2 Objectives Write the general solution of a differential equation. Use indefinite integral notation for antiderivatives.Use basic integration rules to find antiderivatives.Find a particular solution of a differential equation.
3 We will also learn some applications. Think About ItSuppose this is the graph of the derivative of a functionWhat do we know about the original function?Critical numbersWhere it is increasing, decreasingWhat do we not know?f '(x)In this chapter we will learn how to determine the original function when given the derivative.We will also learn some applications.3
4 Antiderivatives Suppose you were asked to find a function F whose derivative is f(x) = 3x2. From your knowledge of derivatives,you would probably say thatThe function F is an antiderivative of f .
5 Anti-Derivatives Derivatives give us the rate of change of a function What if we know the rate of change …Can we find the original function?If F '(x) = f(x)Then F(x) is an antiderivative of f(x)Example – let F(x) = 12x2Then F '(x) = 24x = f(x)So F(x) = 12x2 is the antiderivative of f(x) = 24x5
6 Finding An Antiderivative Given f(x) = 12x3What is the antiderivative, F(x)?Use the power rule backwardsRecall that for f(x) = xn … f '(x) = n • x n – 1That is …Multiply the expression by the exponentDecrease exponent by 1Now do opposite (in opposite order)Increase exponent by 1Divide expression by new exponent6
13 Family of Antiderivatives Consider a family of parabolasf(x) = x2 + n which differ only by value of nNote that f '(x) is the same for each version of fNow go the other way …The antiderivative of 2x must be different for each of the original functionsSo when we take an antiderivativeWe specify F(x) + CWhere C is an arbitrary constantThis indicates that multiple antiderivatives could exist from one derivative13
14 Example y’ = 2x y = (2)(1/2)x1+1 =x2 what are we missing? Could we have started with y = x or y = x ??When we find the antiderivative, we need to remember to account for the constant that could have been in the original function.So we get y = x2 + C
15 Indefinite IntegralThe family of antiderivatives of a function f indicated byThe symbol is a stylized S to indicate summation, we will discuss this further in later sections of this chapter.
19 Properties of Indefinite Integrals The power ruleThe integral of a sum (difference) is the sum (difference) of the integrals19
20 Properties of Indefinite Integrals The derivative of the indefinite integral is the original functionA constant can be factored out of the integral
21 Try It OutDetermine the indefinite integrals as specified below
22 Differential Equations We will study this topic in chapter 6.For now, here’s a taste of it.
23 Differential Equations The family of functions represented by G is the general antiderivative of f, and G(x) = x2 + C is the general solution of the differential equationG'(x) = 2x Differential equationA differential equation in x and y is an equation that involves x, y, and derivatives of y.For instance, y' = 3x and y' = x2 + 1 are examples of differential equations.
24 Example 1 – Solving a Differential Equation Find the general solution of the differential equation y' = 2.Solution:To begin, you need to find a function whose derivative is 2.One such function isy = 2x x is an antiderivative of 2.Now, you can use Theorem 4.1 to conclude that the general solution of the differential equation isy = 2x + C General solution
25 Example 1 – Solutioncont’dThe graphs of several functions of the form y = 2x + Care shown in Figure 4.1.Figure 4.1
26 Notation for Antiderivatives When solving a differential equation of the formit is convenient to write it in the equivalent differential formThe operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) and isdenoted by an integral sign ∫.
27 Notation for Antiderivatives The general solution is denoted byThe expression ∫f(x)dx is read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative.
28 Basic Integration Rules (p.250) The inverse nature of integration and differentiation can be verified by substituting F'(x) for f(x) in the indefinite integration definition to obtainMoreover, if ∫f(x)dx = F(x) + C, then
29 Basic Integration Rules These two equations allow you to obtain integration formulas directly from differentiation formulas, as shown in the following summary.
31 Basic Integration Rules Note that the general pattern of integration is similar to that of differentiation.
32 Example 2 – Applying the Basic Integration Rules Describe the antiderivatives of 3x.Solution:So, the antiderivatives of 3x are of the form where C is any constant.
33 Initial Conditions and Particular Solutions You have already seen that the equation y = ∫f(x)dx hasmany solutions (each differing from the others by aconstant).This means that the graphs of any two antiderivatives of fare vertical translations of each other.
34 Initial Conditions and Particular Solutions For example, Figure 4.2 shows thegraphs of several antiderivativesof the formfor various integer values of C.Each of these antiderivatives is a solutionof the differential equationFigure 4.2
35 Initial Conditions and Particular Solutions In many applications of integration, you are given enoughinformation to determine a particular solution. To do this,you need only know the value of y = F(x) for one value of x.This information is called an initial condition.For example, in Figure 4.2, only one curve passes through the point (2, 4).To find this curve, you can use the following information.F(x) = x3 – x + C General solutionF(2) = Initial condition
36 Initial Conditions and Particular Solutions By using the initial condition in the general solution, you can determine that F(2) = 8 – 2 + C = 4, which implies that C = –2.So, you obtainF(x) = x3 – x – Particular solution
37 Example 7 – Finding a Particular Solution Find the general solution ofand find the particular solution that satisfies the initialcondition F(1) = 0.Solution:To find the general solution, integrate to obtain
38 Example 7 – Solution Using the initial condition F(1) = 0, you cont’dUsing the initial condition F(1) = 0, youcan solve for C as follows.So, the particular solution, as shownin Figure 4.3, isFigure 4.3
39 Homework:Pg #2-44EGreat to try: p.255#1,4,9,11,13,14,22,31,34,35,38,40,43,47,49,50,59,62,63,65,66,68,71,72,87,90,92,94,95This is the bare minimum. It is recommended that you do all of the problems in each section to ensure mastery of the concepts. (this includes the word problems!)