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4.1A Antiderivatives

4.1A Antiderivatives. A function F is the Antiderivative of f if F  ( x ) = f ( x ) for all x. Ex 1 : Find the Antiderivative:. Power Rule:. NOTE: see pg. 250 for more Antiderivative Rules. Ex 2 : Find all functions g such that:. Ex 3 : Evaluate. Ex 4 : Evaluate. y.

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4.1A Antiderivatives

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  1. 4.1A Antiderivatives A function F is the Antiderivative of f if F (x) = f (x) for all x.

  2. Ex 1: Find the Antiderivative:

  3. Power Rule:

  4. NOTE: see pg. 250 for more Antiderivative Rules Ex 2: Find all functions g such that:

  5. Ex 3: Evaluate

  6. Ex 4: Evaluate

  7. y y = f(x) x Ex 5: Make a rough sketch of the antiderivative F, given that F(0) = 2, & the sketch of f.

  8. y y = f(x) x

  9. Ex 6: Find f such that f (0) = 2 and

  10. HW – 4.1A pg. 255 # 9 – 41 odds, # 43 – 47 all, # 55, 59

  11. 4.1B Position, Velocity, & Acceleration NOTE: Acceleration due to gravity is  9.8 m/s2 OR 32 ft/s2

  12. The Total Change Theorem: The integral of a rate of change is the total change.

  13. Ex 1: A particle moves along a line so that its velocity at time t is v(t) = t2 – t – 6 (m/sec). a) Find the displacement from t = 1 to 4 seconds.

  14. Ex 1: A particle moves along a line so that its velocity at time t is v(t) = t2 – t – 6 (m/sec). b) Find the distance traveled during this time period.

  15. Displacement: Total Change in Position Distance:

  16. Ex 2: A ball is thrown upward with a speed of 48 ft/s from the edge of a 432 ft. cliff. a) Find h(t) where h is height in feet & t is time in seconds. b) When does it reach its max height? c) When does it hit the ground?

  17. Ex 3: A particle has an acceleration given by a(t) = 6t + 4. Its initial velocity is v(0) = 6 cm/s and its initial displacement is s(0) = 9 cm. Find the function s(t).

  18. HW – 4.1B pg. 255 #65 – 85 odds, & 94

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