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AP Calculus BC Tuesday , 07 October, 2014

AP Calculus BC Tuesday , 07 October, 2014. OBJECTIVE TSW (1) identify extrema (absolute and local) of a function on an interval, (2) understand and use Rolle’s Theorem, and (3) understand and use the Mean Value Theorem. TODAY’S ASSIGNMENT Sec . 3.2: pp. 176 (1-19 odd, 25-31 odd)

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AP Calculus BC Tuesday , 07 October, 2014

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  1. AP Calculus BCTuesday, 07 October, 2014 • OBJECTIVETSW (1) identify extrema (absolute and local) of a function on an interval, (2) understand and use Rolle’s Theorem, and (3) understand and use the Mean Value Theorem. • TODAY’S ASSIGNMENT • Sec. 3.2: pp. 176 (1-19 odd, 25-31 odd) • Due Friday, 10 October 2014. • ASSIGNMENT DUE TOMORROW/THURSDAY • Sec. 3.1: p. 169 (1-11 odd, 19-35 odd) • LOOKING AHEAD • Tomorrow/Thursday: ASSESSMENT – Rates of Change

  2. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem

  3. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem

  4. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Find the x-intercepts of and show that at some point between the two intercepts. Then find c. x-intercepts and zeros of a function are the same: f (x) = 0  x-intercepts:(–5, 0) and (–4, 0) i. f(x) is continuous on [–5, –4] ii. f(x) is differentiable on (–5, –4) iii. f(–5) = f(–4) = 0  By Rolle’s Theorem  a c in (–5, –4)  f′(c) = 0.

  5. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Find the x-intercepts of and show that at some point between the two intercepts. Then find c. Check to make sure that thec-value is in the open interval.

  6. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Determine if Rolle’s Theorem can be applied. If it can, find all values of c in the interval (–2, 2) such that i. f(x) is continuous on [–2, 2] ii. f(x) is differentiable on (–2, 2) iii. f(–2) = f(2) = 8  Rolle’s Theorem may be applied

  7. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Determine if Rolle’s Theorem can be applied. If it can, find all values of c in the interval (–2, 2) such that

  8. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem "The slope of the tangent line at c equals the slope of the secant line through the end points."

  9. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Determine if the Mean Value Theorem can be applied for Then, find all values of c in the open interval (1, 4) such that i. f(x) is continuous on [1, 4] ii. f(x) is differentiable on (1, 4)  Mean Value Theorem may be applied

  10. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Determine if the Mean Value Theorem can be applied for Then, find all values of c in the open interval (1, 4) such that

  11. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Determine if the Mean Value Theorem can be applied for Then, find all values of c in the open interval (1, 4) such that

  12. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Determine if Rolle’s Theorem can be applied. If it can, find all values of c in the interval [–8, 8] such that Rolle's Theorem cannot be applied becausef (x) is not differentiable at x = 0.

  13. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: A company sells the number of units S of its product according to where t is the time in months. a) Find the average rate of change of S(t) during the 1st year. b) During which month does S′(t) equal the average value during the first year?

  14. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem a) average rate of change (a.r.c.) of S(t) on [0, 12]: Be sure to include units!!!

  15. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem b) During which month does S′(t) equal the average value during the first year?

  16. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Which month? April

  17. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Ex: Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the 1st patrol car, its speed is clocked at 55 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit (55 mph) at some time during the 4 minutes.

  18. Sec. 3.2: Rolle’s Theorem and the Mean Value Theorem Let t = 0 be the time (in hours) that the truck passes the 1st patrol car. The time it passes the 2nd patrol car is t = 4/60 = 1/15 hr. ASSUME THAT THE POSITION FUNCTION IS DIFFERENTIABLE.  At some point during the 4 minutes, the truck must have been traveling at 75 mph. QED QED: quod eratdemonstrandum (Latin: “which was to be demonstrated”)

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