AP Calculus AB Chapter 4, Section 1 Integration

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 3x.

Rewriting Before Integrating Original IntegralRewriteIntegrateSimplify

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, total problems