Section 2.1 Day 3 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

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.

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

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.

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

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→ 0 + 𝑓 0+ℎ −𝑓 0 ℎ =2 2. Limits do not match. Thus the derivative does not exists at that point.

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→ 0 + 𝑓 1+ℎ −𝑓 1 ℎ =3 The limits exist. Thus, the derivative exists at that point

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

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

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

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)

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