Building Understanding of the Derivative of a Function

1. Building Understanding of the Derivative of a Function Jim Rahn www.jamesrahn.com James.rahn@verizon.net

2. Equity & Access The College Board and the Advanced Placement Program encourage teachers, AP Coordinators, and school administrators to make equitable access a guiding principle for their AP programs. The College Board is committed to the principle that all students deserve an opportunity to participate in rigorous and academically challenging courses and programs. All students who are willing to accept the challenge of a rigorous academic curriculum should be considered for admission to AP courses. The Board encourages the elimination of barriers that restrict access to AP courses for students from ethnic, racial, and socioeconomic groups that have been traditionally underrepresented in the AP Program. Schools should make every effort to ensure that their AP classes reflect the diversity of their student Population.

3. Numerical Approach • Students should explore understanding the forward difference quotient, the backwards difference quotient, and the symmetric difference quotient

4. Forward Difference Quotient

5. Backward Difference Quotient

6. Symmetric Difference Quotient

7. We’ll explore these definitions numerically by building table values. • Start with a common function y1 = x2. • Then enter each of the difference quotients in y2, y3, and y4 by entering: • Turn off Y1 because we want to only look at values for y2, y3, and y4.

8. Return to the homescreen. Store 1 in the calculator for h. (1 STO H and press ENTER) • Set your Table Minimum to -3 and the Δtbl = 1. • Generate a set of table values for Y2, Y3, and Y4. • Write equations that can be modeled by these sets of values.

9. Change the h value to 0.1, 0.01, 0.001, and 0.0001. Review the table values for each of these new h values and make a prediction of the equation that can model the data.

10. Each equation may have used either 0.1, 0.01, 0.001, or 0.0001 Rewrite each equation by replacing 0.1, 0.01, 0.001, and 0.0001 with h.

11. Find the following limits Describe at least two ideas you learned by completing this exploration with the three difference quotients.

12. Understanding The Derivative Graphically Using Difference Quotients

13. We’ll study the derivative using three difference quotients

14. We’ll make observations about these definitions graphically by viewing several equations simultaneously. • Again start with a common function y1 = x2, but turn it off. • Enter each of the difference quotients in y2, y3, and y4:

15. Store 1 in the calculator for h. (1 STO H and press ENTER) • Generate graphs for Y2, Y3, and Y4 in a zoom 4 Decimal window. • Write an equation for Y2, Y3, and Y4.

16. Change the h value to 0.1, and 0.01. • After each change re-graph the three equations. • Trace along the graphs to notice the slope and the y-intercept for each equation. • Describe what you see taking place as you let h approach 0. • Find the following limits

17. If y1 = x2 what do you know about: They are all equal to 2x.

18. If f(x)=x2 It would now be appropriate to consider algebraically

21. Change y1 to x3

22. Let h equal 1, 0.1, and 0.01. • After each change re-graph the three equations. • Trace along the graphs to make an observation about y2, y3, and y4. • Describe what you see taking place as you let h approach 0. • Confirm your guess for the function by entering an equation in y5.

23. Find the following limits If y1 = x3 what do you know about: They are all equal to 3x2.

24. If f(x)=x3 It would now be appropriate to consider algebraically

27. Change y1 to sin x

28. Let h equal 1, 0.1, and 0.01. • After each change re-graph the three equations. • Trace along the graphs to make an observation about y2, y3, and y4. • Describe what you see taking place as you let h approach 0. • Confirm your guess for the function by entering an equation in y5.

29. Find the following limits If y1 = x3 what do you know about: They are all equal to cos x.

30. If f(x)=sin(x) It would now be appropriate to consider algebraically

34. Change y1 to cos x

35. Let h equal 1, 0.1, and 0.01. • After each change re-graph the three equations. • Trace along the graphs to make an observation about y2, y3, and y4. • Describe what you see taking place as you let h approach 0. • Confirm your guess for the function by entering an equation in y5.

36. Find the following limits If y1 = x3 what do you know about: They are all equal to cos x.

37. Do you have more understanding for the following limits?

38. Further Investigations