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Continuous Function

A function that can be graphed without breaks, holes, or gaps

Discontinuous Function

A function that is not continuous; 3 types

Infinite Discontinuity

A characteristic of a function in which the absolute value of the function increases or decreases indefinitely as x-values approach c from the left and right

Jump Discontinuity

A characteristic of a function in which the function has two distinct limit values as x-values approach c from the left and right

Removable discontinuity

Occurs when common factors are cancelled in a rational expression.

Nonremovable Discontinuity

Describes infinite and jump discontinuities because they cannot be eliminated by redefining the function at that point

End Behavior

Look at HA

limit

The unique value that a function approaches as x-values of the function approach c from the left and right sides

Definition of Continuity

At a point:

Definition of a Limit

A limit is a number that a function approaches as x approaches a value

lim(h→0)=[f(x+h)-f(x)]/h

Slope of tangent through (x, f(x))

lim(x→a)=[f(x)-f(a)]/(x-a)

Slope of tangent through (a, f(a))

y-y1=m(x-x1)

point-slope form

Existence Theorem

tells you if something exists, but not how to find it

intermediate value theorem (IVT)

suppose f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b). Then there exists a number "e" such that f(e)=N.

1

Limit as x approaches 0 of (sinx/x)/(x/sinx)=

msec

Average rate of change

mtan

Instantaneous rate of change

Derivative=

slope of tangent line!

f(x)

f of x

f'(x)

velocity

f''(x)

acceleration

f''"(x)

jerk

f'(a) exists

A function f is differentiable at a if

zeroes

Mins and maxes on f(x) become ________ on f'(X)

concavity

a graph is concave up is the tangent lines are below the graph; a graph is concave down if the tangent lines are above the graph

when f changes concavity,

f' has a min or a max

When a graph is decreasing,

the slopes of the tangent lines are negative

a point

on a plane is located in terms of two numbers

Instantaneous Rate

The rate of y to x for a given point along the x axis

Secant Line

a line that intersects a circle at two points on the same plane

Tangent Line

A line that only touches one point on a curve

Limit Notation

(the arrow notations is a subscript under lim that I can't do here)

Two-sided limit

One in which the limit is the same whether it is approached from above or below

One-sided limit

a limit that differs depending whether it is approached from above or below

Limit Laws: Sum

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Difference

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Constant Multiple

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Product

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Quotient

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Power

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Fractional Power (or, Root)

Assume lim f(x) and lim g(x) exist for x→a

Squeeze Law

If f(x)≤g(x)≤h(x) for values of x near a, except possibly at a, then

Infinite Limit

If f is defined for all x near a, and f(x) grows arbitrarily large for all x sufficiently close (but not equal) to a, then

Limits at infinity

The dependent variable approaches a finite number as the independent variable becomes arbitrarily large