Transcript POWER SERIES - MATHCHICK.NET

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
xc
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
xc
xc
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.