AP Calculus AB Chapter 4 Section 1 Integration

AP Calculus AB Chapter 4, Section 1 Integration 2013 - 2014

Derivatives vs. Integrals • We use derivatives to find slopes of tangent lines or the __________ of a function at a specific point. • We use integrals to find the _______________. • Integrals are often called ________.

Derivatives vs. Integrals •

Integrals • Is your answer the only possible answer?

General Solution • Remember that a derivative will not include any constants from a given function. So account for the constant that may or may not be present in the original function, we add C to the end of the antiderivative.

Notation for Antiderivatives •

Something to remember • Derivatives and Integrals are basically “inverses” of each other. • Basic Integration Rules on page 250

Applying the Basic Integration Rules • Describe the antiderivatives of 3 x.

Rewriting Before Integrating Original Integral Rewrite Integrate Simplify

Integrating Polynomial Functions •

Integrating Polynomial Functions •

Integrating Polynomial Functions •

Rewriting Before Integrating •

Rewrite Before Integrating •

Initial Conditions and Particular Solutions • The type of antiderivatives you have been learning about are all vertical translations of each other (due to the addition of C). • In many situations, you are given enough information to find the particular solution. To determine the value of C, all you need to know is _____________.

Finding a Particular Solution •

Solve the differential equation •

Solving a Vertical Motion Problem • A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. • Find the position of the function giving the height x as a function of time t. • When does the ball hit the ground? (Use -32 feet per second as the acceleration due to gravity. )

Ch. 4. 1 Homework • Pg 255 – 257, #’s: 1, 7, 9, 13, 15, 23, 29, 35, 39, 45, 55, 61, 69 • 13 total problems