5.4: Fundamental Theorem of Calculus Objectives: Students will be able to… Apply both parts of the FTC Use the definite integral to find area Apply the Mean Value Theorem for integrals

Anti-Derivatives Given a function, f(x), an anti-derivative of f(x) is any function F(x) such that F’(x) = f(x) Derivatives

Examples: Find the antiderivative of the following. 1.f’(x) = sin(x) 2.f’(x) = x 3.f’(x) = 2

4.f’(x) = 1/x 5.f’(x) = sec 2 x 6.f’(x) = 1/sin 2 x

Anti-derivatives for powers If f’(x) = x n, then

Check this out…. Integrals and anti-derivatives are related…. If F is any antiderivative of f, then

This will allow us to evaluate the definite integral. Set x = a so we can solve for C: Now we have: THIS IS AMAZING NEWS!!!!!!!!!!!!!!!!!!!!!!!!!!!

The Fundamental Theorem of Calculus (First Part) If f is continuous at every point of [a,b] and if F is any antiderivative of f on [a,b], then (also called the Integral Evaluation Theorem)

Evaluate each integral.

What should I do with an absolute value???? Oh dear….

Finding total area analytically. 1.Partition [a,b] with the zeros of f. 2.Integrate f over each subinterval. 3.Add the absolute values of the integrals.

Find the total area between the curve and the x axis on the following interval:

If given an initial value, you can use FTC to find another value. ASKED A LOT ON EXAM!!

If f is the antiderivative of, such that f(1) = 0, then f(4) =

Relationship between position, velocity, and acceleration…again If given velocity, v(t): If not dealing with velocity: TOTAL DISPLACEMENT (NET CHANGE) TOTAL DISTANCE NET CHANGE TOTAL CHANGE

-Velocity function is antiderivative of acceleration -Position function is antiderivative of velocity If the acceleration of a rocket launched upward is and the initial velocity is 2 ft/sec, how fast is the rocket traveling at 80 sec?

Fundamental Theorem of Calculus (Second part)  Proves that differentiation and integration are inverse processes EXPLORATION: Let a.)Find F(0), F(1) and F(2) using area. b.)Write a general formula for F(x). c.) Find F’(x).

Take another look….Find F’(x).

FTC: Second Part If f is continuous on [a,b], then the function has a derivative at every point x in [a,b] and

Evaluate Find F’(x) if

Find if 1.) Set u = x 3 2.) Now, y is defined in terms of u, but u is defined in terms of x. 3.) To find dy/dx you need chain rule.

Find F’(x) if F(x) =

General Rule If you have

Another General Rule CAN USE ANY a AS LONG AS f(a) EXISTS

Find dy/dx. 1.2.

1.Find F(4).2. Find F(∏/4)

Find a function y=f(x) with dy/dx=sinx and f(2)=4.