Aim: What is the flip side of the derivative?

1. Aim: What is the flip side of the derivative? Do Now: If f(x) = 3x2 is the derivative a function, what is that function? F(x) = x3 derivative x3 3x2 antiderivative

2. DERIVATIVES ANTIDERIVATIVES Finding the Antiderivative Describe the original function of each of the following derivatives: f(x) = 2x f(x) = x f(x) = x2 f(x) = 1/x2 f(x) = 1/x3 f(x) = cos x F(x) = x2 F(x) = .5x2 F(x) = 1/3x3 x-2 F(x) = -1x-1 x-3 F(x) = -1/2x-2 F(x) = sin x An antiderivative of a function f is a function whose derivative is f; traditionally the antiderivative of f(x) is F(x).

3. 2x 2x 2x Antiderivatives f(x) = 2x F(x) = x2 F(x) = x2 + 6 F’(x) =? F(x) = x2 – 4 F’(x) =? F(x) = x2 + 12/17 F’(x) =? f(x) has an infinite number of anti’s ‘The Family’ If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is in the form G(x) = F(x) + C, for all x in I where C is a constant.

4. Antiderivatives Family of all derivatives of f(x) = 2x G(x) = x2 + C C is a constant DERIVATIVES ANTIDERIVATIVES G(x) = x2 + C G(x) = x2 + C G(x) = x2 + C

5. Variable of integration Constant of Integration Integrand the integral: “the antiderivative of f with respect to x” Notation for Antiderivatives differential equations in x and y is an equation involving x, y, and derivative of y Finding all antiderivative solutions for this differential equation is called antidifferentiation (or the indefinite integral)

6. Integration is the inverse of differentiation Differentiation is the inverse of integration Inverses C kx + C

7. Power Rule Basic Integration Formulas

8. Basic Integration Formulas

9. Model Problem Find the antiderivative of 3x Constant MultipleRule To find the antiderivative (to antidifferentiate) also means to integrate. Rewrite Power Rule (n = 1) Simplify Since C is any constant 3C is still a constant *Check by differentiating answer

10. Rewriting Original Rewrite Integrate Simplify

11. Model Problems

12. Rewrite Model Problem Integrate Simplify

13. Rewrite Model Problem Integrate

14. C = 1 (-1, 1) C = 0 a coordinate point C = -3 Initial Conditions and Particular Solutions General Solution What is that C graphically? y-intercept Often interest is only in finding a particular solution We would need an initial condition C = 1  F(x) = x3– x + 1

15. Model Problem General Solution Particular Solution F(1) = 0

16. Model Problem • A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. • Find the position function giving the height s as a function of the time t. • When does the ball hit the ground? a. s(t) = -1/2gt2 + vot + so position function s(t) = -16t2 + 64t + 80 at t = 0 s(0) = 80 initial height at t = 0 s’(0) = 64 initial velocity s’’(0) = -32 gravitational acceleration

17. s(t) 80 Model Problem • A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. • Find the position function giving the height s as a function of the time t. at t = 0 s(0) = 80 initial height a. at t = 0 s’(0) = 64 initial velocity s’’(0) = -32 gravitational acceleration s’(0) = 64 s(0) = 80

18. Model Problem A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. b. When does the ball hit the ground? b. ball hits ground 5 seconds after it was thrown