1. Problem of the Day If f is continuous for a < x < b, which of the following could be false? A) f(c) = f(b) - f(a) for c, a < c < b b - a B) f '(c) = 0 for c, a < c < b C) f has a minimum value on a < x < b D) f has a maximum value on a < x < b E) ∫ f(x) dx exists a b

2. Problem of the Day If f is continuous for a < x < b, which of the following could be false? A) f(c) = f(b) - f(a) for c, a < c < b b - a B) f '(c) = 0 for c, a < c < b C) f has a minimum value on a < x < b D) f has a maximum value on a < x < b E) ∫ f(x) dx exists a b

3. We have studied 2 major branches of Calculus 1) differential Calculus (tangent line problems) 2) integral Calculus (area problems) They seem unrelated but Isaac Newton and Gottfried Leibniz independently found a connection

4. Differentiation slope = Δy Δx Definite integration Area = Δy Δx There is an inverse relationship

5. 4 Area under the curve using algebraic techniques

6. Area = b. h Area = Σ f(1 + 2i/n)(2/n) i = 1 n Area = 4 Area using Reimann Sums

7. lim Ʃ f(c i )Δx i = ∫f(x) dx i = 1 n n ∞ a b Riemann definite integral

8. lim Ʃ f(c i )Δx i = ∫f(x) dx i = 1 n n ∞ a b

9. All three methods return the same answer Area under the curve = 4

10. The Fundamental Theorem of Cowculus If f is continuous on [a, b] and F is an antiderivative of f on [a, b] then ∫ f(x) dx = F(b) - F(a) = F(x) ] a b a b

11. Example = 20

12. Think back to the definition of absolute value

14. [-¼ + ½ - (0 + 0)]+[(4 - 2) - (¼ - ½) 5/2

15. Mean Value Theorem (applied to area) Somewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is equal to the area under the curve a b c

16. How do we find f(c)? Area = Δy Δx so Δy = Area Δx

17. Average Value Theorem - The value of f(c) given in the mean value theorem is called the average value of f on the interval and is

18. Find the average value of f(x) = 3x 2 - 2x on the interval [1, 4]

19. 16 ( (

20. Jan Feb Mar AprMay Jun Jul Aug Sep Oct NovDec Jan Feb Mar AprMay Jun Jul Aug Sep Oct NovDec Monthly Gas Bills The Budget Payment Plan is a good program for people on fixed incomes, such as retirees. And now is still a good time to sign up for the 1990-91 Budget year. With Budget Payment, you pay the same amount each month. We will review your account in February and, if necessary, adjust the amount up or down to keep your account in line with actual gas use. During the time you are on Budget, we will still read your gas meter every month. Your monthly bill will always show how much gas you have used.

21. If velocity is the derivative of position then position is the _______________ of velocity.

22. For the first minute car 1 goes faster than car 2 and is thus ahead at the end of one minute (area under the curve is greater). Since the area under the curve representing the distance traveled by car 2 is clearly larger than the area for car 1, we know that car 2 has traveled farther than car 1.

23. A car starts at noon and travels with the velocity shown in the figure. A truck starts at 1 pm from the same place and travels at a constant velocity of 50 mph. A) How far away is the car when the truck starts? B) During the period when the car is ahead of the truck, when is the distance between them greatest, and what is that greatest distance? C) When does the truck overtake the car, and dhow far have both traveled then?

24. truck A car starts at noon and travels with the velocity shown in the figure. A truck starts at 1 pm from the same place and travels at a constant velocity of 50 mph. A) How far away is the car when the truck starts? Each rectangle represents 5 miles traveled. The shaded region has 7 rectangles thus 35 miles.

25. truck A car starts at noon and travels with the velocity shown in the figure. A truck starts at 1 pm from the same place and travels at a constant velocity of 50 mph. B) During the period when the car is ahead of the truck, when is the distance between them greatest, and what is that greatest distance? 35 + 50 = 85 miles

26. A car starts at noon and travels with the velocity shown in the figure. A truck starts at 1 pm from the same place and travels at a constant velocity of 50 mph. C) When does the truck overtake the car, and how far have both traveled then? about 85 miles and after 8.3 hours (shaded and unshaded areas are equal) truck

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