1. Section 2.1 Day 2 Derivatives
AP Calculus AB

2. Learning Targets
Use the limit definition to find the derivative of a function
Use the limit definition to find the derivative at a point
Distinguish between an original function and its derivative graph
Draw a graph based on specific conditions
Determine where a derivative fails to exist
Determine one-sided derivatives
Determine values that will make a function differentiable at a point
Define local linearity
Find the slope of the tangent line to a curve at a point
Distinguish between continuity and differentiability

3. Group Work Tasks for each group: 1. Graph the original function
2. Find the derivative function
3. Graph the derivative function in a different color
4. List both function equations on the sheet
Group Work

4. Group Assignments Group 1: 𝒚= 𝒙 𝟐 +𝟒 Group 2: 𝑦=2𝑥−32 and 𝑦=8

5. With your group, look around at all of the examples
With your group, look around at all of the examples. Answer the following questions.
1. What do you notice about the original equation and the derivative equation? What are some patterns you notice?
2. What do you notice about the original graph and the derivative graph? What are some patterns you notice?
Group Observations

6. 1. The derivative graph is one degree less than the original graph
2. The derivative graph shows the slopes for the original graph
3. When the slope is decreasing, this translates to negative for the derivative
4. When the slope is increasing, this translates to positive for the derivative
5. When the slope is zero, the derivative graph has an x-intercept
Key Points

7. Applet Applet to help visualize these topics
GebraCalculus/derivative_elementary_fu nctions.html
Applet

8. Example 1:
Draw the derivative graph for the following function USE HIGHLIGHTERS TO SAHDE IN THE REALTIONSHIOPESSSSSS!!!!!!!

9. Example 2:
Draw the derivative graph for the following function

10. Example 3:
Draw the derivative graph for the following function

11. Example 4:
The following graph is the derivative. What might the original graph look like?

12. Example 5: Draw a graph with the following conditions:
1. lim 𝑥→4 𝑓 𝑥 𝐷𝑁𝐸
2. 𝑓 4 =−1
3. For 𝑥<−1, 𝑓 ′ 𝑥 >0
4. For 𝑥=2, 𝑓 ′ 𝑥 =0

13. Match each graph with its derivative
Exit Ticket : Feedback
Match each graph with its derivative
Original Derivative