Let f be continuous on I containing x=c. If the graph has a tangent line at x=c and the concavity changes over x=c then c is an inflection point

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critical number

Front

the exception is if there is a jump discontinuity

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2nd derivative test

Front

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constant rule

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Back

"infinite limit"

Front

When the function either increases without bound or decreases without bound

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derivative of the natural exponential function

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definition of increasing and decreasing

Front

let f be continuous on [a,b] and differentiable on (a,b).
1) if f'(x)>0 for all x in (a,b) then f is increasing [a,b]
2) if f''(x)<0 for all x in (a,b) then f is decreasing on [a,b]
3) if f'(x)=0 for all x on (a,b) then f is constant on [a,b]

Back

natural log derivative

Front

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local linearity

Front

looks linear even though it is a curve

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squeeze theorem

Front

If one can demonstrate that a function, which cannot be analytically evaluated for a limit, is bound by two other functions and the 2 sided limit of those functions is equal, you can evaluate the limit using the other two functions. sin(x)/x =1 and 1-cos(X)/x =0 come from this theorem.

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chain rule

Front

Back

definition of relative extrema

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hole

Front

removable discontinuity

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sum/difference rule

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power rule

Front

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concave up

Front

increasing slope

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Mean Value Theorem

Front

Back

formal definition of a limit

Front

Let f be defined on an open interval containing the number c (except possibly at c) and let L be a real number.

Back

undefined slope/non-differentiable

Front

function is not locally linear, vertical tangent line, or any discontinuity

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definition of extrema

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concave down

Front

decreasing slope

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vertical asymptotes

Front

let p(x) and q(x) define polynomials for the rational function defined by f(x) = p(x)/q(x), written in lowest terms and for a real number a, if the absolute value of f(X) approaches infinity as x approaches a, then x=a is a vertical asymptote. This occurs where q(x) = 0.

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derivative of sin

Front

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break

Front

jump discontinuity (usually piecewise)

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Rolle's Theorem

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constant multiple rule

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product rule

Front

Back

general definition of the deriviative

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continuity

Front

what does differentiability imply?

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infinite limit

Front

let f be a function that is defined for every real number on some open interval containing a (except possibly a):
1: (see image) this means that for each m>0, there exists a delta>0 such that f(x)>m whenever 0<[x-c]<delta
2:(image, but with negative infinity) this means that for each n>0, there exists a delta>0 such that f(x)<m whenever 0<[x-c]<delta

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e

Front

irrational number, equal to about 2.718281828459045, and is the base of the natural log function

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intermediate value theorem

Front

existence theorem

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derivative at a specific point

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Back

add a - super script for the left, and a + super script for the right

Front

How do you delineate limits from the left and right?

Back

derivative of cosine

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indeterminate forms

Front

0/0 and infinity/infinity

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informal definition of a limit

Front

if f(x) becomes arbitrarily close to some number l as x approaches c from either side, the limit of f(x), as x approaches c, is l.

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quotient rule

Front

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definition of tangent line

Front

if f is defined on an open interval containing a, and if the limit (see image) exists, then the line passing through (a, f(a)) with slope m is the tangent line to the graph of f at the point (a, f(a)).

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continuous function

Front

no breaks or asymptotes. every point exists at its limit

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1st derivative test

Front

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continuity definition

Front

a function is continuous at x=c provided that f(c) is defined, the limit of f(x) as x approaches c is defined, and f(c) equals the limit of f(x) as x approaches c.