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Miss Battaglia AB/BC Calculus

4-4 The Fundamental Theorem of Calculus Objective: Evaluate a definite integral using the Fundamental Theorem of Calculus and understand the MVT for integrals. Miss Battaglia AB/BC Calculus. Fundamental Theorem of Calculus. Connects differentiation and integration.

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Miss Battaglia AB/BC Calculus

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  1. 4-4 The Fundamental Theorem of CalculusObjective: Evaluate a definite integral using the Fundamental Theorem of Calculus and understand the MVT for integrals. Miss Battaglia AB/BC Calculus

  2. Fundamental Theorem of Calculus • Connects differentiation and integration. • Integration & differentiation are inverse operations. If a function is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

  3. Guidelines for Using the Fundamental Theorem of Calculus • Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. • When applying the FTC, the following notation is convenient For instance, to evaluate you can write 3. It is not necessary to include a constant of integration C in the antiderivative because

  4. Evaluate each definite integral a. b. c.

  5. A Definite Integral Involving Absolute Value Evaluate

  6. Using the Fundamental Theorem to Find Area Find the area of the region bounded by the graph of y=2x2-3x+2, the x-axis, and the vertical lines x=0 and x=2.

  7. Thm 4.10 Mean Value Theorem for Integrals If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that

  8. Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a,b], then the average value of f on the interval is

  9. Finding the Average Value of a Function Find the average value of f(x)=3x2-2x on the interval [1,4]

  10. The Speed of Sound At different altitudes in Earth’s atmosphere, sound travels at different speeds. The speed of sound s(x) (in meters per second) can be modeled by Where x is the altitude in kilometers. What is the average speed of sound over the interval [0,80]?

  11. Classwork/Homework • Read 4.4 Page 293 #11,13,15,19,23,27,33,39,45,47

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