Section 2.9What does f ′ say about f ?

Math 1a

February 15, 2008

Announcements

I no class Monday 2/18! No office hours 2/19.

I ALEKS due Wednesday 2/20 (10% of grade).

I Office hours Wednesday 2/20 2–4pm SC 323

I Midterm I Friday 2/29 in class (up to §3.2)

Outline

Cleanup

Increasing and Decreasing functions

Concavity and the second derivative

Last worksheet, problem 2

Graphs of f , f ′, and f ′′ are shown below. Which is which? Howcan you tell?

x

y

SolutionAgain, look at the horizontal tangents. The short-dashed curve hashorizontal tangents where no other curve is zero. So its derivativeis not represented, making it f ′′. Now we see that where the boldcurve has its horizontal tangents, the short-dashed curve is zero, sothat’s f ′. The remaining function is f .

Outline

Cleanup

Increasing and Decreasing functions

Concavity and the second derivative

Definition

I Let f be a function defined on and interval I . f is calledincreasing if

f (x1) < f (x2) whenever x1 < x2

for all x1 and x2 in I .

I f is called decreasing if

f (x1) > f (x2) whenever x1 < x2

for all x1 and x2 in I .

Definition

I Let f be a function defined on and interval I . f is calledincreasing if

f (x1) < f (x2) whenever x1 < x2

for all x1 and x2 in I .

I f is called decreasing if

f (x1) > f (x2) whenever x1 < x2

for all x1 and x2 in I .

Examples: Increasing

Examples: Decreasing

Examples: Neither

Fact

I If f is increasing and differentiable on (a, b), then f ′(x) ≥ 0for all x in (a, b)

I If f is decreasing and differentiable on (a, b), then f ′(x) ≤ 0for all x in (a, b).

Proof.Suppose f is increasing on (a, b) and x is a point in (a, b). Forh > 0 small enough so that x + h < b, we have

f (x + h) > f (x) =⇒ f (x + h)− f (x)

h> 0

So

limh→0+

f (x + h)− f (x)

h≥ 0

A similar argument works in the other direction (h < 0). Sof ′(x) ≥ 0.

Fact

I If f is increasing and differentiable on (a, b), then f ′(x) ≥ 0for all x in (a, b)

I If f is decreasing and differentiable on (a, b), then f ′(x) ≤ 0for all x in (a, b).

Proof.Suppose f is increasing on (a, b) and x is a point in (a, b). Forh > 0 small enough so that x + h < b, we have

f (x + h) > f (x) =⇒ f (x + h)− f (x)

h> 0

So

limh→0+

f (x + h)− f (x)

h≥ 0

A similar argument works in the other direction (h < 0). Sof ′(x) ≥ 0.

Fact

I If f is increasing and differentiable on (a, b), then f ′(x) ≥ 0for all x in (a, b)

I If f is decreasing and differentiable on (a, b), then f ′(x) ≤ 0for all x in (a, b).

Proof.Suppose f is increasing on (a, b) and x is a point in (a, b). Forh > 0 small enough so that x + h < b, we have

f (x + h) > f (x) =⇒ f (x + h)− f (x)

h> 0

So

limh→0+

f (x + h)− f (x)

h≥ 0

A similar argument works in the other direction (h < 0). Sof ′(x) ≥ 0.

Example

Here is a graph of f . Sketch a graph of f ′.

Example

Here is a graph of f . Sketch a graph of f ′.

Fact

I If f ′(x) > 0 for all x in (a, b), then f is increasing on (a, b).

I If f ′(x) < 0 for all x in (a, b), then f is decreasing on (a, b).

The proof of this fact requires The Most Important Theorem inCalculus.

Fact

I If f ′(x) > 0 for all x in (a, b), then f is increasing on (a, b).

I If f ′(x) < 0 for all x in (a, b), then f is decreasing on (a, b).

The proof of this fact requires The Most Important Theorem inCalculus.

Outline

Cleanup

Increasing and Decreasing functions

Concavity and the second derivative

Definition

I A function is called concave up on an interval if f ′ isincreasing on that interval.

I A function is called concave down on an interval if f ′ isdecreasing on that interval.

Definition

I A function is called concave up on an interval if f ′ isincreasing on that interval.

I A function is called concave down on an interval if f ′ isdecreasing on that interval.

Fact

I If f is concave up on (a, b), then f ′′(x) ≥ 0 for all x in (a, b)

I If f is concave down on (a, b), then f ′′(x) ≤ 0 for all x in(a, b).

Fact

I If f is concave up on (a, b), then f ′′(x) ≥ 0 for all x in (a, b)

I If f is concave down on (a, b), then f ′′(x) ≤ 0 for all x in(a, b).

Fact

I If f ′′(x) > 0 for all x in (a, b), then f is concave up on (a, b).

I If f ′′(x) < 0 for all x in (a, b), then f is concave down on(a, b).