Aim: How to Find the Antiderivative Course: Calculus Do Now: Aim: What is the flip side of the derivative? If f(x) = 3x 2 is the derivative a function, what is that function? F(x) = x 3 derivative antiderivative x3x3 3x23x2

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

Aim: How to Find the Antiderivative Course: Calculus Antiderivatives ‘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. f(x) = 2xF(x) = x 2 F(x) = x F’(x) = ? F(x) = x 2 – 4 F’(x) = ? F(x) = x /17 F’(x) = ? 2x2x 2x2x 2x2x f(x) has an infinite number of anti’s

Aim: How to Find the Antiderivative Course: Calculus Antiderivatives Family of all derivatives of f(x) = 2x G(x) = x 2 + C C is a constant G(x) = x 2 + C DERIVATIVESANTIDERIVATIVES

Aim: How to Find the Antiderivative Course: Calculus 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) Integrand Variable of integration Constant of Integration the integral: “the antiderivative of f with respect to x”

Aim: How to Find the Antiderivative Course: Calculus Inverses Integration is the inverse of differentiation Differentiation is the inverse of integration Differentiation FormulaIntegration Formula C kx + C

Aim: How to Find the Antiderivative Course: Calculus Basic Integration Formulas Power Rule

Aim: How to Find the Antiderivative Course: Calculus Basic Integration Formulas

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

Aim: How to Find the Antiderivative Course: Calculus Rewriting OriginalRewriteIntegrateSimplify

Aim: How to Find the Antiderivative Course: Calculus Model Problems

Aim: How to Find the Antiderivative Course: Calculus Model Problem Simplify Rewrite Integrate

Aim: How to Find the Antiderivative Course: Calculus Model Problem Rewrite Integrate

Aim: How to Find the Antiderivative Course: Calculus Initial Conditions and Particular Solutions What is that C graphically? y-intercept C = 0 C = 1 C = -3 Often interest is only in finding a particular solution We would need an initial condition General Solution C = 1  F(x) = x 3 – x + 1 a coordinate point (-1, 1)

Aim: How to Find the Antiderivative Course: Calculus Model Problem General Solution F(1) = 0 Particular Solution

Aim: How to Find the Antiderivative Course: Calculus Model Problem A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. a.Find the position function giving the height s as a function of the time t. b.When does the ball hit the ground? s(t) = -1/2gt 2 + v o t + s o position function s(t) = -16t t + 80 a. at t = 0 s(0) = 80initial height at t = 0 s’(0) = 64initial velocity s’’(0) = -32gravitational acceleration

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

Aim: How to Find the Antiderivative Course: Calculus 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