Lab 9 – Midterm 2 Review
Date: October 25, 2011
Assignment Due Date: November 1, 2011
Goal: In order to prepare for the midterm, today we will use R to complete a problem off the old
midterm posted on Dr. Adler’s website. This problem will explore first and second derivatives along with
the concepts of velocity and acceleration. We will then explore the chain rule using R. All of these
concepts are important to know for the midterm.
Useful Rules for Derivatives
Power Rule
Product Rule
Quotient Rule
Exponential Rule
Chain Rule
Position, Velocity, and Acceleration
Open your script from last week’s lab and save a copy of it as this week’s lab. We will now modify this
script to help us answer the following question found on the old midterm.
Suppose the position of an object follows the equation
Time is measured in seconds and position is measured in meters above the surface of a peculiar
astronomical object, and b is a mysterious positive constant.
a. Find the velocity of this object. Describe the initial height and velocity.
b. Find the acceleration. For which value of b does p(t) most closely match the behavior of an
object on earth?
c. Find the value of b for which the object hits the surface at t=2. Find the velocity and
acceleration when it hits the surface.
d. Describe in words what is happening. What kind of peculiar astronomical object might this
be?
Change your script to plot the function, the exact first derivative, and the exact second derivative side by
side in one figure. You should calculate your exact derivatives by hand first. In R, don’t forget we have
to define a value of b. This value will change as we answer different questions. Add code to your plot to
calculate the approximate first and second derivatives and check to make sure the derivatives you
calculated by hand are correct. Think carefully about how to define the approximate second derivative.
Add straight lines to each of your plot at p(t) = 0 so you can see where p(t), p’(t), and p’’(t) are equal to
zero. Use the functions you have defined and your plots to answer all the questions above and the extra
questions below.
Extra Questions
a. What does it mean if p’(t) is zero? What are these points called?
b. What does it mean if p’(t) is positive or negative?
c. What does it mean if p’’(t) is zero? What are these points called?
d. What does it mean if p’(t) is positive or negative? What does this tell us about the shape of p(t)?
Chain Rule
We will now write a script to explore the chain rule. We will consider the following function.
What three different functions are included in
separate functions as
,
,
? Work from inside to outside and define the
. Then define
. Compute the
derivative by hand using the chain rule and then compute the approximate derivative and check.
Next do the same for
Other things to know for the test
Secant Lines
Tangent Lines
Evaluating Limits
.