(Please Write Your Responses on Separate Sheets of Paper, and Label the Top of Your Papers

Problem Set 6 – Understanding Derivatives

The goal of this section is to make connections between the graph of a function and the graph of the function’s derivative.

1) Look at the two graphs shown below. One is the graph of f (x)  x2 and the other is the graph of its derivative, f '(x)  2x . Every point on the graph of f ' tells you the slope of the corresponding tangent line to the curve f (x)  x2 .

y y This tells you the slope on the graph of. (1,2) (1,1)

x x These are the same

f '(x)  2x

Sketch the line tangent to the graph of f at x  1, and then use the graph of f ' to find the slope of this line. Do the same thing at x = 0.

2) Now let f (x)  x3 . Use the graph of the derivative to find the slope of the tangent line to the graph of f at x = 1, x = 0, and x = -1. Sketch the tangent lines to the graph of f at these points.

f (x)  x3 f '(x)  3x2

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1 Problem Set 6 – Understanding Derivatives

3) Estimate slopes of tangent lines to the graph in the left and use them to sketch the graph of the derivative function on the right.

4) The function f gives the distance (in km) traveled by an emergency vehicle on the Autobahn over a period of 2 hours. Since the derivative is the rate of change of the distance with respect to time, f '(t) represents the velocity.

f (t)  75t3  225t 2 f '(t)

a) Label the axes, identify the correct units and provide a scale for the axes.

b) Compute f '(t) .

c) Use the derivative to find the car’s speed ½ hour into the trip. What does this correspond to in the original graph?

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 2 Problem Set 6 – Understanding Derivatives

d) When was the car going fastest? How can you see this on the graph of the derivative? How is this reflected in the graph of the original function for distance?

5) Marketing director Pam Gilla has done research about the number of 64 oz. boxes of crackers that will be sold at different prices. Let p represent the price in dollars of one box of crackers, and s(p) represent the number of boxes sold (in thousands). Pam found that

s( p)  .014 p 4  .4 p 3  3.5p 2 13.25p 100.

a) Find the derivative and compute s'(8) . Interpret this number in the context of this problem.

b) Sketch the graph of s( p) and the tangent line at p  8 . Provide a graphical interpretation of your answer in a).

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 3