1 / 11

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 .

bridie
Download Presentation

AP Calculus AB Individual Project

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related