Problem Set 2 – Average Rate of Change
Problem Set 2 – Average Rate of Change
2.1 For a linear function, we say the slope is the rate of change of the function. Does a rate of change make sense if the function is not linear?
1) A remote controlled car moves in a line. The graph below shows the distance (in feet) of a remote controlled car from the person controlling it, t seconds after turning it on. We will look at each segment of the graph. Explain what the car was doing during the time interval corresponding to each segment of the graph. (This graph is not plausible – why?)
Fill in the table below.
Segment / Beginning time / Ending time / Beginning distance / Ending distance / VelocityA
B
C
D
E
F
G
Note: Since velocity is the change in distance divided by the change in time, it can be positive or negative. Speed is the absolute value of velocity, and so is always nonnegative.
2) Sheri was running errands for her mother. All of the stores and her house are located along a straight street. She starts from home. On the graph below, her distance from home d is shown and her arrival at the grocery store is marked at point G, her arrival at the hardware store is marked at point H, and her arrival at the auto parts store is marked at point A. Time is in minutes and distance is in miles.
a) What was her average velocity between her home and the grocery store? Would you feel safe as a passenger in Sheri’s car?
b) How long was she at the grocery store?
c) What was her average velocity between her arrival at the grocery store and her arrival at the hardware store?
d) How long was she at the hardware store?
e) What was her average velocity from the time she left the hardware store to the time she arrived at the auto parts store?
f) Find a point P between Sheri’s home and the grocery store where her average velocity before that point was less than her average velocity for the entire trip to the grocery store. What was her average velocity between P and the grocery store?
g) Draw the line segment, and compute its slope. How does this compare to your answer in part e)?
2.2 If a function is not linear, there is no single value that can represent the slope over a given interval. Instead we talk about the average rate of change of the function over a given interval. In Sheri’s situation, her average velocity between the hardware store and the auto parts store is the change in distance from the hardware store to the auto parts store divided by the change in time. Finding the average rate of change of a function over some interval is equivalent to finding the slope of the line that joins the corresponding points.
3) The graph below shows y = h(t), which gives the height (in feet) of a toy rocket t seconds after it was launched.
a) Label the point on the graph that corresponds to the rocket’s launch with O and the highest point it reaches as M. Find the average rate of change of the height over the time intervals given below.
Time 1 / Time 2 / Average Rateof Change
0 / 1
1 / 2
2 / 2.5
2.5 / 4
1 / 4
2 / 3
Sketch the line segments joining the end-points for each of these time intervals.
b) What are the units for the average rate of change?
c) For which time intervals is the average rate of change positive? When is it negative? When is it zero?
d) The coordinates of point A can be represented as (1, ). How can the coordinates of points O, B, M and C be represented using function notation?
Point / CoordinatesO
A / (1, )
B
M
C
e) The slope of the line segment can be written as . How can we write the slope of the line segment using function notation??
4) One of the most common uses of rate of change is speed. What does the speedometer in your car measure?
Copyright 2007. Concepts of Calculus for Middle Level Students. First developed for the La Meta (Mathematics Educators Targeting Achievement) Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 2