Download Absolute Value Exercise

MA108 CALCULUS WITH ALGEBRA I
Tuesday, 9/11/12
Today:
Reading (get the textbook!):
Office hours end at 2:30 today only
Questions on exercises assigned last time?
Absolute Value
Coordinate Geometry
Introduction to functions
Appendix B
1.1
Exercises (not to hand in):
App. A: 1-11 odd, 47-53 odd
App. B: 1-9 odd, 21-27 odd
1.1: 3a-e, 15, 25, 31, 33
Tuesday, 9/11/12, Slide #1
Absolute Value
-x, if x ≥ 0,
-x, if x < 0
|x| is never negative
|x| can be thought of as the distance
from x to 0
Examples:
Definition:
|x| =
To say |x| < 5 means -5 < x < 5
To say |x| ¥ 10 means x ¥ 10 or x § -10
Tuesday, 9/11/12, Slide #2
Exercise
If
|x – 3| > 2, what’s true of x?
A.
x < 1 or x > 5
B. x = 5
C. 1 < x < 5
D. -5 < x < -1
E. x > 2
F. None of the above
Tuesday, 9/11/12, Slide #3
Appendix B: Coordinate Geometry and
Lines
= 1:
The set of all real numbers
Corresponds to points on a line
2:
The set of all ordered pairs, (x, y), of real
numbers x and y
Corresponds to points in the plane
Tuesday, 9/11/12, Slide #4
Points and lines in the plane
Distance between two
points P1(x1, y1) and P2(x2, y2):
Example: What is the distance
from (-2,2) to (1,3)?
Slope of line through
P1 and P2:
Example: What is the slope of
the line through (-2,2) and(1,3)?
Tuesday, 9/11/12, Slide #5
Equations of lines
Point-slope form: Line with slope m,
passing through point P1(x1,y1) has
equation y – y1 = m (x – x1)
Slope-intercept form: Line with slope m,
and with y-intercept (0, b) has equation
y=mx+b
Exercise: Given the two points P1(2, -4)
and P2(5, 0):
Find an equation of the line through P1 and P2
Find the y-intercept of the line.
Tuesday, 9/11/12, Slide #6
Lines as functions
When we write y = m x + b, it’s natural to
view y as a function of x, i.e.,
Start with input a number x
Use the formula y = m x + b
Compute output number y
If we solve an equation for x instead of y,
then we view x as a function of y.
Usually we solve for y in terms of x:
x is the input or “independent variable”
y is the output or “dependent variable”
We often give the function formula a name,
like f(x)
Tuesday, 9/11/12, Slide #7
Example of a linear function
Define
y = f(x) where f(x) = 7 x – 2
What is f(1), f(-1), f(2), f(-2)?
The
“graph” of a function f(x) is the
graph of the equation y = f(x), i.e., in
this example, y = 7 x – 2
What does this graph look like?
Tuesday, 9/11/12, Slide #8
General Concept of Function
A function is a rule which assigns to
every element of an input set (the
domain D) exactly one element of an
output set (the range E).
You can think of a function as a
machine, taking in an input value and
putting out an output value.
Or think of it as arrows pointing from
the domain to the range.
Tuesday, 9/11/12, Slide #9
Two Examples from Daily Life
1. Body Mass
calculator
2. Currency
converter
Tuesday, 9/11/12, Slide #10
Four Ways to Represent a Function
Algebraically
Visually
: Draw a graph.
Graph of above example?
Numerically
: Make a table of values.
Example?
Verbally
: Find a formula.
Example: y = x2 + 1
: Describe it in words.
Example?
Tuesday, 9/11/12, Slide #11
Some Ideas about
Functions of Real Numbers
If not otherwise stated, the domain of a
function in calculus is the set of all real
numbers that can be put into the
function.
1
Example: What’s the domain of
f ( x) =
2x + 4
Vertical Line Test for the graphs: A graph
in the plane is a function of the horizontal
variable if and only if each vertical line
intersects the graph at most once.
Example: What’s a common graph that is not the
graph of a function?
Tuesday, 9/11/12, Slide #12