# Outline of material in Chapters 1 and 2

Outline of material in Chapters 1 and 2

**1.1** – Four ways to represent a function

Examples of functions

Definition of function

Domain, range, in/dependent variable

Ways to picture a function: machine, arrow diagram, graph

4 ways to represent a function:

Verbally (description)

Numerically (table)

Visually (graph)

Algebraically (formula)

Vertical Line Test for a curve to be the graph of a function

Piecewise defined functions

Absolute value function

Even & odd functions

Increasing & decreasing functions

**1.2 **– Mathematical models: a catalog of essential functions

Mathematical model

Linear function

Polynomial ( terms : degree, coefficients , quadratic , cubic)

Power function and its graph

Rational function

Algebraic function

Trigonometric function

Exponential function and its graph

Logarithmic function and its graph

Transcendental function

**1.3** – New functions from old functions

Transformations

Vertical & horizontal shifts

Vertical & horizontal stretching & reflecting

Combinations

**f +-×÷ g**

**f ○ g : **Composition of functions

**1.4** – Graphing Calculators

Viewing rectangle

**1.5** – Exponential functions

Define and graph **a ^{x}** for all positive real numbers

**a**

Cases for

**a : a>1, 0<a<1, a=1**

Laws of exponents

Application to population models

The number

**e**

**1.6** – Inverse functions and logarithms

One-to-one function

Horizontal line test

Definition of inverse function of a one-to-one function (Example 4)

Domain and range of inverse function

“Cancellation” equations

Finding the inverse of a one-to-one function

Relation between graph of **f ^{-1}** and that of

**f**

Logarithm: the inverse of the exponential function

Laws of Logarithms

Natural Logarithms

Change of Base formula

Inverse Trig. functions

**2.1 **– Tangent and velocity problems

Motivating the idea of the derivative (not yet mentioned by name) as a limit:

__ Slope of tangent line__ as limit of slopes of secant lines

__Instantaneous velocity__ as limit of average velocities

**2.2** – The limit of a function

Definition of limit

Guessing limits from a sequence of evaluations

One-sided limits

Limit exists if & only if the two one -sided limits exist and are equal

Vertical asymptote

**2.3 **– Calculating limits using the limit laws

Basic five Limit Laws

Additional Limit Laws #6-11

Direct Substitution Property for polynomial & rational functions (holds for all
continuous functions)

Limit exists if & only if the two one-sided limits exist and are equal (repeated
from 2.2)

Using the above to calculate limits using one-sided limits, e.g., involving
absolute value

Squeeze Theorem

**2.5 **– Continuity

Definition only

**2.6 **– Limits at infinity; horizontal asymptotes

Horizontal asymptote

Calculating of a rational function using
(Example 3)

Examples of with square roots (Examples 4&5)

(Examples 8-11)

(Skip Precise Definitions subsection)

**2.7** – Derivatives

__Slope of tangent line,__ revisited – express as a limit that we now can calculate

Same for __instantaneous velocity__

The mathematics of these two examples is the same, leading to

Definition of the Derivative (two forms: with** h **and with **a**)

Finding equation of tangent line to curve** y=f(x)** at point **( a, f(a) )
**(Example
5)

More general interpretation of the derivative:

rate of change of one variable with respect to another (Example 6)

**2.8** – The derivative as a function

Not on Test #1; on next test.

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