Integrals of Exponential Functions

3 Key excerpts on "Integrals of Exponential Functions"

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  eBook - PDF

  Applied Calculus

  • Geoffrey Berresford, Andrew Rockett(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  c. To find distance, we speed. d. To find acceleration, we speed. 69. For each of the following casual, everyday sentences, fill in the blank with one of the words differentiate or integrate , whichever is more appropriate. a. To find marginal cost, we cost. b. To find profit, we marginal profit. c. To find how fast a stock price is rising, we price. d. To find a population, we how fast it is growing. Introduction In the previous section we defined integration as the reverse of differentiation, and we introduced several integration formulas. In this section we develop integration formulas involving logarithmic and exponential functions. One of these formulas will answer a question that we could not answer earlier—namely, how to integrate x 2 1 , the only power not covered by the Power Rule. The Integral # e ax dx On page 279 we saw that to differentiate e ax , we multiply by a to get ae ax . Therefore, to integrate e ax , the reverse process, we must divide by a. That is, for any a Þ 0 : 5.2 Integration Using Logarithmic and Exponential Functions Integrating an Exponential Function # e ax dx 5 1 a e ax 1 C Brief Example # e 2 x dx 5 1 2 e 2 x 1 C Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 320 Chapter 5 Integration and Its Applications In words: The integral of e to a constant times x is 1 over the constant times the original function (plus C ). As always, we may check the answer by differentiation. d dx a 1 2 e 2 x 1 C b 5 1 2 # 2 # e 2 x 5 e 2 x Using d dx e ax 5 ae ax and canceling the 2’s The result is the integrand e 2x , so the integration is correct.

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  eBook - PDF

  Calculus: An Applied Approach, Brief, International Metric Edition

  • Ron Larson(Author)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Integrating an Exponential Function Find integral.alt1 5 xe -x 2 dx . SOLUTION Let u = -x 2 ; then du H20862 dx = -2 x . You can create the factor -2 x in the integrand by multiplying and dividing by -2. integral.alt1 5 xe -x 2 dx = integral.alt1 parenleft.alt4 -5 2 parenright.alt4 e -x 2 ( -2 x ) d x Multiply and divide by -2. = -5 2 integral.alt1 e -x 2 ( -2 x ) d x Factor -5 2 out of the integrand. = -5 2 integral.alt1 e u du dx d x Substitute u and du dx . = -5 2 e u + C General Exponential Rule = -5 2 e -x 2 + C Substitute for u . Checkpoint 3 Worked-out solution available at LarsonAppliedCalculus.com Find integral.alt1 4 xe x 2 dx . Remember that you cannot introduce a missing variable in the integrand. For instance, you cannot find integral.alt1 e x 2 d x by multiplying and dividing by 2 x and then factoring 1 H20862 ( 2 x ) out of the integrand. That is, integral.alt1 e x 2 dx ≠ 1 2 x integral.alt1 e x 2 ( 2 x ) dx . ALGEBRA T U TOR For help on the algebra in Example 3, see Example 1(d) in the Chapter 5 Algebra Tutor , on page 366. TECH T U TOR If you use a symbolic integration utility to find antiderivatives of exponential or logarithmic functions, you can easily obtain results that are beyond the scope of this course. For instance, the antiderivative of e x 2 involves the imaginary unit i and the probability function called “ERF.” In this course, you are not expected to interpret or use such results. Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 Section 5.3 Exponential and Logarithmic Integrals 333 Using the Log Rule When the Power Rules for integration were introduced in Sections 5.1 and 5.2, you saw that they work for powers other than n = -1.

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  Available until 5 Dec |Learn more

  Basic Mathematics for Economists

  • Mike Rosser, Piotr Lis(Authors)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  14 Exponential functions, continuous growth and differential equations DOI: 10.4324/9781315641713-14

  Learning objectives

  After completing this chapter students should be able to:

  • Use the exponential function and natural logarithms to derive the final sum, initial sum and growth rate when continuous growth takes place.
  • Compare and contrast continuous and discrete growth rates.
  • Set up and solve linear first-order differential equations.
  • Use differential equation solutions to predict values in basic market and macroeconomic models.
  • Comment on the stability of economic models where growth is continuous.

  14.1 Continuous Growth and the Exponential Function

  In Chapter 7 , growth was treated as a process taking place at discrete time intervals. In this chapter we shall analyse growth as a continuous process, but before we do this it is first necessary to understand the concepts of exponential functions and natural logarithms. The term ‘exponential function’ is usually used to describe the specific natural exponential function explained below. However, it can also be used to describe any function in the format

  y =

  A x

  where A is a constant and A > 1

  This is known as an exponential function to base A. When x increases in value this function obviously increases in value very rapidly if A is a number significantly greater than 1. On the other hand, the value of A

  x

  approaches zero if x takes on larger and larger negative values. For all values of A it can be deduced from the general rules for exponents (explained in Chapter 2 ) that A0 = 1 and A1 = A.

  Example 14.1

  Find the values of y = A

  x

  when A is 2 and x takes the following values:

  1. 0.5,
  2. 1,
  3. 3,
  4. 10,
  5. 0,
  6. −0.5,
  7. −1, and
  8. −3

  Solutions

  1. A0.5 = 1.41
  2. A1 = 2
  3. A3 = 8
  4. A10 = 1024
  5. A0 = 1
  6. A−0.5 = 0.71
  7. A−1 = 0.50
  8. A−3 = 0.13

  The natural exponential function

  In mathematics there is a special number which yields several useful results when used as a base for an exponential function and is usually represented by the letter ‘e’. This number is

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