lesson 9: the derivative as a function

Section 2.8The Derivative as a Function

Math 1a

October 15, 2007

Announcements

I Midterm I review session 10/22, 7:30pm in Hall D

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page1of14

Last time: Worksheet problems 3 and 4

ProblemLet f (x) = x1/3. Find f ′(x) and its domain.

Answer

f ′(x) =1

3x−2/3. The domain is all numbers except 0.

ProblemLet f (x) = x2/3. Find f ′(x) and its domain.

Answer

f ′(x) =2

3x−1/3. The domain is all numbers except 0.

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page2of14

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page3of14

Last time: Worksheet problems 3 and 4

ProblemLet f (x) = x1/3. Find f ′(x) and its domain.

Answer

f ′(x) =1

3x−2/3. The domain is all numbers except 0.

ProblemLet f (x) = x2/3. Find f ′(x) and its domain.

Answer

f ′(x) =2

3x−1/3. The domain is all numbers except 0.

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page4of14

Last time: Worksheet problems 3 and 4

ProblemLet f (x) = x1/3. Find f ′(x) and its domain.

Answer

f ′(x) =1

3x−2/3. The domain is all numbers except 0.

ProblemLet f (x) = x2/3. Find f ′(x) and its domain.

Answer

f ′(x) =2

3x−1/3. The domain is all numbers except 0.

Super-continuity

TheoremIf f is differentiable at a, then f is continuous at a.

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page5of14

How can a function fail to be continuous?

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page6of14

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page7of14

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page8of14

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page9of14

Notation

I Newtonian notation

f ′(x) y ′(x) y ′

I Leibnizian notation

dy

dx

d

dxf (x)

df

dx

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page10of14

The second derivative

If f is a function, so is f ′, and we can seek its derivative.

f ′′ = (f ′)′

It measures the rate of change of the rate of change!

Leibnizian notation:

d2y

dx2

d2

dx2f (x)

d2f

dx2

The second derivative

If f is a function, so is f ′, and we can seek its derivative.

f ′′ = (f ′)′

It measures the rate of change of the rate of change!Leibnizian notation:

d2y

dx2

d2

dx2f (x)

d2f

dx2

Worksheet #1

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page13of14

Worksheet #2

Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007

Page14of14