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Warm Up – NO CALCULATOR

Warm Up – NO CALCULATOR. Let f(x) = x 2 – 2x. Determine the average rate of change of f(x) over the interval [-1, 4]. Determine the value of (Check your answer using your calculator). Mean Value Theorem for Integrals Average Value 2 nd Fundamental Theorem of Calculus. a. c. b.

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Warm Up – NO CALCULATOR

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  1. Warm Up – NO CALCULATOR Let f(x) = x2 – 2x. • Determine the average rate of change of f(x) over the interval [-1, 4]. • Determine the value of (Check your answer using your calculator)

  2. Mean Value Theorem for IntegralsAverage Value2nd Fundamental Theorem of Calculus

  3. a c b Mean Value Theorem for Integrals If f is continuous on [a,b] then there is a certain point (c, f(c)) between a and b so if you draw a rectangle whose length is the interval [a,b] and whose height is f(c), the area of the rectangle will be exactly the area beneath the function on [a,b].

  4. In other words… *If f is continuous on [a,b], then there exists a number c in the open interval (a,b) such that . Area under the curve from a to b Area of the rectangle formed

  5. Example 1: Find the value of f(c) guaranteed by MVT for integration for the function f(x) = x3 – 4x2 + 3x + 4 on [1,4] Explain the relationship of this value to the graph of f(x)?

  6. Example 2 Find the value of f(c) guaranteed by MVT for integrals on the interval [1,9] for

  7. The f(c) value you found in both examples is called the average valueof f. Solving for f(c) gives the formula for average value.

  8. Example 3: Find the average value of f(x) = 3x2 – 2x on the interval [1,4] and all values of x in the interval for which the function equals its average value.

  9. Taking the derivative of a definite integral whose lower bound is a number and whose upper bound contains a variable.

  10. The long way…

  11. The 2nd Fundamental Theorem of Calculus: If f(x) is continuous and differentiable,

  12. Here’s what you REALLY do…

  13. Your turn…

  14. If

  15. Let f be defined on the closed interval [-5,5]. The graph of f consisting of two line segments and two semicircles, is shown above. f

  16. Let g be the function given by f Find g(2) Find g’(2) Find g”(2)

  17. g(x)= f On what intervals, if any, is g increasing? Find the x-coordinate of each point of inflection of the graph g on the open interval (-5,5). Justify your answer.

  18. g(x)= f Find the average rate of change of g on the interval [-5,5].

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