Math 1431
Section 14819
TTh 10-11:30am 100 SEC
Bekki George
bekki@math.uh.edu
639 PGH
Office Hours:
11:30am – 12:30pm TTh and by appointment
Class webpage:
http://www.math.uh.edu/~bekki/Math1431.html
POPPER01
1. Which of the following does not have the same value as sin (x)?
2. Compute lim
x 
3.
sin  x 
x
.
Evaluate the following limit:
lim
x 9
x 3
x 3
The Derivative
Measuring how f (x) changes when x changes.
Section 2.1
Slope of a Secant line –
–
slope between two points
average rate of change
Slope of a Tangent line – slope at A point
– instantaneous rate of change
y = f (x)
y = f (x)
y = f (x)
Secant Line
y = f (x)
Secant Line
y = f (x)
Secant Line
y = f (x)
Tangent Line
Slope of the Tangent Line at the point x = a is
lim
h0
f  a  h  f  a
h
Which is
the instantaneous rate of change of f at x = a
Which is
the derivative of f at x = a
PreCal Quick Review:
For f  x   2x  3
f (a) =
f (a + h) =
f (3) =
f (3 +h) =
For f  x   x 2  1
f (a) =
f (a + h) =
f (3) =
f (3 +h) =
Using what we know, find the slope of the tangent line at x = 3 if
f  x   x 2  1.
The Definition of the Derivative
A function f (x) is differentiable at x if and only if
lim
f  x  h  f  x 
h
h0
exists. In this case, we denote
f '  x   lim
h0
f  x  h  f  x 
h
and we refer to f '  x  as the derivative of f at x.
f '  x  can be thought of as the slope function.
It gives the slope of the graph of f  x  at any point x.
d
f x .
You may also see f '  x  written as
dx
If y  f  x  you may see
dy
f '  x   y' 
.
dx
Find f '  x  where f  x   x 2  2x using the definition of the derivative.
Use the previous result to give the equation of the tangent line to the graph
of f  x   x 2  2x at x = –1.
Find f '  x  where f  x   x 3  2 using the definition of the derivative.
1
Find f '  x  where f x =
using the definition of the derivative.
x +1
()
If f is differentiable at x = a, then f is continuous at x = a.
Not every continuous function is differentiable.
Example: The function y=|x| is continuous but not differentiable at x=0.
The Weierstrass function is continuous everywhere and differentiable
nowhere!
How can the graph of a function be used to determine where a function is
not differentiable?
A function is not differentiable at
1. points of discontinuity
2. cusps
3. sharp turns (corners)
POPPER01
1
4. The value of the derivative at x = 2 for f  x  
x 1
is the ______ of the tangent line at x = 2.
5. The derivative of a function y = f (x) is denoted as
Determine if f (x) is differentiable at x = 2.

x 2  1 x  2
f x  

4 x  3 x  2
Determine if f (x) is differentiable at x = 1.

x x  1
f x   2

x x  1
How can we use the derivative to find the slope of the normal line to the
graph of f (x) at x = a?
The normal line to the graph at x = a is
the perpendicular line to the graph at x = a .
That is:
The normal line is
perpendicular to the tangent line at x = a.
Popper01
6. Give the slope of the normal line to the graph of f  x  
at x = 3?
1
x 1
Algebraic Properties of the Derivative
Differentiation Formulas
Section 2.2
If f and g are differentiable and c is a scalar, then f + g, f – g and
(c f ) are differentiable. Furthermore,
Derivative of the sum is the sum of the derivatives.
d
d
d
f x  g x 
f x 
g x
dx
dx
dx


Derivative of the difference is the difference of the derivatives.
d
d
d
f x  g x 
f x 
g x
dx
dx
dx
And the derivative of any scalar times a function is the scalar times the
derivative of the function.


d
d
c f x  c
f x
dx
dx


d
8
dx
d
x
dx
d
5x 

dx
d
5 x  2 

dx
Power Rule
 
d
x n  nx n1 , n  0
dx
Find the derivative of each.
f x  x2
f x  x3
f x  x5  x2
f  x   3x 4  2x 3  4 x
f x 
f x  x
( )
x x
1
2
9
7
5
7
f x = 1
x2
x
Higher Order Derivatives
f ' x,
f ''  x  ,
f '''  x  ,
f
d
f x,
dx
d2
f x,
2
dx
d3
f x,
3
dx
d4
f x
4
dx
4
x
Determine
Determine
d2
dx
2
d3
dx
3

3x 3  5x 2  2x  1


3x 8  2x 5  3x  5

POPPER01
7. The limit lim
1  h 5  1
h
number c. Determine f.
h0
represents the derivative of a function f at a
2
f
(x)
=
6x
- 2x +1, f '(x) =
8.