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Calculus is...

... the study of instantaneous rates of change modeled by the slope of a tangent line

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

Secant Slope

The slope of a line that passes through two points on a curve, showing the average rate of change

Tangent Slope

The slope of a line that touches only one point on a curve, giving the instantaneous rate of change

The Continuity Test

A test to determine if the function f(x) is continuous at the point x=α where the function must pass the following three tests;

horizontal asymptote

a line that the curve approaches as x goes to infinity but never reaches

Point Discontinuity

A discontinuity caused by a "hole" in the function, ie. an undefined value of f(x) when x=α

Step Discontinuity

A discontinuity caused by a step function that prevents the existence of limits at certain intervals

Infinity Discontinuity

A discontinuity caused by an asymptote where the left and right sides of the function diverge to infinity

Oscillating Discontinuity

A discontinuity caused by the oscillation of a function (esp. trigonometric) that prevents an infinity limit

Removable discontinuity

Occurs when common factors are cancelled in a rational expression.

Intermediate Value Theorem

If f(a) > 0 and f(b) < 0 then there is a real zero between a and b

Horizontal Tangent

The value of f(x) where mtan is equal to zero and the tangent drawn to the point is a horizontal line

Derivitive

An instantaneous rate of change

Derivative Shortcut

For f(x)=(z)(xⁿ), mtan=(n)(z)(xⁿ⁻¹)

No

f(x) has a hole; is it continuous?

No

f(x) has a jump; is it continuous

No

f(x) has an asymptote(s); is it continuous

A function is continuous if

it's graph is a single UNBROKEN curve (or line) -- you could draw it without lifting your pencil.

Domain

The ______________________ is all the possible INPUT, X, values.

Range

The difference between the greatest number and the least number in a set of data.

Is a function's continuity dependent on it's domain?

Yes a function's continuity can be dependent on it's domain. Ex: for a function with holes or jumps --

f(x) has a hole at x=c; is f(x) defined at x=c?

NO; if there is no value for f(x) at x=c then f(x) is UNDEFINED at x=c for a hole

f(x) has a hole at x=c; does f(x) have a limit at x=c?

YES, limit exists; f(x) approaches the same value at x=c (where the hole is) from both the left and the right sides

f(x) has a jump at x=c; does f(x) have a limit at x=c?

NO, limit does NOT exist; f(x) approaches different values at x=c (where the jump is) from the left versus the right

What makes a function differentiable?

A function is differentiable at a point x if the limit exists at that point -- this means the limit of f(x) approaches a value from the left and limit of f(x) approaches that SAME value from the right.

Can a function be differentiable but NOT continuous?

NO - the function MUST be continuous in order for it to be differentiable.

Yes - example:

Can a function be continuous but NOT differentiable?

Vertical Asymptotes

zeroes of denominator

Horizontal Asymptotes

y=0 when degree of denominator > degree of numerator

indeterminate forms

All these are indeterminate (which means that if any of these are a limit of something then that limit is undefined):

Facts about ∞

0^∞ = 0

Linear approximation

L(x)=f(a)+f '(a)*(x-a)

(Formal form)

lim (as h-->0) [f(x+h)-f(x)]/h

(Alternate form)

lim (as x-->a) [f(x)-f(a)]/(x-a)

L'Hopital's Rule

If lim (as x-->a) f(x)/g(x) is any of the following: