Antiderivatives and Indefinite Integration Lesson 5.1

Reversing Differentiation An antiderivative of function f is a function F which satisfies F’ = f Consider the following: We note that two antiderivatives of the same function differ by a constant

Reversing Differentiation General antiderivatives f(x) = 6x2 F(x) = 2x3 + C because F’(x) = 6x2 k(x) = sec2(x) K(x) = tan(x) + C because K’(x) = k(x)

Differential Equation A differential equation in x and y involves x, y, and derivatives of y Examples Solution – find a function whose derivative is the differential given

Differential Equation When Then one such function is The general solution is

Notation for Antiderivatives We are starting with Change to differential form Then the notation for antiderivatives is "The antiderivative of f with respect to x"

Basic Integration Rules Note the inverse nature of integration and differentiation Note basic rules, pg 286

Practice Try these

Finding a Particular Solution Given Find the specific equation from the family of antiderivatives, which contains the point (3,2) Hint: find the general antiderivative, use the given point to find the value for C

Assignment A Lesson 5.1 A Page 291 Exercises 1 – 55 odd

Slope Fields Slope of a function f(x) Suppose we know f ‘(x) Example at a point a given by f ‘(a) Suppose we know f ‘(x) substitute different values for a draw short slope lines for successive values of y Example

Slope Fields For a large portion of the graph, when We can trace the line for a specific F(x) specifically when the C = -3

Finding an Antiderivative Using a Slope Field Given We can trace the version of the original F(x) which goes through the origin.

Vertical Motion Consider the fact that the acceleration due to gravity a(t) = -32 fps2 Then v(t) = -32t + v0 Also s(t) = -16t2 + v0t + s0 A balloon, rising vertically with velocity = 8 releases a sandbag at the instant it is 64 feet above the ground How long until the sandbag hits the ground What is its velocity when this happens? Why?

Note Spreadsheet Example Rectilinear Motion A particle, initially at rest, moves along the x-axis at acceleration At time t = 0, its position is x = 3 Find the velocity and position functions for the particle Find all values of t for which the particle is at rest Note Spreadsheet Example

Assignment B Lesson 5.1 B Page 292 Exercises 57 – 93, EOO