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 .