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Calculus

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Section 1

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Slant asymptote

Front

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Calculus Math

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6 years ago

Date created

Mar 1, 2020

Cards (14)

Section 1

(14 cards)

Slant asymptote

Front

Slant asymptotes occur in a rational function when the degree of the numerator is one greater than the degree of the denominator. Find the slant asymptotes may be found by dividing and discarding the remainder term.

Back

Extreme Value Theorem

Front

If f(x) is continuous on [a,b], there must be values of f(x) obtained on the interval that are greater than or equal to all other values of f(x) on the interval and less than or equal to all the other values obtained on an interval. Put another way, a continuous function will have absolute extrema on a closed interval.

Back

concavity

Front

A function f is concave up when f' is increasing A function f is concave down when f' is decreasing

Back

PPOI

Front

where f'' is equal to zero or undefined but is defined in the original function there may be a POI here

Back

Mean Value Theorem

Front

If a function f(x) is continuous on [a,b] and differentiable on (a,b), then there exists at least one x value c on the interval (a,b) such that f'(c)= (f(b)-f(a))/b-a

Back

The POI test

Front

this is a sign test of 2nd derivative, not the 2nd derivative test. Sign test f'' between PPOI and places where f is discontinuous (undefined) This determines whether PPOI are in fact POI's. This determines where f'' is + or - (where f is CU or CD)

Back

Relative Extrema

Front

x=m is a relative extrema of f if f(m) is greater than or less than all other function values on some open interval containing m. Relative extrema must occur at critical points To determine if a critical point is a relative max or min use the first derivative test (sign change on the derivative at a CP) or the second derivative test. If continuous at a relative extrema, the function graph will turn there.

Back

Graphing functions with radicals

Front

Consider the domain when a function involves an even radical such as (square root of f(x)). Find the domain by solving the inequality f(x)>-0. Any boundary values may be endpoints of the graph.

Back

POI

Front

f has POI's at defined points where the concavity changes f'' changes sign at a POI f has POI's everywhere the graph of f'(x) has relative extrema

Back

Increasing, Decreasing

Front

f is increasing where f'>0 f is decreasing where f'<0

Back

Critical point

Front

A CP is an x value where f is defined and f' is equal to zero or is undefined. There may be a relative max or min at a CP.

Back

First Derivative Test

Front

Sign test the first derivative between CP's and points of discontinuity of the original function (designate points of discontinuity with open circles) Accomplishes two things, determines where the function is increasing and decreasing and whether a CP is a relative mac, min or neither. If f' changes from + to - at a CP, it is a relative max. If f' changes from - to + at a CP, it is a relative min.

Back

2nd derivative test

Front

may determine if specific CP's are relative mas or min's based on their concavity plug a critical point x=c into the second derivative if f''(c)>0 f has a relative min at x=c if f''(c)<0 f has a relative max at x=c If a CP makes the 2nd derivative 0 or undefined, use the first derivative test.

Back

Candidates Test

Front

A test for the absolute extrema of a continuous function on a closed interval [a,b]. Find critical points of the function on the interval. Compare the function values at the endpoints and critical points occurring on the interval to determine absolute extrema.

Back