Assignment 4
Section 3.1
The Derivative and Tangent Line
Problem
The Basic Question is…
• How do you find the equation of a line that
is tangent to a function y=f(x) at an
arbitrary point P?
• To find the equation of a line you need:
a point and a slope
How do you find the slope when
the line is a tangent line?
First, we approximate with the
secant line.
f ( x  h)  f ( x )
m 
h
sec
How do we make the
approximation better?
• Choose h smaller…
• And smaller…
•
And smaller…
•
And smaller…
• How close to zero can it get?
• Infinitely
Definition of slope of the tangent
line
If f(x) is defined on an open interval (a,b)
then the slope of the tangent line to the
graph of y=f(x) at an arbitrary point (x,f(x))
is given by:
f ( x  h)  f ( x )
m  lim
h
h 0
Example:
• #6—Find the slope of the tangent line to the
graph of the function at the given point.
•
2
•
(-2, -2)
g ( x)  5  x
The limit that is the slope of the
tangent line is actually much more..
• Definition of the Derivative of a Function
The derivative of f at x is given by
f ( x  h)  f ( x )
f ' ( x)  lim
h 0
h
Provided the limit exists. For all x for which the limit
exists, f ' is a function of x.
Notations for derivative
f ' ( x)
dy
dx
y'
d
[ f ( x )]
dx
Dx [ y ]
Find the derivative by the limit
process.
#20
f ( x)  x  x
3
2
#24
f ( x) 
4
x
Find an equation of the tangent line
to th graph of f at the given point.
• #26
f ( x)  x  2 x  1
2
»
( - 3, 4)
#34
Find an equation of the line that is tangent to the
graph of f and parallel to the given line.
f ( x)  x  2
3
3x  y  4  0
Sketch the graph of f’
#46
What destroys the derivative at a
point?
a) Cusps
b) Corners
c) Vertical tangents
And…
Points of Discontinuity
Fact: If a function is differentiable at x=c,
then f is continuous at x=c