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THE FUNDAMENTAL THEOREM OF CALCULUS

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14 diapositives · 444 Views · Nov 20, 2014

THE FUNDAMENTAL THEOREM OF CALCULUS. Section 4.4. When you are done with your homework, you should be able to…. Evaluate a definite integral using the Fundamental Theorem of Calculus Understand and use the Mean Value Theorem for Integrals

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THE FUNDAMENTAL THEOREM OF CALCULUS
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  1. THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4

  2. When you are done with your homework, you should be able to… • Evaluate a definite integral using the Fundamental Theorem of Calculus • Understand and use the Mean Value Theorem for Integrals • Find the average value of a function over a closed interval • Understand and use the Second Fundamental Theorem of Calculus

  3. Galileo lived in Italy from 1570-1642. He defined science as the quantitative description of nature—the study of time, distance and mass. He invented the 1st accurate clock and telescope. Name one of his advances. • He discovered laws of motion for a falling object. • He defined science. • He formulated the language of physics.. • All of the above.

  4. THE FUNDAMENTAL THEOREM OF CALCULUS • Informally, the theorem states that differentiation and definite integrals are inverse operations • The slope of the tangent line was defined using the quotient • The area of a region under a curve was defined using the product • The Fundamental Theorem of Calculus states that the limit processes used to define the derivative and definite integral preserve this relationship

  5. Theorem: The Fundamental Theorem of Calculus • If a function f is continuous on the closed interval and F is an antiderivative of f on the interval , then

  6. 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 Fundamental Theorem of Calculus, the following notation is convenient: • It is not necessary to include a constant of integration in the antiderivative because

  7. Example: Find the area of the region bounded by the graph of , the x-axis, and the vertical lines and .

  8. Find the area under the curve bounded by the graph of , , and the x-axis and the y-axis. • 9/4 • 0.0

  9. THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on the closed interval , then there exists a number c in the closed interval such that

  10. So…what does this mean?! Somewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve.

  11. AVERAGE VALUE OF A FUNCTION • If f is integrable on the closed interval , then the average value of f on the interval is

  12. Find the average value of the function • 13.0 • 0.0

  13. THE SECOND FUNDAMENTAL THEOREM OF CALCULUS If f is continuous on an open interval I containing c, then, for every x in the interval,