Ultimate AB Calc Study Brain Basher 2000

d/dx (x^n)

nx^(n-1)

d/dx (sin x)

cos(x)

d/dx (cos x)

-sin(x)

d/dx [tan x]

sec^2(x)

d/dx [cot x]

-csc^2(x)

d/dx [sec x]

sec(x)tan(x)

d/dx [csc x]

-csc(x)cot(x)

d/dx [ln u] (use option-8)

1/u•du/dx

d/dx [e^u] use option-8

e^u•du/dx

d/dx [arcsin x]

1/(1-x^2)^0.5

d/dx [arccos x]

-1/(1-x^2)^0.5

d/dx [arctan x]

1/(1+x^2)

d/dx [arccot x]

-1/(1+x^2)

d/dx [arcsec x]

1/[|x|(x^2-1)^0.5]

d/dx [arccsc x]

-1/[|x|(x^2-1)^0.5]

d/dx [a^x] use option-8

a^x•ln(a)

d/dx [log base "a" of (x)]

1/[xln(a)]

Chain rule of d/dx[f(u)] use option-8

f'(u)•du/dx

product rule of d/dx • [uv]

u'v+uv'

quotient rule of d/dx • [f/g]

(gf'-fg')/g^2

Extrema

The biggest/smallest y-values over a certain range

(if you typed something like it, then override quizlet and say you were correct)

Intermediate Value Theorem (IVT)

If f(x) is continuous on [a, b] then and N is a value between f(a) and f(b), then there must be a value "c" in (a, b) such that f(c) \= N

Mean Value Theorem (MVT)

If f(x) is continuous on [a, b] and differentiable on (a, b) then there must be "c" in (a, b) such that f'(c) \= [f(b)-f(a)]/(b-a)

Rolle's Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b) and f(a) \= f(b), then there is at least one critical number in (a, b)

Is an Extrema an x or y value

y-value

critical number

the x-values where extrema occur

critical point

(critical number, extrema) in (x,y) format

Absolute Maximum

Largest y-value in the entire function

Absolute Minimum

smallest y-value in the entire function

Local/Relative Maximum

Largest y-value in a certain area of the function marked by bounds

Local/Relative Minimum

Smallest y-value in a certain interval of the function marked by bounds

Where do critical numbers occur?

endpoints of interval, f'(x)\=0, f'(x) does not exist

You forgot something on your indefinite integral. What is it?

+C

You forgot something on your indefinite and definite integral. What was it?

dx

Concave up like a cup is when

f'(x) is increasing

Concave down like a frown is when

f'(x) if decreasing

inflection points occur when

concavity changes

Law of Sines with angles A, B, and C and corresponding sides a, b, and c

\[sin (A)]/a \= [sin (B)]/b \= [sin (C)]/c

Law of Cosines with angles A,B, and C and sides a, b, and c

c^2 \= a^2+b^2-2ab•cos(C) *note* you can do this for any angle as long as it corresponds with the opposite side

L'Hospital's Rule (pronounced Lo-Pay-Tall) \-- option-5 keyboard shortcut and shift - option +

If the limit as x approaches a of f(x)/g(x) \= 0/0, ±∞/±∞, 1^∞, of ∞^0, then it becomes the limit as x approaches a of f'(x)/g'(x)

derivatives are

tangent slopes

definite integrals model for

area under a curve

integral of a rate (ie: integral of v(t)) \=

total amount (ie: in the example, it would be the total distance

local minimums occur where f'(x) goes

(-,0,+) or (-, undefined, +)

local minimums also occur where f''(x) is

greater than zero (up like a cup)

local maximums occur where f'(x) goes

(+,0,-) or (+, undefined, -)

local maximums also occur where f''(x) is

less than zero (down like a frown)

points of inflection (concavity changes) occur when f''(x) goes

(+,0,-), (-,0,+), (+, undefined, -), or (-, undefined, +)

interval of continuity for a polynomial

(-∞,∞)

interval of continuity for f(x) \= ln(x)

(0,∞)

interval of continuity for f(x) \= e^x

(-∞,∞)

interval of continuity for f(x) \= arctan(x)

(-∞,∞)

interval of continuity for f(x) \= x^0.5

\[0,∞)

interval of continuity for f(x) \= sin(x)

(-∞,∞)

interval of continuity for f(x) \= cos(x)

(-∞,∞)

factor a^2 - b^2 (difference of squares)

(a-b)(a+b)

factor a^3 - b^3 (difference of cubes)

(a-b)(a^2+ab+b^2)

factor a^3 + b^3 (sum of cubes)

(a+b)(a^2-ab+b^2)

range of arcsin (use option - p)

\[-π,π]

range of arccos

\[0,π]

range of arctan

\[-π,π]

It is an even function if

f(x) \= f(-x)

It is an odd function if

f(-x) \= -f(x)

is sin(x) even or odd

odd

is cos(x) even or odd?

even

Fun fact that you should know on related rates (generally distance problems), the minimums and maximums are the same for

f(x) and [f(x)]^0.5 Now you can differentiate without the evil radical in the denominator.

volume of a sphere

(4πr^3)/3

surface area of a sphere

4πr^2

volume of a cone (bonus hint for related rates problems\-- use similar triangles to relate the radius and height)

π•r^2•h/3

volume of a cylinder

π•r^2•h

area of an equilateral triangle in terms of side s

3^0.5•s^2/4 (root 3 times s squared over 4\-- useful in integration volume problems)

what is revenue (R)

how much an item sells for

what is cost (C)

how much it takes to make an item

what is profit (P)

net gain from selling an item

Profit formula (Cost C, profit P, and revenue R)

P\=R-C

revenue formula (related rates/optimization)

(units sold)(price per unit)

average cost formula (denoted with C with a line above it)

total cost/units sold

marginal revenue

the derivative of the revenue function

marginal profit

the derivative of the profit function

marginal cost

the derivative of the cost function

what does marginal revenue represent

approximately the revenue generated from selling one more item

what does marginal profit represent

approximately the profit of selling one more item

what does marginal cost represent

approximately the cost of making one more item

| a | equals

a or -a

| ab | \=

|a|•|b|

|a-b| \=

|b-a|

(a^2)^0.5 \=

|a|

(f+g)(x) \=

f(x) + g(x)

(f-g)(x)

f(x)-g(x)

(fg)(x) \=

f(x)-g(x)

(f/g)(x) \=

f(x)/g(x)

Arsec(x) in terms of arccos so you can do it on a calculator

arccos(1/x)

Arccsc x in terms of arcsin so you can do it on a calculator

arsin(1/x)

Arccot x in terms of arctan

π/2 - arctan x

Log base a of x in terms of ln

ln(x)/ln(a)

double angle sin(2x) \=

2sinxcosx

cos(2x) \=

1-2sin^2x \= cos^2 (x) - sin^2 (x)

tan(2x) \=

2tanx/(1-tan^2x)

half angle sin(x/2) \=

± square root of (1-cosx)/2