Antiderivatives An antiderivative of f(x) is any function F(x) such that F’(x) = f(x)

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Presentation transcript:

Antiderivatives An antiderivative of f(x) is any function F(x) such that F’(x) = f(x)

Example Find antiderivatives of f(x) = x 2

Example Find antiderivatives of f(x) = 2x

Example Find antiderivatives of f(x) = 1/x

Theorem If F(x) is an antiderivative of f(x) then F(x) + C is an antiderivative of f(x) for any constant C

Antiderivatives Graphically Match the function to its antiderivative f(x) F(x) 1) 2) 3) 4) D C B A

The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in, and

First Fundamental Theorem: 1. Derivative of an integral.

2. Derivative matches upper limit of integration. First Fundamental Theorem: 1. Derivative of an integral.

2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. First Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. New variable. First Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. The long way: First Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

Example If find F’(x)

The upper limit of integration does not match the derivative, but we could use the chain rule.

The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits.

Example If f(x) =find f’(x)

Neither limit of integration is a constant. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.) We split the integral into two parts.

HW: p. 287/37-42

The Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of, and if F is any antiderivative of f on, then (Also called the Integral Evaluation Theorem) To evaluate an integral, take the anti-derivatives and subtract.

Antiderivatives Antiderivatives are also called indefinite integrals They are sometimes written Note that there are no limits on the integral Do not confuse with definite integrals!

Common Antiderivatives

Evaluate the integral using FTC2

Rewrite then evaluate the integral using FTC2

Evaluate the integral involving trigonometric functions using FTC2

Special Example: absolute value

Area using Integrals Find the zeros of the function over the interval [a,b] integrate over each subinterval add the absolute value of the integrals

Example: Find the area using integrals

Using the GC to find the integral hit MATH then 9 fnInt( will come up on the screen type in the function, comma, x, comma, -a, comma, b) then hit ENTER Ex:

Examples: Use GC

Area using GC To find the area under the curve f(x) from [a,b] type fnInt(abs(f(x)),x,a,b) Example: Find the area under the curve y = xcos 2 x on [-3, 3]

HW: FTC 2 wksheet