math theory questions

Four approaches to the derivative

(20 points)

Per usual, I am glad to offer assistance, but the more I help you the less credit you will receive. Have fun! ☺ (if you skip steps and do not show all of your work you will not receive full credit)

(write all answers in pencil!)

The following are ways in which one can determine the derivative at a point (specifically the derivative of at :

• Using the difference quotient over the interval for smaller and smaller values of .

• Using the definition of the derivative and taking the limit as for .

• Finding the derivative function using the definition , then evaluating at .

• Finding the slope of the tangent line at .

Think:

Consider the above procedures for finding the instantaneous rate of change of at . Do you think that each of these procedures will have the same result or different results? Explain.

Experiment: (conduct the following experiments with your partner)

• Use the difference quotient over the interval for smaller and smaller values of (i.e. , etc.) for .

• Use the definition of the derivative on and take the limit as for .

• Find the derivative function for using the definition , then evaluate at .

• Find the slope of the tangent line at for . (you will need to draw the graph of and use a straight edge to draw the line that is tangent to the point ).

Share:

• Did your experiment agree with your hypothesis in the “think” portion of this activity? Explain.

• Are there any conclusions that you can make about limits, the difference quotient, and the definition of the derivative after conducting this experiment.

Name ____________________________ points _________

Date _____________________________