2.1 Tangent Line Problem
Tangent Line Problem
The tangent line can be
found by finding the
slope of the secant line
through the point of
tangency and a point on
the curve
Point A is the point of tangency
Tangent Line Problem
• How to find slope of a curve at a point?
y  f (x)
Secant Line
Tangent Line
x
x + Δx
y  f (x)
x  0
Setting up a limit!
Slope of the
Tangent Line
x
x + Δx
1.) Find slope of the secant line
y  f (x)
Secant Line
x
x + Δx
y  f (x)
Conclusion:
mtan  lim msec
x 0
x + Δx
x
mtan
f ( x  x)  f ( x)
 lim
x 0
x
Called the difference quotient
Definition of the Derivative
For a function f(x) the average rate of
change along the function is given by:
lim
f ( x  x)  f ( x)
m  f ' ( x) 
x  0
x
f ( x  h)  f ( x )
or
lim
h
h 0
Which is called the derivative of f
Notation of the Derivative
The derivative of a function f at x is
given by:
mtan
f ( x  x)  f ( x)
 f ' ( x)  lim
x 0
x
**Provided the limit exists
Notation: f (x )
dy
dx
d [ f ( x )]
dx
Dx [y]
2.) Find the slope of the tangent line to the
2
curve y  x at (2,6)
mtan
f ( x  h)  f ( x )
 lim
h 0
h
First, find the Slope at any point
Terminology
• Differentiation (Differentiate) – the process
of finding the derivative
• Differentiable – when a functions
derivative exists at x
When Derivatives Fail
1. Cusp or sharp point:
cusp
When Derivatives Fail
2. Vertical asymptotes:
3. When one sided limits fail
When Derivatives Fail
4.
Removable discontinuity
When Derivatives Fail
5. Corners or vertical tangents
3.) Differentiate f ( x)  2 x  3(if possible)
4.) Differentiate f ( x)  x  x  1 (if possible)
2
5.) Differentiate f ( x) 
x
if possible
xh  x
xh  x
f ( x)  lim

h 0
h
xh  x
2
6.) Find the derivative of f ( x ) 
x 1
2
2

( x  h)  1 x  1
f ( x)  lim
h 0
h
HOMEWORK
Page 104
# 5 – 21 (odd), 61 and 62, 83-88 (all). Find
where f(x) is not differentiable and state
the type of discontinuity