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Level 513

Continuity & Limits

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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
An instantaneous rate of change
Derivative Shortcut
For f(x)=(z)(xⁿ), mtan=(n)(z)(xⁿ⁻¹)
f(x) has a hole; is it continuous?
f(x) has a jump; is it continuous
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.
The ______________________ is all the possible INPUT, X, values.
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: