Section 2.1 Day 2 Derivatives AP Calculus AB

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

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

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

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

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

Applet Applet to help visualize these topics http://webspace.ship.edu/msrenault/Geo GebraCalculus/derivative_elementary_fu nctions.html Applet

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

Example 2: Draw the derivative graph for the following function

Example 3: Draw the derivative graph for the following function

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

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

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