62_CONSTRUCTING_ANTIDERIVATIVESANALYTICALLYWhat_Is_an

6.2 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY What Is an Antiderivative of f (x) = 0 ? A function whose derivative is zero everywhere on an interval must have a horizontal tangent line at every point of its graph, and the only way this can happen is if the function is constant. Alternatively, if we think of the derivative as a velocity, and if the velocity is always zero, then the object is standing still; the position function is constant. A rigorous proof of this result using the definition of the derivative is surprisingly subtle. (See the Constant Function Theorem.)

()

What Is the Most General Antiderivative of f ? We know that if a function f has an antiderivative F, then it has a family of antiderivatives of the form F (x) + C, where C is any constant. You might wonder if there are any others. To decide, suppose that we have two functions F and G with F′ = f and G′ = f: that is, F and G are both antiderivatives of the same function f. Since F′ = G′ we have (G − F)′ = 0. But this means that we must have G − F = C, so G(x) = F (x) + C, where C is a constant. Thus, any two antiderivatives of the same function differ only by a constant.

()

The Indefinite Integral All antiderivatives of f (x) are of the form F (x) + C. We introduce a notation for the general antiderivative that looks like the definite integral without the limits and is called the indefinite integral:

() It is important to understand the difference between

() The first is a number and the second is a family of functions. The word “integration” is frequently used for

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the process of finding the antiderivative as well as of finding the definite integral. The context usually makes clear which is intended.

What Is an Antiderivative of f (x) = k ? If k is a constant, the derivative of kx is k, so we have

() Using the indefinite integral notation, we have If k is constant,

()

Finding Antiderivatives Finding antiderivatives of functions is like taking square roots of numbers: if we pick a number at random, such as 7 or 493, we may have trouble finding its square root without a calculator. But if we happen to pick a number such as 25 or 64, which we know is a perfect square, then we can find its square root exactly. Similarly, if we pick a function which we recognize as a derivative, then we can find its antiderivative easily. For example, to find an antiderivative of f (x) = x, notice that 2x is the derivative of x2; this tells us that x2 is an antiderivative of 2x. If we divide by 2, then we guess that

() To check this statement, take the derivative of x2/2:

() What about an antiderivative of x2? The derivative of x3 is 3x2, so the derivative of x3/3 is 3x2/3 = x2. Thus,

()

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The pattern looks like

() (We assume n ≠ −1, or we would have x0/0, which does not make sense.) It is easy to check this formula by differentiation:

() In indefinite integral notation, we have shown that

() What about when n = −1? In other words, what is an antiderivative of 1/x? Fortunately, we know a function whose derivative is 1/x, namely, the natural logarithm. Thus, since

() we know that

() If x < 0, then ln x is not defined, so it can't be an antiderivative of 1/x. In this case, we can try ln(−x):

() so

()

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This means ln x is an antiderivative of 1/x if x > 0, and ln(−x) is an antiderivative of 1/x if x < 0. Since |x| = x when x > 0 and |x| = −x when x < 0, we can collapse these two formulas into:

() Therefore

() Since the exponential function is its own derivative, it is also its own antiderivative; thus

() Also, antiderivatives of the sine and cosine are easy to guess. Since

() we get

()

Example 1 Find ∫ (3x + x2) dx. Solution We know that x2/2 is an antiderivative of x and that x3/3 is an antiderivative of x2, so we expect

() You should always check your antiderivatives by differentiation—it's easy to do. Here

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()

The preceding example illustrates that the sum and constant multiplication rules of differentiation work in reverse:

Theorem 6.1: Properties of Antiderivatives: Sums and Constant Multiples In indefinite integral notation,

1.∫ (f (x) ± g(x)) dx = ∫ f (x) dx ± ∫ g(x) dx 2.∫ cf(x) dx = c ∫ f (x) dx.

In words, 1.An antiderivative of the sum (or difference) of two functions is the sum (or difference) of their antiderivatives. 2.An antiderivative of a constant times a function is the constant times an antiderivative of the function.

These properties are analogous to the properties for definite integrals given in Section 5.4, even though definite integrals are numbers and antiderivatives are functions.

Example 2 Find ∫ (sin x + 3 cos x) dx. Solution We break the antiderivative into two terms:

() Check by differentiating:

()

Using Antiderivatives to Compute Definite Integrals As we saw in Section 5.3, the Fundamental Theorem of Calculus gives us a way of calculating definite integrals. Denoting F (b) − F (a) by , the theorem says that if F′ = f and f is continuous, then

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()

To find , we first find F, and then calculate F (b) − F (a). This method of computing definite

integrals gives an exact answer. However, the method only works in situations where we can find the antiderivative F (x). This is not always easy; for example, none of the functions we have encountered so far is an antiderivative of sin(x2). Example 3

Compute using the Fundamental Theorem.

Solution Since F (x) = x3 is an antiderivative of f (x) = 3x2,

() gives

()

Notice in this example we used the antiderivative x3, but x3 + C works just as well because the constant C cancels out:

()

Example 4

Compute exactly.

Solution We use the Fundamental Theorem. Since F (θ) = tan θ is an antiderivative of f (θ) = 1/cos2 θ, we get

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()

Exercises and Problems for Section 6.2 EXERCISES 1. If p′(x) = q(x), write a statement involving an integral sign giving the relationship between p(x) and q(x).

2. If u′(x) = v(x), write a statement involving an integral sign giving the relationship between u(x) and v(x).

3. Which of (I)-(V) are antiderivatives of f (x) = ex/2?

I.ex/2

II.2ex/2

III.2e(1+x)/2

IV.2ex/2 + 1 V.

4. Which of (I)-(V) are antiderivatives of f (x) = 1/x?

I.ln x II.−1/x2

III.ln x + ln 3 IV.ln(2x) V.ln(x + 1)

5. Which of (I)-(V) are antiderivatives of

()

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I.−2 sin x cos x II.2 cos2 x sin2 x III.sin2 x IV.−cos2 x V.2 sin2 x + cos2 x

In Exercises 6-21, find an antiderivative. 6. f (x) = 5

7. f (t) = 5t

8. f (x) = x2

9. g(t) = t2 + t

10.

11.

12.

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13. h(t) = cos t

14.

15.

16. f (z) = ez

17. g(t) = sin t

18. f (t) = 2t2 + 3t3 + 4t4

19.

20.

21.

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In Exercises 22-33, find the general antiderivative. 22. f (t) = 6t

23. h(x) = x3 − x

24. f (x) = x2 − 4x + 7

25.

26. r(t) = t3 + 5t − 1

27. f (z) = z + ez

28. g(x) = sin x + cos x

29. h(x) = 4x3 − 7

30.

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31. p(t) = 2 + sin t

32.

33.

In Exercises 34-41, find an antiderivative F (x) with F′(x) = f (x) and F (0) = 0. Is there only one possible solution? 34. f (x) = 3

35. f (x) = 2x

36. f (x) = −7x

37. f (x) = 2 + 4x + 5x2

38.

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39. f (x) = x2

40.

41. f (x) = sin x

In Exercises 42-55, find the indefinite integrals. 42. ∫ (5x + 7) dx

43.

44. ∫ (2 + cos t) dt

45. ∫ 7ex dx

46. ∫ (3ex + 2 sin x) dt

47. ∫ (4ex − 3 sin x) dx

48.

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49. ∫ (x + 3)2 dx

50.

51.

52. ∫ (ex + 5) dx

53. ∫ t3(t2 + 1) dt

54.

55.

In Exercises 56-65, evaluate the definite integrals exactly [as in ], using the Fundamental Theorem,

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and numerically : 56.

57.

58.

59.

60.

61.

62.

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63.

64.

65.

In Exercises 66-75, decide if the statement is True or False by differentiating the right-hand side. 66.

67.

68.

69.

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70.

71.

72.

73.

74.

75.

PROBLEMS 76. Use the Fundamental Theorem to find the area under f (x) = x2 between x = 0 and x = 3.

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77. Find the exact area of the region bounded by the x-axis and the graph of y = x3 − x.

78.

Calculate the exact area above the graph of and below the graph of y = cos x. The curves intersect at .

79. Find the exact area of the shaded region in Figure 6.27 between y = 3x2 − 3 and the x-axis.

Figure 6.27

80. (a) Find the exact area between f (x) = x3 − 7x2 + 10x, the x-axis, x = 0, and x = 5.

(b)

Find exactly and interpret this integral in terms of areas.

81. Find the exact area between the curve y = ex − 2 and the x-axis for 0 ≤ x ≤ 2.

82. Find the exact area between the curves y = x2 and y = 2 − x2.

83. Find the exact area between the x-axis and the graph of f (x) = (x − 1)(x − 2)(x − 3).

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84. The area under on the interval 1 ≤ x ≤ b is equal to 6. Find the value of b using the Fundamental Theorem.

85. Use the Fundamental Theorem to find the value of b if the area under the graph of f (x) = 8x between x = 1 and x = b is equal to 192. Assume b > 1.

86. Find the exact positive value of c which makes the area under the graph of y = c(1 − x2) and above the x-axis equal to 1.

87. Sketch the parabola and the curve y = sin x, showing their points of intersection. Find the exact area between the two graphs.

88. Find the exact average value of on the interval 0 ≤ x ≤ 9. Illustrate your answer on a graph of

89. (a) What is the average value of f (t) = sin t over ? Why is this a reasonable answer?

(b) Find the average of f (t) = sin t over .

90.

Let where Q(3) = 12. Given that , find Q(8).

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91. Water is pumped into a cylindrical tank, standing vertically, at a decreasing rate given at time t minutes by

() The tank has radius 5 ft and is empty when t = 0. Find the depth of water in the tank at t = 4.

92. A car moves along a straight line with velocity, in feet/second, given by

() (a) Describe the car's motion in words. (When is it moving forward, backward, and so on?)

(b) The car's position is measured from its starting point. When is it farthest forward? Backward?

93. In drilling an oil well, the total cost, C, consists of fixed costs (independent of the depth of the well) and marginal costs, which depend on depth; drilling becomes more expensive, per meter, deeper into the earth. Suppose the fixed costs are 1,000,000 riyals (the riyal is the unit of currency of Saudi Arabia), and the marginal costs are

() where x is the depth in meters. Find the total cost of drilling a well x meters deep.

94. A helicopter rotor slows down at a constant rate from 350 revs/min to 260 revs/min in 1.5 minutes. (a) Find the angular acceleration (i.e. change in rev/min) during this time interval. What are the units of this acceleration?

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(b) Assuming the angular acceleration remains constant, how long does it take for the rotor to stop? (Measure time from the moment when speed was 350 revs/min.)

95.

Use the fact that (xx)′ = xx(1 + ln x) to evaluate exactly: .

96. Assuming that ∫g(x) dx = G(x) + C, where G(4) = 9, G(6) = 4, and G(9) = 6, evaluate the definite integral: (a)

(b)

(c)

For Problems 97-99, let ∫g(x) dx = G(x) + C. Which of (I)-(III), if any, is equal to the given integral? 97. ∫g(2x) dx

I.0.5G(0.5x) + C II.0.5G(2x) + C III.2G(0.5x) + C

98.

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∫cos(G(x)) g(x) dx I.sin(G(x)) g(x) + C II.sin(G(x)) G(x) + C III.sin(G(x)) + C

99. ∫ xg(x) dx

I.G(x2) + C II.xG(x) + C III.

Strengthen Your Understanding In Problems 100-101, explain what is wrong with the statement.

100.

101.

For all n, .

In Problems 102-103, give an example of: 102. Two different functions F (x) and G(x) that have the same derivative.

103. A function f (x) whose antiderivative F (x) has a graph which is a line with negative slope.

Are the statements in Problems 104-112 true or false? Give an explanation for your answer. 104. An antiderivative of is 2(x + 1)3/2.

105.

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An antiderivative of 3x2 is .

106. An antiderivative of 1/x is ln |x| + ln 2.

107. An antiderivative of is .

108. ∫ f (x) dx = (1/x) ∫ xf(x) dx.

109. If F (x) is an antiderivative of f (x) and G(x) = F (x) + 2, then G(x) is an antiderivative of f (x).

110. If F (x) and G(x) are two antiderivatives of f (x) for −∞ < x < ∞ and F (5) > G(5), then F (10) > G(10).

111. If F (x) is an antiderivative of f (x) and G(x) is an antiderivative of g(x), then F (x) ⋅ G(x) is an antiderivative of f (x) ⋅ g(x).

112. If F (x) and G(x) are both antiderivatives of f (x) on an interval, then F (x) − G(x) is a constant function.

Additional Problems In Problems 113-114, evaluate the integral using f (x) 4x−3.

AP113.

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AP114.

AP115. If An is the area between the curves y = x and y = xn, show that as n → ∞ and explain this result graphically.

AP116. Consider the area between the curve y = ex − 2 and the x-axis, between x = 0 and x = c for c > 0. Find the value of c making the area above the axis equal to the area below the axis.

AP117. The origin and the point (a, a) are at opposite corners of a square. Calculate the ratio of the areas of the two parts into which the curve divides the square.

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Prepare a 12-15 slide PowerPoint presentation that outlines your proposal to the stakeholders who would approve or deny your plan. Include the following:

1. Present the prevailing pressure that threatens organizational success.

2. Summarize your strategies for successfully implementing the change plan in order to address the internal or external pressure.

3. Propose strategies for effectively communicating to internal and external stakeholders. Identify what information needs to be disseminated and to which stakeholders. Discuss the communication skills and language necessary to be effective with both internal and external stakeholders. Describe the communication process and the strategies used to disseminate information throughout the implementation process.

4. Present a strategic plan for sustaining the change after the plan has been implemented.

5. Predict potential obstacles or unintended consequences and recommend strategies to overcome or respond to them.

While APA style is not required for the body of this assignment, solid academic writing is expected, and documentation of sources should be presented using APA formatting guidelines, which can be found in the APA Style Guide, located in the Student Success Center.

This assignment uses a rubric. Please review the rubric prior to beginning the assignment to become familiar with the expectations for successful completion. (BELOW)

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Benchmark: Presentation – Communicating and Sustaining Change

	1 Unsatisfactory 0.00%	2 Less than Satisfactory 65.00%	3 Satisfactory 75.00%	4 Good 85.00%	5 Excellent 100.00%
100.0 %Content
5.0 %Prevailing Pressures Presented	Environmental and organizational pressures are not presented.	At least one environmental or organizational pressure is referenced.	Environmental and organizational pressures are described generally.	Key environmental and organizational pressures are described.	Major and minor environmental and organizational pressures are described.
5.0 %Implementation Strategies Presented	Implementation strategies are not presented.	The specific steps needed to implement the change vision are missing. No strategies are described clearly.	General steps for implementation are presented. One or two key strategies are presented for implementing change.	The steps and strategies for implement are outlined.	The steps for implement are clearly described. Strong strategies for the implementation of each step are proposed.
10.0 %Effective Communication Strategies Presented 6.2: Evaluate methods for communicating and implementing change within organizations.	Communication strategies are not presented.	The need for communication is acknowledged, but clear communication strategies are not presented.	A general communication strategy is presented, but the process for disseminating information is unclear. Strategies do not include both internal and external stakeholders.	Strategies for communication are presented. A realistic process for disseminating information is proposed. Strategies include both internal and external stakeholders.	Strong strategies for communication are proposed. A clear and well-developed process for disseminating information is proposed. Strategies effectively include both internal and external stakeholders.
10.0 %Strategic Plan for Sustaining Change Presented	A strategic plan for sustaining change is not presented.	A plan for sustaining change is acknowledged. No strategy is described clearly.	A general plan for sustaining change is discussed, but it is incomplete or lacks important details.	A plan for sustaining change is described. The plan addresses most major areas for sustaining change within the organization.	A clear plan for sustaining change is described. The plan addresses all major areas for sustaining change within the organization.
10.0 %Prediction of Obstacles and Unintended Consequences Presented	Obstacles and unintended consequences are not predicted.	A vague prediction for obstacles and unintended consequences is presented.	A general prediction for obstacles and unintended consequences is presented.	A realistic prediction for obstacles and unintended consequences is presented.	A sound prediction for obstacles and unintended consequences is presented.
30.0 %Presentation of Content	The content lacks a clear point of view and logical sequence of information. Includes little persuasive information. Sequencing of ideas is unclear.	The content is vague in conveying a point of view and does not create a strong sense of purpose. Includes some persuasive information.	The presentation slides are generally competent, but ideas may show some inconsistency in organization and/or in their relationships to each other.	The content is written with a logical progression of ideas and supporting information exhibiting a unity, coherence, and cohesiveness. Includes persuasive information from reliable sources.	The content is written clearly and concisely. Ideas universally progress and relate to each other. The project includes motivating questions and advanced organizers. The project gives the audience a clear sense of the main idea.
10.0 %Layout	The layout is cluttered, confusing, and does not use spacing, headings, and subheadings to enhance the readability. The text is extremely difficult to read with long blocks of text, small point size for fonts, and inappropriate contrasting colors. Poor use of headings, subheadings, indentations, or bold formatting is evident.	The layout shows some structure, but appears cluttered and busy or distracting with large gaps of white space or a distracting background. Overall readability is difficult due to lengthy paragraphs, too many different fonts, dark or busy background, overuse of bold, or lack of appropriate indentations of text.	The layout uses horizontal and vertical white space appropriately. Sometimes the fonts are easy to read, but in a few places the use of fonts, italics, bold, long paragraphs, color, or busy background detracts and does not enhance readability.	The layout background and text complement each other and enable the content to be easily read. The fonts are easy to read and point size varies appropriately for headings and text.	The layout is visually pleasing and contributes to the overall message with appropriate use of headings, subheadings, and white space. Text is appropriate in length for the target audience and to the point. The background and colors enhance the readability of the text.
10.0 %Language Use and Audience Awareness (includes sentence construction, word choice, etc.)	Inappropriate word choice and lack of variety in language use are evident. Writer appears to be unaware of audience. Use of primer prose indicates writer either does not apply figures of speech or uses them inappropriately.	Some distracting inconsistencies in language choice (register) or word choice are present. The writer exhibits some lack of control in using figures of speech appropriately.	Language is appropriate to the targeted audience for the most part.	The writer is clearly aware of audience, uses a variety of appropriate vocabulary for the targeted audience, and uses figures of speech to communicate clearly.	The writer uses a variety of sentence constructions, figures of speech, and word choice in distinctive and creative ways that are appropriate to purpose, discipline, and scope.
5.0 %Mechanics of Writing (includes spelling, punctuation, grammar, language use)	Slide errors are pervasive enough that they impede communication of meaning.	Frequent and repetitive mechanical errors distract the reader.	Some mechanical errors or typos are present, but they are not overly distracting to the reader.	Slides are largely free of mechanical errors, although a few may be present.	Writer is clearly in control of standard, written, academic English.
5.0 %Documentation of Sources (citations, footnotes, references, bibliography, etc., as appropriate to assignment and style)	Sources are not documented.	Documentation of sources is inconsistent or incorrect, as appropriate to assignment and style, with numerous formatting errors.	Sources are documented, as appropriate to assignment and style, although some formatting errors may be present.	Sources are documented, as appropriate to assignment and style, and format is mostly correct.	Sources are completely and correctly documented, as appropriate to assignment and style, and format is free of error.
100 %Total Weightage

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