2 Definition of an Antiderivative In many cases, we would like to know what is the function F(x) whose derivative is a given function f(x). Or,F’(x) = f(x)For instance, if a given function f(x) = –sinx, then the derivative of F(x) = cosx + 4 is the given function f(x).
3 Definition of an Antiderivative Notice that even we could find the antiderivative of a given function, the answer is not unique!If f(x) = sinx + cosxthen F1(x) = sinx – cosx – 1F2(x) = sinx – cosx + 4F3(x) = sinx – cosx – 9This is why F(x) is called an antiderivative of f(x), rather then the antiderivative of f(x). In fact, all antideritives of f(x) differs at a constant.
4 Antiderivatives and Indefinite Integration Differentiable FunctionsALL functionsDderivative operatorantiderivative operator/ indefinite integral operatorThis expression denotes (all) the antiderivatives of f(x).is read as “the antiderivative of f with respect to x” or “the indefinite integral of f with respect to x”.
5 Theorem 4.1 Representation of Antiderivatives This Theorem tells us that we can represent the entire family of antideritives of a given function by adding a constant to a known antiderivative.If f(x) = sinx + cosxthen F(x) = sinx – cosx + C
6 Theorem 4.1 Representation of Antiderivatives The constant C is called the constant of integration. The family of functions represented by G is the general antiderivative of f.If f(x) = sinx + cosxthen F(x) = sinx – cosx + C is the general solution of the differential equation F ’(x) = sinx + cosx
7 Notation of Antiderivatives Let F(x) be an antiderivative of f(x), and y be all antiderivatives f(x), then(*)We differentiate (*), thenThe operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) denoted by the sign
8 Notation of Antiderivatives Variable of IntegrationConstant of IntegrationIntegrandsingle functionbag of functions
9 Indefinite Integration Example Find the antiderivatives of 5xSolutionsingle functionbag of functionsiff F’(x) = f(x)
10 ExampleExample Find the antiderivatives of 3Solutionconstant
11 Example 3 Find the antiderivatives of the following andSolutionm is not necessary to be an integer. mψR
22 Initial Conditions and Particular Solutions In the beginning of this section, we have already known that all antideritives of f(x) differs at a constant. This means that the graphs of any two antiderivatives of f are VERTICAL translation of each other.C=2C=1C=0C=–1C=–2
23 Initial Conditions and Particular Solutions Graph must pass through point (0, –2).C=2So, x = 0, y = –2 is the solution of the equationC=1C=0C=–1Or,C=–2
24 (To find a particular solution, we need an initial condition) Example 7 Find f(x) if f ’(x) = 6x – 4 and f(1) = 2.differential equationinitial conditionuse this to get a bag of functions containing f(x)use this to reach into bag and pull out a particular f(x)
25 Example 7 Find f(x) if f’ (x) = 6x – 4 and f(1) = 2. Solut1ioncheck:
26 Example Example 8 Find f(x) if Solut1ion Answer: differential equation initial conditionSolut1ionAnswer:
30 Example Example 10 Find the antiderivatives of the following Solut1ion check:
31 Motion Along a Line s(t) position function v(t) = s’ (t) v(t) velocity a(t) = v’ (t)a(t)accelerationv(t) = s’ (t)a(t) = v’ (t)a(t) = s’’ (t)
32 Free Falling ObjectsNOTE Velocity is negative when falling (or positive when thrown up)acceleration due to gravity
33 ExampleExample The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as function of the time t in the seconds. How long will it take the rock to hit the canyon floor?Solut1ionLet h(t), v(t) and a(t) be the height, the velocity, and the acceleration of the rock at time t. Then
34 ExampleSolut1ionLet h(t), v(t) and a(t) be the height, the velocity, and the acceleration of the rock at time t. ThenSo
35 ExampleSolut1ionWhen h(t) = 0, the rock will hit the canyon floor. So(The negative root will be discard)