Worksheet on Rates of change

  1. Consider the following graph giving the distance of a bus from its starting point.

  1. How far did the bus travel in the first hour? first one and a half hours? first 2 hours? between the second and third hour?

    2. In the first hour, the bus traveled 20 miles.

    In the first one and a half hour the bus traveled 40 miles.

    In the first two hours it traveled 60 miles.

    Between the second and third hour it traveled 120 – 60 = 60 miles.
  2. What was the average speed of the bus during the first hour?

    3. The average speed of the bus during the first hour was

  3. Connect the points on the graph where time = 0 and where time = 1 with a straight line. This is called a secant line. Find the slope of the secant line you drew. How does this slope relate to the average speed of the bus during the first hour?

    4. The slope of the secant line is the same as the average speed.
  4. Repeat parts b and c for the first hour and a half, the first 2 hours, and for the time between the second and third hour.

    5. The average rates of change are respectively.
  5. Carefully sketch in the tangent lines to the graph at the points where time is 1 hour, 2 hours and 3 hours.

    6.  

     

     

  6. Find the slope of each of the tangent lines that you drew. What do these slopes represent. (Hint: How does the bus driver think about speed?)

    7. The slopes should be near 30, 50, and 70 respectively.

    These represent the instantaneous speed of the bus at exactly 1 hour, 2 hours, and 3 hours, respectively, after the trip started. (That is, 1 hour into the trip the speedometer registers 30 m.p.h.)
  7. The function drawn above is actually d(t) = 10t2+10t. Compute the following limits:

    8. How do these values relate to those in part f? They should be the same (although

    they probably aren’t due to errors in estimation in f)
  8. Now find:

.

How do these values relate to those in part f? Again, they should be the same.