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7.1: Antiderivatives

DERIVATIVES. 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. Up until this point, we have done problems such as: f(x)= 2x +7, find f’(x)

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7.1: Antiderivatives

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  1. DERIVATIVES 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

  2. 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

  3. Warm Up: • Find a function that has the derivative f’(x)=3x2+2x • Find a function that has the derivative f’(x)=x4-4x3+2

  4. DEFINITION: ANTIDERIVATIVE If F’(x) = f(x) then F(x) is an antiderivative of f(x) F’(x) = 2x, then F(x) = x2 is the antiderivative of 2x (it is a function whose derivative is 2x) Find an antiderivative of 6x5

  5. 2 antiderivatives of a function can differ only by a constant: f’(x) = 2x g’(x)=2x F(x) = x2 +3 G(x)=x2-1 F(x)-G(x)= C The constant, C, is called an integration constant

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

  7. 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!!!!!!!

  8. Example: Find the indefinite integral.

  9. Rules of Integration

  10. Examples: Find the indefinite integral 1. 2.

  11. 3. 4.

  12. 5.

  13. More Rules…..

  14. Examples: Find the Indefinite Integral

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

  16. Find an equation of the curve whose tangent line has a slope of f’(x)=x2/3given the point (1, 3/5) is on the curve.

  17. Applications 1. An emu is traveling on a straight road. Its acceleration at time t is given by a(t)=6t+4 m/hr2. 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.

  18. (Acceleration due to gravity= -32 ft/sec2) 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?

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