Name:____________________________
Date:_____________ Class:__________
Introduction to Limits/Average Rate of Change
BC Calculus
1. Consider the following story: Teresa leaves for work at noon. She is in a hurry but needs
to stop and grab lunch on the way. For the first 5 minutes, she travels at 30 mph, for the
next 5 minutes she travels at 45 mph, and the next 5 minutes, 60mph. She stops at a food
truck for 10 minutes, then travels at 45 mph for 10 minutes, 30 mph for 5 minutes, and 60
mph for 20 minutes. (Assume she is driving a magic car that takes her from one speed to
another without reaching the intermediary speeds).
a. Draw a graph of distance vs. time for Teresa’s drive to school.
b. Draw a graph of velocity vs. time for Teresa’s drive to school.
c. Consider the first 15 minutes of the drive. Suppose we were able to model the
position with a smooth curve (which would make more sense). Would the curve
be concave up or concave down? Explain using the velocity vs. time graph.
d. What is the distance from Teresa’s starting point at t = 35?
e. What is Teresa’s velocity at t = 35? Explain using the definition of the limit.
f. What was Teresa’s average velocity over the entire trip?
2. Consider a company making a popular sandal. Their profit function wince the company
began ( for t  0 ) is closely modeled by the function f ( x)  x 3  9 x 2  27 x  23 where x
is the number of years since they opened in 2005 and f(x) is the profit in 1000’s of
dollars.
We want to compare how the profit is changing in 2007 with how the profit is changing
in 2008.
a. Explain in words how the profit is changing in 2007 and 2008 and justify your
reasoning.
b. Using the blank tables provided below, approximate the rate of change of profit in
2007 and 2008.
2007
x1 to x 2
2008
RoC
x1 to x 2
RoC
c. Using the approximation you found in part (b), explain what each rate of change
means (for 2007 and 2008) in the context of the problem.
d. Sketch an accurate graph of the profit function f(x) on the interval 0  t  5 (or
2005 – 2010)
e. Using the gridlines on the graph approximate each rate of change for the profit in
2007 and 2008.
f. If you know any other method for approximating the rate of change, please
describe below and approximate the rate of change for 2007 and 2008 using this
method.
g. Which method is the easiest and most accurate?
h. Write a paragraph (at least 4 sentences) on why limits are important. (Include
limits at infinity and limits at a point).