Limits Applied to Derivatives Problem: Suppose you start driving off from a traffic

QUESTION:

Limits Applied to Derivatives Problem: Suppose you start driving off from a traffic light. Your distance, d(t), in feet, from where you started is given by d(t) = 3t2 where t is time, in seconds, since you started. a. Figure 2-2h shows d(t) versus t. Write the average speed, m(t), as an algebraic fraction for the time interval from 4 seconds to t seconds. Figure 2-2h b. Plot the graph of function m on your grapher. Use a friendly window that includes t = 4. What feature does this graph have at the point t = 4? Sketch the graph. c. Your speed at the instant t = 4 is the limit of your average speed as t approaches 4. What does this limit appear to equal? What are the units of this limit? d. How close to 4 would you have to keep t for m(t) to be within 0.12 unit of the limit? (This is an easy problem if you simplify the algebraic fraction first.) e. Explain why the results of this problem give the exact value for a derivative.