1. W-up Get out note paper Find 12.3 notes
Write the 3 steps to determine if a function is continuous
1) F is defined at c; that is, c is in the domain of f so that f(c) equals a #.
2) lim 𝑥→ 𝑐 − 𝑓 𝑥 =𝑓(𝑐)
3) lim 𝑥→ 𝑐 + 𝑓 𝑥 =𝑓(𝑐)
*So the same output is approached from both sides.

2. 14.1 Horizontal and Vertical Tangent Lines; Continuity and Differentiability
Find horizontal and vertical tangent lines
Discuss the graph of a function f where the derivative of f does not exist

3. Horizontal tangent line
If function is differentiable at point c, horizontal tangent line occurs when f ‘ (c) = 0
Ex1: At what points is the tangent line of
f(x)=x3 + 3x2 – 24x horizontal?
Find derivative: f ‘(x) = 3x2 + 6x – 24
Set f ‘(x) = 0 and solve
0 = 3x2 + 6x – 24 factor
3(x2 + 2x – 8) = 0
(x + 4 ) (x – 2) = 0
x = -4 and x = 2
Horizontal tangent lines will occur at (-4,80) and (2, -28)
Plug x values into ORIGINAL equation to find the y-value where tangent line is horizontal
f(-4) = (-4)3 + 3(-4)2 – 24(-4) = 80
f(2) = (2)3 + 3(2)2 – 24(2) = -28

4. Conditions for Vertical Tangent Lines
If a vertical tangent line is present at a point (c, f(c)) on a continuous function then
1) x = c is in the domain
2) f’(x) must be unbounded at x = c
Unbounded will be where the denominator of the derivative = 0

5. Ex1: find horizontal and vertical tangent lines of f(x) = -3x2 + 12x
Step 1 state domain – all real numbers
Step 2- find derivative
f ‘(x) = -6x + 12
To find horizontal set f ‘(x) = 0 and solve
0 = -6x + 12
x = 2
Plug 2 into original to find y
f(2) = -3(2)2 + 12(2) = 12
Horizontal tangent line at (2, 12)
When will there be a vertical tangent line?
NO VERTICAL TAN LINE – there is no denominator (no unbounded points)

6. Ex2: find horizontal and vertical tangent lines of
1) FIND domain 𝑥≠1
2) find the derivative – good times
Get common denominator

7. Find horizontal tangent line set derivative = 0 and solve
A fraction will be zero when numerator is zero
x=-2
Find y by plugging x into original equation
Horizontal tangent line at (-2,.53)
To find vertical tangent line set denominator of derivative = 0 and solve
Set each factor = 0 and solve
x = 0 and x = 1
Throw out 1 NOT IN DOMAIN
Find y by plugging x into original equation
Vertical tangent line a (0,0)

8. Continuity and Differentiability
If c is a number in the domain of f and f is differentiable at the number c, then f is continuous at c
If it is not continuous it is not differentiable.
Converse is not always true – if a graph is continuous at c, it may not be differentiable

9. Given: a) determine if it is continuous at x = 0
Is it defined
find left and right side limits
Yes continuous all equal zero
B) does f’(0) exist?
Plug in zero
Will this be a VERTICAL tangent line or NO tangent line?

10. Given: a) determine if it is continuous at x = 1
Is it defined f(1) = = 6
find left and right side limits
Yes continuous all equal six

11. B) does f’(c) exist? GRAPH
f(x) = 6x; x< 1
f(x) = x2+5; x> 1 1 6 -1 -6 1 6 2 9 3 14
f ‘(1) does not exist, there are two different slopes at 1
Find the derivative of each part
f’ (x) = 6 f’(x) = 2x
f’(1) = 6 f’ (1) = 2

12. Homework
14.1 # 1 – 33 odds
Don’t forget to graph # 27-33