POWER SERIES - MATHCHICK.NET
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THE DERIVATIVE AND THE
TANGENT LINE PROBLEM
Section 2.1
When you are done with your
homework, you should be able
to…
– Find the slope of the tangent line
to a curve at a point
– Use the limit definition to find the
derivative of a function
– Understand the relationship
between differentiability and
continuity
The Tangent Line Problem
How do we find an equation of the tangent
line to a graph at point P?
We can approximate this slope using a
secant line through the point of tangency
and a second point on the curve.
f c x
f c
x
c
c x
f c x f (c)
Find the equation of the secant
2
line to the function f x x 5
at x 2 and x 3.
A. Y = -5x + 19
B. Y = 5x - 11
C. There is not enough information to
solve this problem.
A secant line represents the
A. Instantaneous rate of change of a
function.
B. The average rate of change of a
function.
C. Line tangent to a function.
Definition of the
Derivative of a Function
The derivative of f at x is given by
f x x f x
f x lim
x 0
x
provided the limit exists. For all x
for which this limit exists, f’ is a
function of x.
Definition of Tangent
Line with Slope m
• If f is defined on an open interval containing c,
and if the limit
f c x f c
y
lim
lim
m
x 0 x
x 0
x
exists, then the line passing through f with slope
m is the tangent line to the graph of at the point
c, f c.
• The slope of the tangent line to the graph of f at
the point c is also called the slope of the graph
of f at c, f c .
2
f
x
x
5
Find the slope of the graph of
at x 2.
A.
B.
C.
D.
4
9
1
Does not exist
Alternative
f
x
f
c
limit form of f c lim
x c
xc
the derivative
The existence of the limit in this alternative form
requires that the following one-sided limits
f x f c
f x f c
lim
and lim
x c
x c
xc
xc
exist and are equal.
These one-sided limits are called the derivatives from
the left and from the right, respectively. It follows
that f is differentiable on the closed interval a, b if
it is differentiable on a, b and if the derivatives from
the right at a and the derivative from the left at b
both exist.
Evaluate the
derivative of
A.
B.
C.
D.
f x 5 x at x 4
-1
0
1
Does not exist
THEOREM: Differentiability
Implies Continuity
If f is differentiable at x c,
then f is continuous at x c.