Lecture 1 admin & representing fcts

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Math 138

Section 003

Professor Brown

Fall 2014

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Prof Brown Contact Info

ronbrown@njit.edu (best)

973-642-4096 (I don’t check that often)

Office Hours:

Tuesday 10 to 1

Wednesday 11:30 to 12:30

CULM 212A (back of the adjunct office)

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Course Information

Start with Moodle page if you have any questions

Class Expectations

Academic Integrity Policy

Attendance Policy

Quiz Policy

Exam Policy

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Grading

Quizzes 15%Midterm 1 25%Midterm 2 25%Final 35%

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Resources

Textbook

My office hours

Math Learning Center (CULM 214)

Internet (Kahn Academy, etc)

Four Ways to Represent a Function1.1

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What is a Function?

Functions arise whenever one quantity depends on another.

We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function.

The number f (x) is the value of f at x and is read “f of x.” The range of f is the set of all possible values of f (x) as x varies throughout the domain.

A symbol that represents an arbitrary number in the domain of a function f is called an independent variable.

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What is a Function?

It’s helpful to think of a function as a machine

DomainIndependent Variable

RangeDependent Variable

or y

Key Idea – each input has only one output

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Representations of Functions

There are four possible ways to represent a function:

verbally (by a description in words)

numerically (by a table of values)

algebraically (by an explicit formula)

visually (by a graph)

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Verbal Representation

• Revenue is $10 for every unit sold

• Force required to stretch/compress a spring is

proportional to the distance spring is

stretched/compressed

• Voltage across a capacitor decays exponentially

with a time constant RC

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Numerical RepresentationThe human population of the world P depends on the

time t. The table gives estimates of the world population P(t) at time t, for certain years. For instance,

P(1950) 2,560,000,000

But for each value of the time tthere is a corresponding value

of P, and we say that P is a function of t.

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Algebraic Representation

The area of a circle A = r

2

Height of projectile: h(t) = -16t2+ vt + h0

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Graphical RepresentationThe graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5.

Figure 5

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Example 1 – Reading Information from a Graph

The graph of a function f is shown in Figure 6.

(a) Find the values of f (1) and f (5).

(b) What are the domain and range of f ?

Figure 6

The notation for intervals is given in Appendix A.

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Example 1 – Solution

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Vertical Line TestThe graph of a function is a curve in the xy-plane. But the question arises: Which curves in the xy-plane are graphs of functions? This is answered by the following test.

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Representations of FunctionsThe reason for the truth of the Vertical Line Test can be seen in Figure 13.

If each vertical line x = a intersects a curve only once, at (a, b), then exactly one functional value is defined by f (a) = b. But if a line x = a intersects the curve twice, at (a, b) and (a, c), then the curve can’t represent a function because a function can’t assign two different values to a.

Figure 13

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Vertical Line Test

Draw a graph that fails the vertical line test at x=3

Draw a graph that passes the vertical line test.

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Finding Domains

• Sometimes explicitly given

• Sometimes implied – word problem describing

area of a circle as a function of radius, implies

radius is positive number

• Exclude values that would “break” the function:• No division by zero• No even roots of negative numbers• No logs of non-positive numbers

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Finding Domains - Examples

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Finding Domains - Examples

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Finding Domains - Examples

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Finding Domains - Examples