Definition of a limit

lim (x -> a) f(x) = L iff lim (x -> a-) f(x) = L and lim (x -> a+) f(x) = L

lim (x -> a) [f(x) ± g(x)]

lim (x -> a) f(x) ± lim (x -> a) g(x)

lim (x -> a) [f(x)g(x)]

lim (x -> a) f(x) * lim (x -> a) g(x)

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

lim (x -> a) f(x) / lim (x -> a) g(x), provided lim (x -> a) g(x) ≠ 0

Definition of continuity

A function f is continuous at x = a iff (1) f(a) exists, (2) lim (x -> a) f(x) exists, (3) lim (x -> a) f(x) = f(a)

line x=a if lim x->a +/- f(x) = +/-infinite

y=a if lim x-> +/- infinite f(x) = a

ln^d a < x^c < x^c ln^d < x^c+d < a^x < x! < x^x

lim x-> ∞ g(x)/f(x) = 0 lim x-> ∞ f(x)/g(x) = ∞

1. if f is continuous on closed interval (a,b) 2. then it has absolute max and absolute min on interval (a,b)

Definition of critical points

Critical point of f is a #c (x-value) such that 1. f'(c) = 0 OR f'(c) DNE

Candidates for extreme values on closed interval

1. find y-values of critical points in interval (a,b) 2. find y-values of endpoints, a and b 3. largest y-value = max, smalled = min

there exists a #cIf f continuous on (a,b) + differentiable on (a,b) Then there exists a #c b/t a and b such that f'(c) = f(b)-f(a)/b-a slope of tangent = slope of secant at some point b/t a + b

If f continuous on (a,b), differentiable on (a,b) and f(a)=f(b), Then there is atleast one #c on (a,b) such that f'(c) = 0

relative/local min/maxes must occur at critical points

1st derivative test for relative extrema

f(c) must exist 1. if f' changes from + to - at c, then f(c) = relative max 2. if f' changed from - to + at c, then f(c) = relative min 3. if f' doest change signs at c, then f(c) neither max/min

point on f graph where concavity of f changes

functio f has inflection point at (c,f(c)) if

1. f'' changes signs at x=c 2. f'' = 0 where f'' DNE

2nd deriv test for relative extrema

given f'(c) = 0, 1. if f''(c)<0 = max 2. if f''(c)>0 = min 3. if f''(c) = 0, test inconclusive

Mathematical Concepts and Formulas

Flashcard

•

Mathematics

•

12th Grade

•

Practice Problem

•

Easy

•

CCSS

HSF.TF.A.4, HSF.TF.C.9, HSF.TF.A.2

Standards-aligned

Ash Udgirkar

Used 25+ times

FREE Resource

Student preview

58 questions

Show all answers

FLASHCARD QUESTION

Front

sin(x)/cos(x)

Back

tan(x)

Tags

CCSS.HSF.TF.A.4

FLASHCARD QUESTION

Front

cos(x)/sin(x)

Back

cot(x)

Tags

CCSS.HSF.TF.A.4

FLASHCARD QUESTION

Front

sec(x)

Back

1/cos(x)

Tags

CCSS.HSF.TF.A.2

FLASHCARD QUESTION

Front

csc(x)

Back

1/sin(x)

Tags

CCSS.HSF.TF.A.2

FLASHCARD QUESTION

Front

sin^2(x) + cos^2(x)

Back

FLASHCARD QUESTION

Front

sec^2(x) - tan^2(x)

Back

FLASHCARD QUESTION

Front

2sin(x)cos(x)

Back

sin(2x)

Tags

CCSS.HSF.TF.C.9

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