Section 2.9 What does f 0 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)
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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).