Chapter 2

Tangent

A tangent line is a line that touches a curve at one point near a Point P.

A secant line intersects a curve at more than one point.

Equation of a line

Ax+By = C

y = mx+b

Point Slope Form

y-y1 = m (x-x1)

y - f (x1) = m (x-x1)

The limit of a function

If f(x) is defined when x is near the number.

f(x) does NOT have to equal L.

One sided Limits

Left-hand limit of f(x) as x approaches a

Intuitive Definition of an Infinite Limit

The vertical linex = a is called a vertical asymptote

Calculating Limits using Limit laws

Direct Substitution Property

If f is a polynomial or rational function and a is in the domain of f the limit function = f (a)

Theorem: The limit exists if and only if the limit from the left equals the limit from the right

The Squeeze Theorem

Horizontal Aysmptotes

The line y = L

Velocity

The average velocity is the distance traveled over an interval of time.

s (t) = change in position / change in time

The instantaneous velocity is the velocity at a specific time

"velocity"

limit

Precise Defnition of Limit

if for every number e > 0 there is a number > 0

Continuity

a function is continuous at a number a

3 requirements

#1 f (a) is defined at a

#2 the limit exists

#3 the limit is f(a)

if a function is define on an open interval containing a, except possibly at a, we say f is discontinuous at a if f is not continuous at a

A point is a removable discontinuous if we can redefinef(x) at a to a single number

A point a is an infinite discontinuous if the function goes to infinity (or negative infinity) at a.

f (a) may or may not

A point a is a jump discontinuous if the limits form the right and the left exist but are not equal

Theorem

Polynomials

rational functions

root functions

trigonometric functions

Derivatives

the derivative of f is f (x + h) + f (x) / h

provided the limit exists and a is in the doamin of f . If f' (x) exists, we say f is differentable at a.

If f is differentiable on every point of an interval I, we say that f is differentiable on I

Differentiability

a function will fail to be differentiable at a point a if

1. There is a cusp or "kink" at a

The function is discontinuous at a

There is a vertical tangent at a

The function does not exist at a