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AP Calculus AB Individual Project

AP Calculus AB Individual Project. Malgorzata Stapor Period 2/3 . Citations. www.apcentral.collegeboard.com. Question 6 From Official 2006 AP Calculus AB Exam .

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AP Calculus AB Individual Project

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  1. AP Calculus AB Individual Project Malgorzata Stapor Period 2/3

  2. Citations • www.apcentral.collegeboard.com

  3. Question 6 From Official 2006 AP Calculus AB Exam • The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function f. In the figure shown, f(t) = -(1/4) t³+(3/2)t²+1 for 0 ≤ t ≤ 4 and f is a piecewise linear for 4 ≤ t ≤ 24.

  4. Part A Find f’(22). Indicate the units of measure. What we know: F’ symbolizes the derivative of the function. In math, a derivative stands for the slope! This question is asking us to find the slope. We do this through calculating the change of Y divided by the change of X. (3 – 15) (-12) ---------- = ------ = -3 calories per minute (24 – 20) (4)

  5. Part B For the time interval 0 ≤ t ≤ 24, at what time t is f increasing at its greatest rate? Show the reasoning that supports your answer.

  6. SOLUTION From the graph, f, it can be determined that the function is increasing on the intervals 0 ≤ t ≤ 4 and 12 ≤ t ≤ 16

  7. How? • On the interval 12 ≤ t ≤ 16 the rate at which the function is increasing can be determined by : (15 -9) 6 --------- = ----- = 1.5 (16-12) 4 • On the interval 0 ≤ t ≤ 4 the rate at which the function is increasing can be determined by : f(t) = -(1/4) t³+(3/2)t²+1 F’(t) = -(3/4) t²+3t F’’(t) = -(6/4)t+3  -(3/2)t+3

  8. When does f’’(t) = 0? -(3/2)t+3 = 0 -(3/2)t = -3 3/2t = 3 t = 2 • Find the maximum at f’(t) at t=2 : f’2) = -(3/4)(2)²+3(2) = 3 3 > 1.5 therefore, f is increasing at it’s greatest rate at t=2

  9. Part C Find the total number of calories burned over the time interval 6 ≤ t ≤ 18 ?

  10. What to do? We must solve the given function with the use of integrals in the interval 6 ≤ t ≤ 18 Total number of calories burned : big rectangle : 12x9 = 108 triangle: (1/2)(4)(6) = 12 small rectangle: 2x6 = 12 Total: 12+12+108 = 132 calories burned

  11. Part D The setting on the machine is now changed so that the person burns f(t)+c calories per minute . For this setting, find c so that average of 15 calories per minute is burned during the time interval 6 ≤ t ≤ 18. • What to do: The average amount of calories before the settings changed: So: = (132/12) + c = 15 = 11 + c = 15  c = 4

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