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Definition of an Antiderivative Definition of an Antiderivative For a function f , an antiderivative of f is a function F for which F 0 (x ) = f (x ) Clint Lee Math 122 Lecture 1: Antiderivatives 2/11 Antiderivatives Are Not Unique Suppose that F and G are both antiderivatives of the same function f . Then G 0 (x ) = F 0 (x ) = f (x ) So that 0 G 0 (x ) − F 0 (x ) = 0 ⇒ [G (x ) − F (x )] = 0 By the Mean Value Theorem the only differentiable functions whose first derivative is zero for all x in their domains, are constant functions. Hence, there is a constant C for which G (x ) − F (x ) = C ⇒ G (x ) = F (x ) + C Clint Lee Math 122 Lecture 1: Antiderivatives 3/11 The Most General Antiderivative The Most General Antiderivative I If F is an antiderivative of a function f , then any other antiderivative of f is of the form G (x ) = F (x ) + C for some constant C. I Hence, if F is any antiderivative of a function f , then the most general antiderivative of the function f is F (x ) + C where C is an arbitrary constant. Clint Lee Math 122 Lecture 1: Antiderivatives 4/11 The Indefinite Integral The Indefinite Integral If F is an antiderivative of a function f , then the indefinite integral of f is Z f (x ) dx = F (x ) + C The operator Z . . . dx is an instruction to find any antiderivative of whatever is between the (integral) sign and the dx, and then add an arbitrary constant. Clint Lee Math 122 Lecture 1: Antiderivatives R 5/11 Some Antiderivative (Integration) Formulas Every differentiation formula gives a corresponding antiderivative (integration) formula. Some Integration Formulas Z Z Z x n dx = eax 1 Z x n +1 + C , n 6 = − 1 n+1 1 ax dx = e + C a Z sec2 x dx = tan x + C Z x −1 dx = Z 1 dx = ln |x | + C x sin x dx = − cos x + C 1 1 + x2 dx = arctan x + C Verify any of these formulas by differentiating the righthand side to obtain the function under the integral sign, the integrand. Clint Lee Math 122 Lecture 1: Antiderivatives 6/11 The General Solution to a Differential Equation An antiderivative of a function f is a solution to the differential equation dy = f (x ) dx If F is an antiderivative of the function f , then one possible solution is y = F (x ). The general solution is the most general antiderivative of the function f , that is, y = F (x ) + C where C is an arbitrary constant. Clint Lee Math 122 Lecture 1: Antiderivatives 7/11 A Family of Functions Suppose that the graph of one solution to the differential equation y dy = f (x ) dx looks like this. The graph of any other solution is the graph of F shifted vertically by the constant C. Further, for any x the slopes of the curves are equal. In either case, the curves are parallel. Clint Lee F(x) + 1 F(x y F(x) F x x Math 122 Lecture 1: Antiderivatives 8/11 A Family of Functions The graphs of all possible solutions to the differential equation y dy = f (x ) dx F(x) + 1 F(x y F(x) F x x form a family of functions, all of whose graphs are parallel. At each x the slopes of all the graphs are equal. Clint Lee Math 122 Lecture 1: Antiderivatives 9/11 A Particular Solution The graphs in the family of functions representing the general solution to the differential equation y F(x) + 1 F(x y F(x) dy = f (x ) dx F y0 are all parallel. So they do not intersect. Hence, to select a particular solution it is only necessary to specify one point on the graph, say (x0 , y0 ). This information is sufficient to determine a value for the arbitrary constant C in the general solution. Clint Lee x Math 122 Lecture 1: Antiderivatives x0 x 10/11 The Initial Value Problem The Initial Value Problem For a function f , the differential equation dy = F 0 (x ) = f (x ) dx together with the initial condition F (x0 ) = y0 determines exactly one, a unique, particular solution. The problem F 0 (x ) = f (x ) where F (x0 ) = y0 is an initial value problem. Clint Lee Math 122 Lecture 1: Antiderivatives 11/11