1. Section 2.1 Day 3 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. In your groups,
Draw a picture of where you think the derivative might not exist. (PUT THIS PICTURE ON THE WHITE BOARD) Remember, the derivative is just a “fancy” limit.

4. Derivative fails to exist: Case 1
At a corner or cusp
Notice that the slopes on both sides of the cusp or corner are different. Thus, the limit doesn’t exist, so the derivative fails to exist

5. Derivative fails to exist: Case 2
At a vertical tangent
A vertical tangent implies that the derivative at that point resulted with zero in the denominator.

6. Derivative fails to exist: Case 3
At a discontinuity
It could be removable or non-removable

7. Example 1: One-Sided Limits
Determine if the derivative of the function exists at 𝑥=0 for 𝑓 𝑥 ={ 𝑥 2 , 𝑥≤0 2𝑥, 𝑥>0
1. Find lim ℎ→ 0 − 𝑓 0+ℎ −𝑓 0 ℎ =0 and lim ℎ→ 𝑓 0+ℎ −𝑓 0 ℎ =2
2. Limits do not match. Thus the derivative does not exists at that point.

8. Example 2: One-Sided Limits
Determine if the derivative of the function exists at 𝑥=1 for 𝑓 𝑥 ={ 𝑥 2 +𝑥, 𝑥≤1 3𝑥−2, 𝑥>1
Find lim ℎ→ 0 − 𝑓 1+ℎ −𝑓 1 ℎ =3 and lim ℎ→ 𝑓 1+ℎ −𝑓 1 ℎ =3
The limits exist. Thus, the derivative exists at that point

9. Example 3
Let 𝑓 𝑥 ={ 3 𝑥 2 , 𝑥≤1 𝑎𝑥+𝑏, 𝑥>1 . Find the values of 𝑎 and 𝑏 so that 𝑓 will be differentiable at 𝑥=1
1. Needs to be continuous: Thus, 𝑎+𝑏=3.
2. One – sided derivative limits need to match: 6𝑥=𝑎
3. At x = 1, 𝑎=6. Thus, 𝑏=−3

10. Differentiability & Continuity
Differentiability implies continuity.
More specifically, if 𝑓 has a derivative at 𝑥=𝑎, then 𝑓 is continuous at 𝑥=𝑎

11. Differentiability, Continuity, & Limits Examples
Continuous & differentiable
Continuous & not differentiable
Limit Exists & differentiable
Limit Exists & not differentiable
Not continuous, not differentiable, & limit does not exist

12. Example 5 Construct a graph with the following criteria:
The function is differentiable from (−2, 1)
The function is not differentiable at 𝑥=1
𝑓 1 =−4
lim 𝑥→−2 𝑓(𝑥) =0
𝑓 ′ 𝑥 >0 for (−∞, 0)

13. Exit Ticket for Homework
Worksheet