 when x is kept close to 2, even though f(2) itself is undefined. The number 3 fits the verbal definition of limit you learned in Section 1-2: You can keep f(x) arbitrarily close to 3 by keeping x close enough to 2 but not equal to 2. In this section you’ll use two Greek letters: (lowercase epsilon) to specify how arbitrarily close f(x) must be kept to the limit L, and (lowercase delta) to specify how close x must be kept to c in order to do this. The result leads to a formal definition of limit.

to the Definition of

Solution

f(x) = 5

c. If x is close to 3 on the left, then f(x) is close to 10. If x is close to 3 on the right, then f(x) is close to 8. Therefore, there is no one number you can keep f(x) close to just by keeping x close to 3 but not equal to 3. The fact that f(3) exists and is equal to 10 does not mean that 10 is the limit.

f(x) does not exist.

Figure 2-2a

Note: The discontinuity at x = 1 in Figure 2-2a is called a removable discontinuity. If f(1) were defined to be 2, there would no longer be a discontinuity. The discontinuity at x = 3 is called a step discontinuity. You cannot remove a step discontinuity simply by redefining the value of the function.

Figure 2-2b

Figure 2-2b shows the graph of function f for which f(2) is undefined.

a. What number does f(x) equal? Write the definition of this limit using proper limit terminology.

b. If = 0.6, estimate to one decimal place the largest possible value of you can use to keep f(x) within units of the limit by keeping x within units of 2 (but not equal to 2). c. What name is given to the missing point at

x = 2?

Solution

Figure 2-2c

a. f(x) = 4

4 = f(x) if and only if Use separate lines for the various parts of the definition.

for any number > 0, no matter how small, there is a number > 0 such that

if x is within units of 2, but x 2, then f(x) is within units of 4.

b. Because = 0.6, you can draw lines 0.6 unit above and below y = 4, as in Figure 2-2c. Where these lines cross the graph, go down to the x-axis and estimate the corresponding x-values to get x 1.6 and x 2.8.

So, x can go as far as 0.4 unit to the left of x = 2 and 0.8 unit to the right. The smaller of these units, 0.4, is the value of .

c. The graphical feature is called a removable discontinuity.

EXAMPLE 2

Section 2-2: Graphical and Algebraic Approaches to the Definition of Limit © 2005 Key Curriculum Press 35 a. If x is kept close to 1 but not equal to 1, you can make f(x) stay within units of 2, no matter how small is. This is true even though there is no value for f(1).

f(x) = 2

b. If x is kept close to 2, you can make f(x) stay within units of 5 no matter how small is. The fact that f(2) = 5 has no bearing on whether f(x) has a limit as x approaches 2.

Figure 2-2d shows the graph of

f(x) = (x – 3)1/3_{+ 2. The limit of f(x) as}

x approaches 3 is L = 2, the same as the value of f(3).

a. Find graphically the largest value of δ for which f(x) is within = 0.8 unit of 2 whenever x is kept within δ units of 3. b. Find the value of δ in part a algebraically. c. Substitute (3 + δ) for x and (2 + ) for f(x).

Solve algebraically for δ in terms of . Use the result to conclude that there is a positive value of δ for any positive value of , no matter how small is.

Solution _{a. Figure 2-2e shows the graph of f with horizontal lines plotted 0.8 unit}

above y = 2 and 0.8 unit below. Using the intersect feature of your grapher, you will find

x = 2.488 for y = 1.2 and x = 3.512 for y = 2.8 6

Figure 2-2e

Each of these values is 0.512 unit away from x = 3, so the maximum value of δ is 0.512.

b. By symmetry, the value of δ is the same on either side of x = 3. Substitute 2.8 for f(x) and (3 + δ) for x.

[(3 + δ) – 3]1/3₊₂₌_2.8

δ1/3₌_0.8

δ = 0.83₌_0.512

which agrees with the value found graphically. c. _{[(3 + δ) – 3]}1/3_{+ 2}₌_{2 +}

δ1/ 3₌

δ = 3

there is a positive value of δ for any positive number , no matter how small is.

Let f(x) = 0.2(2x_).

a. Plot the graph on your grapher. b. Find f(x)..

c. Find graphically the maximum value of δ you can use for = 0.5 at x = 3. d. Show algebraically that there is a positive value of δ for any 0, no

matter how small.

Solution _{a. Figure 2-2f shows the graph.}

b. By tracing to x = 3, you will find that f(3) = 1.6, which is the same as f(x). c. Plot the lines y = 1.1 and y = 2.1, which are = 0.5 unit above and below

1.6, as shown in Figure 2-2f.

EXAMPLE 3

EXAMPLE 4

Figure 2-2f

Use the intersect feature of your grapher to find the x-values where these lines cross the graph of f.

x = 2.4594... and x = 3.3923..., respectively The candidates for are 1= 3 – 2.4594... = 0.5403... and

2= 3.3923... – 3 = 0.3923... .

The largest possible value of is the smaller of these two, namely = 0.3923... .

d. Because the slope of the graph is positive and increasing as x increases, the more restrictive value of is the one on the positive side of x = 3.

Substituting (1.6 + ) for f(x) and (3 + ) for x gives 0.2(23 +δ ) = 1.6 +

(23+δ ₎₌_{8 + 5}

log(23+δ ) ₌_{log(8 + 5 )}

(3 + ) log 2 = log(8 + 5 )

Because (log 8) / (log 2) = 3, because 8 + 5 is greater than 8 for any positive number , and because log is an increasing function, the expression

log(8 + 5 ) / (log 2) is greater than the 3 that is subtracted from it. So will be positive for any positive number , no matter how small is.

Problem Set 2-2

Q1. Sketch the graph of y = 2x_.

Q2. Sketch the graph of y = cos x.

Q3. Sketch the graph of y = –0.5x + 3.

Q4. Sketch the graph of y = –x2_.

Q5. Sketch the graph of a function with a removable discontinuity at the point (2, 3).

Q6. Name a numerical method for estimating the value of a definite integral.

Q7. What graphical method can you use to estimate the value of a definite integral?

Q8. Write the graphical meaning of derivative.

Q9. Write the physical meaning of derivative.

Q10. If log3x =y, then

A. 3x₌_y _{B. 3}y₌_x _{C. x}3₌_y

D. y3₌_x _{E. x}y₌₃

1. Write the definition of limit without looking at the text. Then check the definition in this section. If any part of your definition is wrong, write the entire definition over again. Keep doing this until you can write the definition from memory without looking at the text.

2. What is the reason for the restriction “ . . . but x ≠ c . . .” in the definition of limit?

For Problems 3–12, state whether the function has a limit as x approaches c; if so, tell what the limit equals.

3. 4.

9. 10.

11. 12.

For Problems 13–18, photocopy or sketch the graph. For the point marked on the graph, use proper limit notation to write the limit of f(x). For the given value of , estimate to one decimal place the largest possible value of that you can use to keep f(x) within units of the marked point when x is within units of the value shown.

13. x = 3, = 0.5 14. x = 2, = 0.5

15. x = 6, = 0.7 16. x = 4, = 0.8

17. x = 5, = 0.3 18. x = 3, = 0.4

For Problems 19–24,

a. Plot the graph on your grapher. How does the graph relate to Problems 13–18? b. Find the limit of the function as

x approaches the given value.

c. Find the maximum value of that can be used for the given value of at the point. d. Calculate algebraically a positive value of

for any > 0, no matter how small. 19. f(x) = 5 – 2 sin(x – 3) x = 3, = 0.5 20. f(x) = (x – 2)3_{+ 3} x = 2, = 0.5 21. f(x) = 1 + x = 6, = 0.7 22. f(x) = 1 + 24–x x = 4, = 0.8 23. x = 5, = 0.3 24. f(x) = 6 – 2(x – 3)2/ 3 x = 3, = 0.4

25. Removable Discontinuity Problem 1: Function

Figure 2-2g

a. Show that f(2) has the indeterminate form 0/0. What feature does the graph of f have at x = 2? Do an appropriate calculation to show that 5 is the limit of f(x) as x approaches 2. b. Find the interval of x-values close to 2, but

not including 2, for which f(x) is within 0.1 unit of 5. Keep at least six decimal places for the x-values at the ends of the interval. Based on your answer, what is the largest value of for which f(x) is within = 0.1 unit of 5 when x is kept within unit of 2? c. Draw a sketch to show how the numbers L,

c, , and in the definition of limit are related to the graph of f in this problem. 26. Removable Discontinuity Problem 2: Function

is undefined at x = 2.

a. Plot the graph of f using a friendly window that includes x = 2 as a grid point. What do you notice about the shape of the graph? What feature do you notice at x = 2? What does the limit of f(x) appear to be as x approaches 2? b. Try to evaluate f(2) by direct substitution.

What form does your result take? What is

the name for an expression of the form taken by f(2)?

c. Algebraically find the limit of f(x) as x approaches 2 by factoring the numerator, then canceling the (x – 2) factors. How does the clause “ . . . but x ≠ c . . .” in the definition of limit allow you to do this canceling? d. “If x is within —?— unit of 2, but not equal

to 2, then f(x) is within 0.001 unit of the limit.” What is the largest number that can go in the blank? Show how you find this. e. Write the values for L, c, , and in the

definition of limit that appears in part d. 27. Limits Applied to Derivatives Problem: Suppose

you start driving off from a traffic light. Your distance, d(t), in feet, from where you started is given by

d(t) = 3t2

where t is time, in seconds, since you started. a. Figure 2-2h shows d(t) versus t. Write the average speed, m(t), as an algebraic fraction for the time interval from 4 seconds to t seconds.

Figure 2-2h

b. Plot the graph of function m on your grapher. Use a friendly window that includes t = 4. What feature does this graph have at the point t = 4? Sketch the graph.

c. Your speed at the instant t = 4 is the limit of your average speed as t approaches 4. What does this limit appear to equal? What are the units of this limit?

d. How close to 4 would you have to keep t for m(t) to be within 0.12 unit of the limit? (This is an easy problem if you simplify the algebraic fraction first.)

e. Explain why the results of this problem give the exact value for a derivative.

cancel the (x – 2) factors, and the equation becomes f(x) = x2_{– 6x + 13, x}_≠₂

So f is a quadratic function with a removable discontinuity at x = 2 (Figure 2-2g). The y-value at this missing point is the limit of f(x) as x approaches 2.

2-3 The Limit Theorems

f(4) is undefined because it has an indeterminate form.

Because the numerator is also zero, there may be a limit of f(x) as x approaches 4. Limits such as this arise when you try to find exact values of derivatives.

Simplifying the fraction before substituting 4 for x gives

=3x + 12,provided x ≠ 4

From Section 2-2, recall that 0/0 is called an indeterminate form. Its limit can be different numbers depending on just what expressions go to zero in the numerator and denominator. Fortunately, several properties (called the limit theorems) allow you to find such limits by making substitutions, as shown above. In this section you will learn these properties so that you can find exact values of derivatives and integrals the way Isaac Newton and Gottfried Leibniz did more than 300 years ago.

Limit of a Product or a Sum of Two Functions

Suppose that g(x) = 2x + 1 and h(x) = 5 – x. Let function f be defined by the product of g and h.

f(x) = g(x) · h(x) = (2x + 1)(5 – x)

Figure 2-3a

You are to find the limit of f(x) as x approaches 3. Figure 2-3a shows the graphs of functions f, g, and h. Direct substitution gives

f(3) = (2 · 3 + 1)(5 – 3) = (7)(2) = 14

The important idea concerning limits is that f(x) stays close to 14 when x is kept close to 3. You can demonstrate this fact by making a table of values of x, g(x), h(x), and f(x).

OBJECTIVE For the properties listed in the property box in this section, be able to state them, use them in a proof, and explain why they are true.

Surprisingly, you can find the limit by substituting 4 for x in the simplified expression

Suppose that f(x) is given by the algebraic fraction

You may have seen this fraction in Problem 27 of Problem Set 2-2. There is no value for f(4) because of division by zero. Substituting 4 for x gives

2.97 6.94 2.03 14.0882 When g(x) and h(x) are close

to 7 and 2, respectively, f(x) is close to14. 2.98 6.96 2.02 14.0592 2.99 6.98 2.01 14.0298 3.01 7.02 1.99 13.9698 3.02 7.04 1.98 13.9392 3.03 7.06 1.97 13.9082 3.04 7.08 1.96 13.8768

You can keep the product as close to 14 as you like by keeping x close enough to 3, even if x is not allowed to equal 3. From this information you should be able

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