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Integration

Integration. b y Olivia Kazior. The question 2009 #1.

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Integration

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  1. Integration by Olivia Kazior

  2. The question2009 #1 Caren rides her bicycle along a straight road from home to school, starting at time t=0 minutes and arriving at school at time t=12 minutes. During the time interval 0 ≤ t ≤ 12 minutes, her velocity v(t), in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.

  3. Part a • (a) Find the acceleration of Caren’s bicycle at time t=7.5 minutes. Indicate units of measure. Since the graph of v(t) gives values of Caren’s velocity, the rate of change, or slope, the provided graph will give the acceleration at a given point. Given that f(8)=0.2 miles/minute and that f(7)=0.3 miles/minute, these values can be used to find the acceleration between the given times. == -0.1 miles/minute²

  4. Part b • (b) Using correct units, explain the meaning of in terms of Caren’s trip. Find the value of . Since the antiderivative of the velocity function is position, represents the total distance, in miles, that Caren travelled during the time interval between t=0 and t=12. Because of the absolute value sign, the integral becomes the total distance since it includes negative positions, or positions that were going in opposite directions.

  5. Part B • (b) Using correct units, explain the meaning of in terms of Caren’s trip. Find the value of . To solve for the value of , find the area under the curve for each shape and add them together. Since there is an absolute value sign in the integral, all negative values must be turned positive. =()+()+()+()+()+()+()=1.8 miles

  6. Part C • (c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer. Caren turns around to get her calculus homework at t=2 minutes. A change in direction is evident when the velocity graph, i.e. the graph provided, changes sign. At time t=2 minutes, Caren’s velocity changes from positive to negative.

  7. Part D • (d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled by the function w given by , where w(t) is in miles per minute for 0 ≤ t ≤ 12 minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer.

  8. Part D • (d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled by the function w given by , where w(t) is in miles per minute for 0 ≤ t ≤ 12 minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer. To determine who lives closer to school, the distance travelled for both Caren and Larry must be determined. The distance travelled for Caren can be solved by finding the area under the curve of velocity function without the absolute value sign, since the question is asking for the distance between Caren’s home and school, or the displacement, and not the total distance she travelled. The distance travelled for Larry can be determined by integrating the velocity function w(t) on the interval 0 ≤ t ≤ 12 minutes with a calculator.

  9. Part D • (d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled by the function w given by , where w(t) is in miles per minute for 0 ≤ t ≤ 12 minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer. Caren: )=1.4 miles Larry: =1.6 miles Caren lives closer to school.

  10. References • AP Central - College Board • Desmos Graphing Calculator • Ms. Zhao’s AP Calculus Website

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