1. Tangents and Differentiability

2. 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

3. 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?

4. 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.

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

6. 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?

7. 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.

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

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

10. 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!

11. 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)

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

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

14. 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

15. 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

16. 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

17. 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?

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