Antiderivatives Chapter 4.9

Antiderivatives The antiderivative problem: Given the derivative of a function, f ‘(x), can you guess the original function, f(x)? I’m thinking of a function. Its derivative is . Which of the following could be my function? A. 12x2 – 2 B. x4 – x2 C. x4 – x2 + 1 D. x4 – x2 + x – 12

Definition An antiderivative of f(x) is a function F(x) such that F’(x) = f(x). Question: Which of the following is an antiderivative of f(x) = 6x2 ? A. 2x3 B. 2x3 – 5 C. 2x3 + 18 D. all of the above

The family of antiderivatives of y = 2x

Terminology and Notation The antiderivative of a function is also called the indefinite integral. The symbol for the antiderivative of f(x) is We often write F(x) as an antiderivative of f(x), and so

Antiderivative Rules Antiderivative of 0: Antiderivative of a constant: Antiderivative of a power of x: p is a real number, p ≠ -1 The Power Rule

Voting Questions 1. A. 0 B. C C. 12x D. 12x + C 2. A. B. C. D.

Constant Multiples, Sums, Differences Constant multiple rule: Antiderivative of a sum: Antiderivative of a difference:

Voting Question A. B. C. D.

Voting Question A. 3x2 + C B. 3x2 + 5x + C C. D.

Products, Quotients There is no product rule for antiderivatives! There is no quotient rule for antiderivatives! Consequently, antiderivatives of products and quotients cannot be directly computed by using a general rule.

A. B. C. D.

1. A. B. C. D. 2. A. B. C. D.

A Trig Antiderivative We know that , so working in reverse, we have More generally, Working in reverse, we get Dividing both sides by a gives

Antiderivatives Involving Trig Functions

Examples 1. 2.

Particular Antiderivatives When the value of an antiderivative F(x) is given for a particular value of x, we can determine the value of C. Example: Find the antiderivative of f (x) = 8x3 – 2x-2 that satisfies F(1) = 5.

Position, velocity, acceleration Position, y(t) Velocity, y ’ (t) Acceleration, y ’’ (t)

Example A ball is thrown into the air from an initial height of 80 ft with initial velocity 20 ft/sec. Near the surface of the earth, acceleration due to gravity is about -32 ft/sec2. Find the position of the ball as a function of time (t), and determine the height of the ball after 2 seconds.