Studied by 113 peopled/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
Note
Flashcard