Tangents and Differentiability

Tangent Lines Def: (we already know this) To find the equation of a line tangent to a curve, you need a point and the derivative to find the slope. Ex 1. Find the equation of the tangent line to the curve 𝑓 𝑥 =3 𝑥 2 −5𝑥+2 at the point where 𝑥=2

Horizontal and Vertical Tangents What is unique about a horizontal tangent line? What is the slope of a horizontal tangent line? Can you sketch a graph where the tangent line is horizontal?

Horizontal and Vertical Tangents Def: (we already know this) The graph of 𝑓(𝑥) has a horizontal tangent at points where 𝑓 ′ 𝑥 =0. Steps to finding HT: Find 𝑓′(𝑥) Set 𝑓 ′ 𝑥 =0 and solve for x. Find 𝑦 value by plugging answers back into 𝑓(𝑥). If 𝑓(𝑥) is undefined at that point, then it is extraneous.

Ex 2. Find points where 𝑓 𝑥 =2 𝑥 3 +3 𝑥 2 −36𝑥 has a horizontal tangent.

Horizontal and Vertical Tangents What is unique about a vertical tangent line? What is the slope of a horizontal tangent line? Can you sketch a graph where the tangent line is horizontal?

Def: The graph of 𝑓 𝑥 has a vertical tangent at points where 𝑓′(𝑥) is not defined, i.e. where the denominator of 𝑓 ′ 𝑥 =0. Steps to finding VT: Find 𝑓 ′ 𝑥 Set denominator of 𝑓 ′ 𝑥 =0 and solve for x. Find 𝑦 value by plugging answers back into 𝑓(𝑥). If 𝑓(𝑥) is undefined at that point, then it is extraneous.

Ex 3. Find the point(s) where 𝑔 𝑥 = 4− 𝑥 2 +3 has a vertical tangent.

III. Differentiability What does it mean for a function to be differentiable? Can you sketch the situations where a function is not differentiable?

III. Differentiability (We already know) Derivatives do not exist when the function is not continuous, there is a vertical tangent line, or there is sharp turn or edge. (We already know) If a function is differentiable, then it is continuous. Not necessarily the other way around!

IMPORTANT DEFINITION A function is differentiable at 𝑥=𝑎 if: I. 𝑓(𝑥) is continuous at 𝑥=𝑎 i.) 𝑓(𝑎) is defined. ii.) exists. iii.) lim 𝑥→𝑎 𝑓(𝑥) =𝑓(𝑎) lim 𝑥→𝑎 𝑓(𝑥) II. exists. lim 𝑥→𝑎 𝑓′(𝑥) i.) lim 𝑥→ 𝑎 − 𝑓′(𝑥) = lim 𝑥→ 𝑎 + 𝑓′(𝑥)

Ex 5. Determine if the following function is differentiable at 𝑥=1. 𝑔 𝑥 = 8𝑥−3, 𝑥≤1 4 𝑥 2 +5, 𝑥>1

Ex 6. Determine whether the function is differentiable at 𝑥=3 ℎ 𝑥 = 𝑥 2 −4𝑥+8, 𝑥≤3 2𝑥−1, 𝑥>3

Ex 7. Find values of 𝑎 and 𝑏 that make the function differentiable at 𝑥=2. 𝑓 𝑥 = 𝑥 2 , 𝑥≤2 𝑎𝑥+𝑏, 𝑥>2

Ex 8. Find values of 𝑎 and 𝑏 that make the function differentiable at 𝑥=2. 𝑓 𝑥 = 𝑎 𝑥 3 , 𝑥≤2 𝑥 2 +𝑏, 𝑥>2

𝑓 ′ 𝑥 = lim ℎ→0 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ Refresher! Limit Definition of Derivative 𝑓 ′ 𝑥 = lim ℎ→0 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ 𝑓 ′ 𝑥 = lim 𝑥→𝑎 𝑓(𝑥)−𝑓(𝑎) 𝑥−𝑎

Does this look familiar? What about this one? IV. Derivatives Disguised as Limits lim ℎ→0 ln 2+ℎ − ln 2 ℎ lim 𝑥→2 𝑒 𝑥 − 𝑒 2 𝑥−2 Does this look familiar? What about this one?

IV. Derivatives Disguised as Limits lim ℎ→0 sin 𝜋 3 +ℎ − 3 2 ℎ lim ℎ→0 𝑒 2+ℎ − 𝑒 2 ℎ