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04/20/23 Perkins
AP Calculus AB
Day 1Section 2.1
-2 2 4 6 8
4
3
2
1
-1
-2
Find the slope of the tangentline to f(x) at the given point.
x , f x f(x)
x
B
The Derivative (Definition of the Derivative)
or 0
' limx
f x x f xf x
x
0
' limh
f x h f xf x
h
The Alternative Form of the Definition of the Derivative
' lim
x c
f x f cf c
x c
Gives the numeric value of the derivative at a given point, x = c.
Gives a generic expression into which we can substitute any x to get a numeric value of the derivative at that point.
1. Find using the definition of the derivative.
f x x
'f x
0
' limx
f x x f xf x
x
0limx
x x x
x
x x x
x x x
0limx
x x x
x x x x
0limx
x
x x x x
0
1limx x x x
1
2 x
2. Find the equation of the tangent line to at (1,5). Use the alternative form of the definition of the derivative.
23 2f x x
1
1' 1 lim
1x
f x ff
x
2
1
3 2 5lim
1x
x
x
2
1
3 3lim
1x
x
x
1
3 1 1lim
1x
x x
x
1
lim 3 1x
x
6D.S.
165y x
6 1y x
3. Find the slope of the tangent line to at x = 2. 2f x x
4
2
-2
-4
-5 5
This function has a sharp turn at x = 2.
Functions are not differentiable at
a. Discontinuities
b. Sharp turns
c. Vertical tangents
Therefore the slope of the tangent line at x = 2 does not exist.
4. Find any values where is not differentiable. 1
3f x
x
This function has a V.A. at x = 3.
Theorem:
If f is differentiable at x = c,
then it must also be continuous at x = c.
Therefore the derivative at x = 3 does not exist.
4
2
-2
-4
-5 5
Perkins
AP Calculus AB
Day 1Section 2.1
-2 2 4 6 8
4
3
2
1
-1
-2
Find the slope of the tangentline to f(x) at the given point.
x , f x f(x)
x
B
The Derivative (Definition of the Derivative)
or 0
' limx
f x x f xf x
x
0
' limh
f x h f xf x
h
The Alternative Form of the Definition of the Derivative
' lim
x c
f x f cf c
x c
1. Find using the definition of the derivative.
f x x
'f x
2. Find the equation of the tangent line to at (1,5). Use the alternative form of the definition of the derivative.
23 2f x x
3. Find the slope of the tangent line to at x = 2. 2f x x
4. Find any values where is not differentiable. 1
3f x
x
