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4.1 Antiderivatives and Indefinite Integration Objective: Use indefinite integral notation for antiderivatives and use basic integration rules to find antiderivatives . Miss Battaglia AP Calculus AB/BC. Definition of Antiderivative - PowerPoint PPT Presentation
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4.1 Antiderivatives and Indefinite Integration
Objective: Use indefinite integral notation for antiderivatives and use basic integration rules to find antiderivatives.
Miss BattagliaAP Calculus AB/BC
Definition of Antiderivative
A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I.
Representation of Antiderivatives
If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x)=f(x)+C, for all x in I where C is a constant.
A differential equation in x and y is an equation that involves x, y, and derivatives of y.
Find the general solution of the differential equation y’=2
Solving a Differential Equation
When solving a differential equation of the form
it is convenient to write it in the equivalent differential form
The general solution is denoted by
Notation for Antiderivatives
*Refer to handout
Basic Integration Rules
Describe the antiderivatives of 3x.
Applying the Basic Integration Rules
Original Rewrite Integrate Simplify
Original Integral
Rewrite Integrate Simplify
Rewriting Before Integrating
a. b. c.
Integrating Polynomial Functions
Rewriting Before Integrating
Rewriting Before Integrating
Find the general solution of
and find the particular solution that satisfies the initial condition F(1)=0.
Finding a Particular Solution
A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 ft.a) Find the position function giving the height s as
a function of the time t.b) When does the ball hit the ground?
Solving a Vertical Motion Problem
Original Integral
Rewrite Integrate Simplify
Rewriting Before Integrating
Read 4.1 Page 255 #5,6,10,11,12,14,27,29,33,35,37,38,49,60,61,64,84
Classwork/Homework
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