A Type of Limit & Its Purpose in Studying Change 9/23/11
Quick Review of Limits & Continuity
To ensure you are working on and understanding the work with limits and their application in
comprehending behavior of a function as well as scenarios involving continuity, answer the
following questions concerning limits.
Perform these without the aid of a calculator – only use the calculator to “check” your answer
x2 k 2
1. Given the following limit: lim
x 3 x 3
a. Find a value of the constant, k, such that the limit exists and briefly justify your
Hint: when working with rational functions, think about whether a “hole” or a “vertical
asymptote” would allow a limit to exist at a point. Use this to guide you.
b. Let f ( x) . Is this function f ( x) continuous over all real numbers? If not, at
what point is there a discontinuity and report why the function is not continuous
at that point. Be sure to consider all values of a.
e x , 1 x 0
2. Given the function g ( x) 1, x 0
cos x, 0 x 1
Make and support an argument regarding the continuity of the function at x 0 . Your
argument should address the definition of continuity in some fashion using limits.
Rate of Change (an exploration)
The following table and graph provides information about the position of an alien creature at a
specific time. Label the axes with the appropriate units.
Point A B C D E F G H I J K L
0 1 2 4 5 7 8 9 10 11 12 13
5 11 16 20 23 28 32 31 29 26 19 16
a. We assume that this creature travels along a continuous path (go ahead and connect the
position points to create this path). What meaning would you give to calculating the
slope of the line connecting points B & I.
b. Suppose we want to calculate the instantaneous velocity of the alien creature at a specific
value, say F. Why does this appear to be an impossible task? Can you “estimate” this
instantaneous velocity? If so, provide your best estimate and briefly explain why this is
a good estimate.
Rates of Change (defined more formally)
Suppose we are given a function, s (t ) , that describes the position of an object over time. In this
particular context, the rate of change in the position of an object over time is velocity.
Q: How would you define the average velocity (in term of s) of the particle from t a to t b ?
Average Velocity =
Provide a visual representation of average velocity, given a graph of the function s (t ) :
Average Velocity = Slope of the ______________ line (when graphing position vs. time)
Q: How would you define the instantaneous velocity (in term of s) of the particle at t a ?
Instantaneous Velocity =
Provide a visual representation of instantaneous velocity, given a graph of the function s (t ) :
Instantaneous Velocity = Slope of the ______________ line (when graphing position vs. time)
Suppose you are given the position function of a particle, s(t ) 4t 2 3 .
a) Calculate the average velocity of the particle over the time interval t 1to t 3 .
b) Only using your knowledge of the function that describes the path of this particle,
do you think this average velocity is equal to, greater, or less than the instantaneous
velocity of the particle at t 2 ? Briefly explain your reasoning.
c) Calculate the instantaneous velocity of the particle at t 2 .
HW: Pg. 71/#1, 2, 5, 14