NUMERICAL

DERIVATIVES

Section 3.2b

The “Do Now”Find all values of x for which the given function is differentiable.

3 3 6 5h x x This function is differentiable except possibly where

3 6 0x Check for differentiability at x = 2:

2x

0

2 2limk

h k h

k

3

0

3 2 6 5 5limk

k

k

3

0

3limk

k

k 3

2 30

13 limk k

The function has avertical tangent at x = 2.It is differentiable for allreals except x = 2.

The “Do Now”Find all values of x for which the given function is differentiable.

3cosQ x xWhat type of symmetry does the cosine function have???

Cosine is an even function (symmetric about the y-axis), so

3cosQ x x 3cos xThis 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.

C x x x Let’s rewrite C asa piecewise function…

This function is differentiable for all x except possibly at x = 0:

0

0 0limh

C h C

h

The function is differentiable for all reals.(support graphically?)

2

2

, 0

, 0

x x

x x

0

0limh

h h

h

0limhh

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 thesymmetric difference quotient:

2

f a h f a h

h

This is what our calculator uses to find the numerical derivativeof a function, denoted NDERf x

We only need an h value of about 0.001 to get accurate valuesfor derivatives most calculators use

0.001 0.001NDER

0.002

f a f af a

The Symmetric Difference Quotient Graphically

2 2

f a h f a hm

h

a h

1

f a h f am

h

a a h

Tangent line

Which approximationis better???

Practice Problems

Find the derivative of the cubing function:

23f x x Look to your notes!!!

3f x x

What is the value of this derivative at x = 2? 2 12f Compute , the numerical derivative of the cubingfunction at x = 2.

3NDER ,2x

3 3

3 2.001 1.999NDER ,2

0.002x

12.000001

With your calculator:

3NDER , ,2x x 12.000001

Practice Problems

Compute the numerical derivative of the absolute value functionat x = 0.

0

0 0NDER ,0 lim

2h

h hx

h

00

lim2h

h h

h

Do you get the same answer with your calculator?

0

0lim2h h

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 ProblemsUse NDER to graph the derivative of the given function. Can youguess what function the derivative is by analyzing its graph?

lnf x x In your calculator:

1 NDER ln , ,y x x x

2,4 by 1,3 Use the window

1f x

x

What function does the derivative look like???

Practice ProblemsUse NDER to graph the derivative of the given function. Can youguess what function the derivative is by analyzing its graph?

2f x x In your calculator:

21 NDER , ,y x x x

10,10 by 10,10 Use the window

2f x x

What function does the derivative look like???

Practice ProblemsUse NDER to graph the derivative of the given function. Can youguess what function the derivative is by analyzing its graph?

sinf x x In your calculator:

1 NDER sin , ,y x x x

2 ,2 by 2,2 Use the window

cosf x x

What function does the derivative look like???