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Calculus I

Calculus I. Section 5.3 The Fundamental Theorem of Calculus FTC. Warm-up. A vehicle is in motion and it’s velocity is recorded every 12 seconds. Plot the data using t (seconds) as the independent variable and v (ft/sec) as the dependent.

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Calculus I

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  1. Calculus I Section 5.3 The Fundamental Theorem of CalculusFTC

  2. Warm-up • A vehicle is in motion and it’s velocity is recorded every 12 seconds. Plot the data using t (seconds) as the independent variable and v (ft/sec) as the dependent. • Draw in rectangles that represent right hand sums and explain why the area of each rectangle is the estimate of the distance traveled over that interval. • Estimate the total distance traveled.

  3. The Area Function

  4. Example • Let f(t) = 2t3 – 18t2 + 52t – 48. Then the area function from 2 to x isUsing your calculator, find A(3). Show it graphically. • Find A(4) and show it graphically. Interpret your result. • Find A(5) and show it graphically.

  5. Let’s Do It • Generalize this to any function f(t) and find the area out to x. Then move a distance of h from x and find the area at x + h.

  6. The FTC part 1 • If f is continuous on [a, b], then the function A defined byis continuous on [a, b] and differentiable on (a, b), and A’(x) = f(x)

  7. Or

  8. Yowza! • What this says is that integration and differentiation are inverse operations of each other.

  9. Examples

  10. Example • Evaluate

  11. FTC part II • If f is a continuous function on [a, b], thenWhere F’ = f

  12. Let A(x) be as defined earlier. Then we know that A’(x) = f(x). That is A is an antiderivative of f. If F is any other antiderivative, then we know thatF(x) = A(x) + C. Now we know that F and A are continuous on [a, b] so plugin end points and we get F(b) – F(a) = [A(b) + C] – [A(a) + C]

  13. Examples • Evaluate the following and give a sketch of the net change as the area captured between the integrand and the x–axis.

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