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Calculus Fundamentals: Maximize and Evaluate Functions

Learn how to maximize and evaluate functions using the Fundamental Theorem of Calculus. Find average height and solve equations step by step.

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Calculus Fundamentals: Maximize and Evaluate Functions

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  1. Upper Sum with n=5

  2. Upper Sum with n=5 • (2+2+1+1+2) p/2

  3. Upper Sum with n=5 • 12.56 • 1.0

  4. .

  5. .

  6. .

  7. .

  8. 8.85 • 0.4

  9. Fundamental Theorem of Calculus If f is continuous on [a, b] and F is any antiderivative of f, then

  10. Maximize =-x(x-3) = 0 when

  11. Maximize =-x(x-3) = 0 when x = 0 or x = 3 f(1) = 2 > 0 => F(x) is increasing

  12. Maximize

  13. Maximize

  14. Maximize = [-54 + 81]/6 = 27/6

  15. Maximize = [-54 + 81]/6 = 27/6 = 9/2 = 4.5 sq. in.

  16. Find the average heighty=-x(x-3) on [0, 3] AH = 4.5 sq. in. / 3 in = 1.5 in

  17. -cos x |

  18. -cos x | =-cos(2p)-(-cos(0))

  19. -cos x | =-0-(-0)

  20. -cos x | =-1 - (-1)=0

  21. 0

  22. F(x) = + 12 • f(x) = F’(x) =

  23. y = [sin x] ½cosx

  24. Evaluate

  25. Evaluate • 0.6667 • 0.01

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