Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 1: Functions and their Representations
1. Section 1.1
Functions
V63.0121.006/016, Calculus I
January 19, 2010
Announcements
Syllabus is on the common Blackboard
Office Hours TBA
. . . . . .
2. Outline
What is a function?
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
. . . . . .
3. Definition
A function f is a relation which assigns to to every element x in a
set D a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { f(x) | x ∈ D } is called the range of f.
. . . . . .
4. Outline
What is a function?
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
. . . . . .
5. The Modeling Process
. .
Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
. . . . . .
7. The Modeling Process
. .
Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
S
. hadows F
. orms
. . . . . .
8. Outline
What is a function?
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
. . . . . .
9. Functions expressed by formulas
Any expression in a single variable x defines a function. In this
case, the domain is understood to be the largest set of x which
after substitution, give a real number.
. . . . . .
10. Example
x+1
Let f(x) = . Find the domain and range of f.
x−1
. . . . . .
11. Example
x+1
Let f(x) = . Find the domain and range of f.
x−1
Solution
The denominator is zero when x = 1, so the domain is all real
numbers excepting one. As for the range, we can solve
x+1 y+1
y= =⇒ x =
x−1 y−1
So as long as y ̸= 1, there is an x associated to y.
. . . . . .
12. No-no’s for expressions
Cannot have zero in the denominator of an expression
Cannot have a negative number under an even root (e.g.,
square root)
Cannot have the logarithm of a negative number
. . . . . .
13. Piecewise-defined functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.
. . . . . .
14. Piecewise-defined functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.
Solution
The domain is [0, 2]. The range is [0, 2). The graph is piecewise.
. .
2 .
. .
1 . .
. . .
0
. 1
. 2
.
. . . . . .
17. Example
Is this a function? If so, what is the range?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6
. . . . . .
18. Example
Is this a function? If so, what is the range?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6
Yes, the range is {4, 5, 6}.
. . . . . .
26. Example
Here is the temperature in Boise, Idaho measured in 15-minute
intervals over the period August 22–29, 2008.
.
1
. 00 .
9
.0.
8
.0.
7
.0.
6
.0.
5
.0.
4
.0.
3
.0.
2
.0.
1
.0. . . . . . . .
8
. /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29
8 8 8 8 8 8 8
. . . . . .
27. Functions described graphically
Sometimes all we have is the “picture” of a function, by which
we mean, its graph.
.
.
. . . . . .
28. Functions described graphically
Sometimes all we have is the “picture” of a function, by which
we mean, its graph.
.
.
The one on the right is a relation but not a function.
. . . . . .
29. Functions described verbally
Oftentimes our functions come out of nature and have verbal
descriptions:
The temperature T(t) in this room at time t.
The elevation h(θ) of the point on the equator at longitude θ.
The utility u(x) I derive by consuming x burritos.
. . . . . .
30. Outline
What is a function?
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
. . . . . .
31. Monotonicity
Example
Let P(x) be the probability that my income was at least $x last
year. What might a graph of P(x) look like?
. . . . . .
32. Monotonicity
Example
Let P(x) be the probability that my income was at least $x last
year. What might a graph of P(x) look like?
. .
1
. .5 .
0
. . .
$
.0 $
. 52,115 $
. 100K
. . . . . .
33. Monotonicity
Definition
A function f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2
for any two points x1 and x2 in the domain of f.
A function f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2
for any two points x1 and x2 in the domain of f.
. . . . . .
34. Examples
Example
Going back to the burrito function, would you call it increasing?
. . . . . .
35. Examples
Example
Going back to the burrito function, would you call it increasing?
Example
Obviously, the temperature in Boise is neither increasing nor
decreasing.
. . . . . .
36. Symmetry
Example
Let I(x) be the intensity of light x distance from a point.
Example
Let F(x) be the gravitational force at a point x distance from a
black hole.
. . . . . .
39. Definitions
Definition
A function f is called even if f(−x) = f(x) for all x in the
domain of f.
A function f is called odd if f(−x) = −f(x) for all x in the
domain of f.
. . . . . .