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Ch 4.4 – The Fundamental Theorem of Calculus (day 2). Target Goals: Find the value of c guaranteed by the Mean Value Theorem for Integrals Find the average value of the function over an interval. Mean Value Theorem for Integrals.

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ch 4 4 the fundamental theorem of calculus day 2

Ch 4.4 – The Fundamental Theorem of Calculus (day 2)

Target Goals:

Find the value of c guaranteed by the Mean Value Theorem for Integrals

Find the average value of the function over an interval.

mean value theorem for integrals
Mean Value Theorem for Integrals
  • If f is continuous on [a, b], then there exists a number c in the interval [a, b] such that:
average value of a function
Average Value of a Function
  • The value of f (c) given by the MVT for integrals is the average value of f on [a, b].
  • If f is integrable on [a, b], then the average value of f on the interval is:
example 1
Example 1
  • Find the value of c guaranteed by the MVT for integrals:
visual representation of average value of a function and mvt
Visual Representation of Average Value of a Function and MVT
  • http://archives.math.utk.edu/visual.calculus/5/average.1/
  • The area of the region under the graph of f is equal to the area of the rectangle whose height is the average value.
example 2
Example 2
  • Find the average value and all values of x in the interval for which the function equals its average value.

Average value