1 Lesson 11: Antiderivatives of Elementary Functions


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1 1 Lesson 11: Antiderivatives of Elementary Functions Chapter 6 Material: pages in the textbook: The material in this lesson covers The definition of the antiderivative of a function of one variable. Development of algebraic formulas for computing antiderivatives. In particular, linearity when computing antiderivatives is presented. Development of simple formulas for antiderivatives based on differentiation formulas is presented The differentiation formula approach results in the determination of a general way to compute antiderivatives of polynomials. However, the use of differentiation formulas for all elementary functions will not work. Notation for the antiderivative using the integral sign if presented in this lesson. Prerequisite Content for Lesson 9 Students should understand all of the content through Chapter 4 of the textbook. The techniques developed in Chapter 4 are used development of general antiderivatives in the first two sections of this chapter. Students will need to use the linearity of the derivative operation when determining forms for antiderivatives. To be successful in this lesson, students must know how to algebraically manipulate polynomials, exponential and logarithm functions, hyperbolic trigonometric functions, and simple rational functions of polynomials. This lesson does not cover methods of integration. The lesson involves inverting the differentiation of elementary functions in the simplest of cases. So, the algebraic work and use of differentiation formulas involves noticing a relationship between a function and a derivative. Goals and Objectives for Lesson 11 Students who complete this lesson should be able to: identify the computation of an antiderivative as reversing differentiation, compute antiderivatives of many elementary functions, use the linearity of integration methods in the determination of general forms of antiderivatives, develop a table of antiderivatives of elementary functions, determine the general form of an antiderivative and understand the addition of a constant of integration, and determine the value of the constant of integration in the case when an additional or auxiliary condition is specified in the statement of the problem. The goals and objectives of this lesson involve a type of pattern matching due to the development of antiderivatives through differentiation formulas. The idea is that since students know certain formulas for differentiation, antiderivatives for some functions can be constructed using this relationship. 1
2 1.1 Lecture Notes for Lesson 11  Day 1 Lesson 11 begins the process of undoing differentiation. The techniques in this lesson are based on noticing relationships between functions and their derivatives. The lesson does not cover integration via substitution, integration by parts, or integration via partial fractions. These are covered in later lesson(s). The example (page 2378) in the introduction to the chapter is simple, but illustrates the nonuniqueness of antiderivatives in that there are three different functions that result in the same derivative. The difference is that each of the three functions has a different constant value. The constant shows up in the definition of the antiderivative of a function and is an important part of the result. Definition of an Antiderivative The definition of an antiderivative given in the textbook is in terms of the relationship that if F (x) = f(x) then F (x) is an antiderivative of f(x). This seems to work best for the type of presentation used in a Business Calculus course like Math 1100 at USU. The antiderivative approach allows students the ability to use integration without the use of summations. Note that if an antiderivative is computed that the result can be tested by differentiating the result. If the differentiation produces the original function then the antiderivative is probably ok. Students should get in the habit of checking an antiderivative using differentiation. Antiderivatives Based on Differentiation Formulas The basic idea of this section is to use known formulas for derivatives to build tables of antiderivatives. Most of the initial formulas are easy to apply due to the simple forms. The initial results involve the computation of antiderivatives of polynomials and exponential and logarithm functions. Students should understand that more formulas will be developed in the next lessons. However, the methods of integration presented in the next lesson will require a lot more work. Algebraic Rules for Antiderivatives Students must understand the algebraic rules for computing antiderivatives. In this lesson the rules related to linearity are presented. That is a constant multiple of a derivative is the same as the derivative of the constant times the the function and the sum of derivatives of two functions is the derivative of the sum of the functions. 2
3 1.2 Lecture Notes for Lesson 11  Day 2 The material in this lecture should include the determination of antiderivatives for basic elementary functions. The process used in each of the examples is to use differentiation formulas to directly determine an antiderivative from a given function. Note that initially, the formulas developed are relatively simple. The first formulas are simple monomials that are referred to as power functions. Once these formulas are developed, antiderivative formulas for polynomials can be developed. After this, more complicated formulas will continue to be developed until almost all types of elementary functions have been treated. Power Rules for Antiderivatives There is basically one rule for computing antiderivatives of powers. This is relatively easy and should be easy to explain to students. The rule does a basic inversion of the rule associated with differentiation of powers of an independent variable. Antiderivatives of Polynomials This is a simple, but important section. Students need to be able to work with polynomials in many applications in business and economics courses. Students should be able to compute antiderivatives of polynomials term by term very quickly. Exponential and Logarithm Rules for Antiderivatives Without the chain rule it is a bit difficult to do a lot with exponential and logarithm functions. Students should be able to guess the form of the antiderivative if the function is not too complicated. This is not the best way to compute antiderivatives. However, in the next lesson the use of substitution will be addressed which will take out all of the guess work in most cases. 3
4 1.3 Lecture Notes for Lesson 11  Day 3 Hyperbolic Trigonometric Rules for Antiderivatives Use the tables of formulas for derivatives of the hyperbolic trigonometric functions to determine antiderivatives of simple combinations of these functions. As with exponential functions, unless the form is pretty simple, it is difficult to guess an antiderivative. The methods in the next lesson will help with a formal determination of antiderivatives of hyperbolic trigonometric functions. Antiderivative Forms for Simple Rational Functions There are only a few forms that can be used in this section. The only types of functions that can be used at this point involve simple linear factors in the denominator. However, once partial fraction decomposition is introduced, students will be able to do a lot more. A Simple Notation for Antiderivatives The notation to emphasize is F = f which tells us to compute an antiderivative of f. following. Students should also get used to using notation like the F (x) = x f(t) dt if F (x) = f(x). Not that t is a variable of integration and could be replaced with any integration variable. The notation will be reinforced in the next two lessons. Antiderivatives for Other Elementary Functions Students should be made aware that the determination of antiderivatives to this point has involved undoing simple differentiation formulas. There are a lot of elementary functions that cannot be integrated using the simple formulas. Make sure that students know that other methods of integration will also be covered. One of these methods is substitution or making a change of variables which is used to invert the chain rule in differentiation. In addition, integration by parts which inverts the product rule will be developed. Finally, partial fraction decomposition will be used to integrate all rational functions of polynomials in its most general form. 4
5 1.4 Homework Verifying the Antiderivative of a Function Problem 1: In each of the problems verify that the derivative of the function F (x) matches the given function, f(x). a. F (x) = x 3 x 2 + x 1, f(x) = 3 x 2 2 x + 1 b. F (x) = x e 2 x + x 1, f(x) = e 2 x (1 + 2 x) + 1 c. F (x) = ln(2 x) + cosh 2 (3 x) + 42, f(x) = sinh(3 x) cosh(3 x) 2 x 5
6 Nonuniqueness of Antiderivatives Problem 2: Show that each of the following functions is an antiderivative of the function, f(x) = 4x 3 2x 2 +x 1 a. F (x) = x x x2 x + 6 b. F (x) = x x x2 x 42 6
7 Specification of the Constant of Integration Problem 3: The marginal revenue function R (x) = x x 2 in a given mathematical model is given. If the revenue for producing x = 120 items is 10, 000, determine the unique antiderivative that includes this specific given condition. 7
8 Antiderivatives of Powers Problem 4: Compute antiderivatives of the following power functions. a. f(x) = x 38 b. f(x) = x 4/3 c. f(x) = x
9 Antiderivatives of Polynomials Problem 5: Compute antiderivatives for the following polynomials. a. f(x) = 5 x 3 6 x 7 b. f(x) = 15 4 x + 16 x 2 8 x 3 c. f(x) = e x 3 π x 2 9
10 Antiderivatives of Exponential Functions Problem 6: Compute antiderivatives for the following exponential functions a. f(x) = 2 x 10 x b. f(x) = e x+1 c. f(x) = e 2 x 10
11 Antiderivatives and Logarithm Functions Problem 7: For each of the functions given below, compute the antiderivative based on the logarithm differentiation formulas. Remember to use linearity and other properties. a. f(x) = 1 2x b. f(x) = x 11
12 Using a Formula to Compute Antiderivatives Problem 8: Given a function f(x) = ln(a x) + b a rule for computing the antiderivative of f is given as F (x) = x ln(a x) x + b x + B where B is the constant of integration. Verify that the formula is correct by computing the derivative of F (x) and comparing F (x) to f(x). Then compute the antiderivative of f(x) = ln(3 x) + 7 using the general formula. 12
13 Antiderivatives of Hyperbolic Trigonometric Functions Problem 9: Compute antiderivatives of the following hyperbolic trigonometric functions. a. f(x) = cosh(x) sinh(x) b. f(x) = sech 2 (x) + 42 c. f(x) = sech(x) cosh(x) 13
14 Antiderivatives of Simple Rational Functions Problem 10: Compute antiderivatives for each of the following simple rational functions of polynomials. a. f(x) = 7 x + 3 b. f(x) = 5 4 x c. f(x) = 2 x 11 14
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