Section 3.2b
The “Do Now”
Find all values of x for which the given function is differentiable.
h  x   3x  6  5
3
This function is differentiable except possibly where
3x  6  0  x  2
Check for differentiability at x = 2:
3 3 2  k  6  5  5



h  2  k   h  2

lim
 lim 
k 0
k 0
k
k
3
The function has a
3k 3
1
vertical tangent at x = 2.  lim
 3 lim 2 3  
k 0
k 0 k
k
It is differentiable for all
reals except x = 2.
The “Do Now”
Find all values of x for which the given function is differentiable.
Q  x   3cos  x 
What type of symmetry does the cosine function have???
 Cosine is an even function (symmetric about the y-axis), so
Q  x   3cos  x   3cos x
This is just a vertical stretch of the basic cosine function
 It is differentiable for all reals.
The “Do Now”
Find all values of x for which the given function is differentiable.
 x , x  0
C  x  x x   2
 x , x  0
2
Let’s rewrite C as
a piecewise function…
This function is differentiable for all x except possibly at x = 0:
C  0  h   C  0
h h 0
lim

lim
h

lim
h 0
h 0
h 0
h
h
The function is differentiable for all reals.
(support graphically?)
0
The difference quotient:
f a  h  f a
h
For small values of h, this is a good approximation of
f a
To get an even better approximation, we can use the
symmetric difference quotient: f a  h  f a  h




2h
This is what our calculator uses to find the numerical derivative
of a function, denoted NDERf x
 
We only need an h value of about 0.001 to get accurate values
for derivatives  most calculators use
f  a  0.001  f  a  0.001
NDERf  a  
0.002
The Symmetric Difference Quotient Graphically
f a  h  f a  h
m2 
2h
f a  h  f a 
m1 
h
Tangent line
Which approximation
is better???
ah
a
ah
Practice Problems
Find the derivative of the cubing function:
f  x  x
3
2

f  x   3x
Look to your notes!!!
f   2  12
What is the value of this derivative at x = 2?
3
NDER
x
Compute
 , 2 , the numerical derivative of the cubing
function at x = 2.
NDER  x , 2 
3
2.001  1.999 


With your calculator:
3
0.002
3
 12.000001
NDER  x 3 , x, 2   12.000001
Practice Problems
Compute the numerical derivative of the absolute value function
at x = 0.
NDER  x , 0   lim
h 0
 lim
h 0
0h  0h
2h
hh
2h
0
 lim
h 0 2h
0
Do you get the same answer with your calculator?
Does this answer make sense?
 Your calculator can be fooled!!! (It uses the
symmetric difference quotient, which never
detects the corner of this graph at x = 0…)
Practice Problems
Use NDER to graph the derivative of the given function. Can you
guess what function the derivative is by analyzing its graph?
f  x   ln x
In your calculator:
y1  NDER  ln x, x, x 
Use the window
2, 4 by 1,3
What function does the derivative look like???
1
f  x 
x
Practice Problems
Use NDER to graph the derivative of the given function. Can you
guess what function the derivative is by analyzing its graph?
f  x  x
2
In your calculator:
y1  NDER  x , x, x 
2
Use the window
10,10 by 10,10
What function does the derivative look like???
f   x   2x
Practice Problems
Use NDER to graph the derivative of the given function. Can you
guess what function the derivative is by analyzing its graph?
f  x   sin x
In your calculator:
y1  NDER  sin x, x, x 
Use the window
2 , 2  by 2, 2
What function does the derivative look like???
f   x   cos x