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4.1A Antiderivatives

This guide covers essential concepts of antiderivatives and their applications in physics. It defines a function F as an antiderivative of f if F'(x) = f(x) for all x, allowing for calculations using the Power Rule for finding antiderivatives. The guide provides examples for evaluating antiderivatives, examining position, velocity, and acceleration in motion problems. Key exercises involve calculating displacement and distance, determining maximum heights, and using the Total Change Theorem for understanding motion. Perfect for calculus students seeking foundational knowledge.

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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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