7.1: Antiderivatives Objectives: To find the antiderivative of a function using the rules of antidifferentiation To find the indefinite integral To apply.

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Presentation transcript:

7.1: Antiderivatives Objectives: To find the antiderivative of a function using the rules of antidifferentiation To find the indefinite integral To apply the indefinite integral DERIVATIVES

Up until this point, we have done problems such as: f(x)= 2x +7, find f’(x) Now, we are doing to do problems such as: f’(x)=2, find f(x) We do this through a process called antidifferentiation

Warm Up: 1.Find a function that has the derivative f’(x)=3x 2 +2x 2.Find a function that has the derivative f’(x)=x 4 -4x 3 +2

DEFINITION: ANTIDERIVATIVE If F’(x) = f(x) then F(x) is an antiderivative of f(x) F’(x) = 2x, then F(x) = x 2 is the antiderivative of 2x (it is a function whose derivative is 2x) Find an antiderivative of 6x 5

2 antiderivatives of a function can differ only by a constant: f’(x) = 2xg’(x)=2x F(x) = x 2 +3G(x)=x 2 -1 F(x)-G(x)= C The constant, C, is called an integration constant

INDEFINITE INTEGRAL!!!!! integral sign f(x) integrand dx change in x ( remember differentials?!?!) Be aware of variables of integration…

If F’(x) = f(x), then = F(x) + C, for any real number C F(x) is the antiderivative of f(x) This is a big deal!!!!!!!

Example: Find the indefinite integral.

Rules of Integration Power Rule Constant Multiple Rule (k has to be a real #, not a variable) Sum or Difference Rule

Examples: Find the indefinite integral 1.2.

3.4.

5.

More Rules…..

Examples: Find the Indefinite Integral

Initial Value Problems Find the function, f(x), that has the following:

Find an equation of the curve whose tangent line has a slope of f’(x)=x 2/3 given the point (1, 3/5) is on the curve.

Applications 1. An emu is traveling on a straight road. Its acceleration at time t is given by a(t)=6t+4 m/hr 2. Suppose the emu starts at a velocity of -6 mph (crazy…its moving backwards) at a position of 9 miles. Find the position of the emu at any time, t.

(Acceleration due to gravity= -32 ft/sec 2 ) A stone is dropped from a 100 ft building. Find, as a function of time, its position and velocity. When does it hit the ground, and how fast is it going at that time?