Terms  Definitions 

∫cscx 
lncscx+cotx

∫cotx 
lnsinx+C

position 
x(t)

1tanh^2(x) 
sech^2(x)

d/dx tanx 
sec^2x

d/dx cscx 
cscxcotx

Marginal Cost 
C'(q)

d/dx(tanh x) 
sech^2(x)

d/dx(csc^1 x) 
1/[x*√(x^21)]

d/dx(csch x) 
csch(x)*coth(x)

d/dx(sech x) 
sech(x)tanh(x)

volume of sphere 
4/3πr^3

lim (x>0) cosx1/x 
0

lim (x>0) sinx/x 
1

d/dx(cosh x) 
sinh x

sinh^1(x) 
ln[x+√(x^2 + 1)]

surface area of sphere 
4πr^3

d/dx logau 
1/u lna du/dx

d/dx a^u 
a^u (lna) du/dx

optimization 
minimum or maximum problem

d/dx(cosh^1 x) 
1/√(x^2  1)

d/dx(csch^1 x) 
1/[abs(x)*√(x^2 + 1)]

d/dx(coth^1 x) 
1/(1  x^2)

d/dx sinh x 
cosh x

∫udv aka integration by parts 
uv∫vdu

∫[1/(x^2 + a^2)dx] 
(1/a)*tan^1(x/a) + C

critical point 
local minimum or maximum

Volume of shell 
2π (radius) (altitude) (thickness)

sinh x 
e^x  e^x / 2

sinh x 
e^x  e^x / 2

derivative 
lim (h>0) f(x1+h)  f(x1) / h

sigma notation 
definite integralnumber at the top tells the number of intervalsnumber at the bottom tells where to begin

Profit 
Revenue  cost
usually written as pi 
continuity terms 
1. f(c) exists
2. lim (x>c) f(x) exists 3. lim (x>c) f(x) = f(c) 
L'Hopital's Rule 
if the derivative is in intederminate form then use which is 0/0 and infinity/infinitylim x>a f(x)/g(x) = lim x>a f'(x)/g'(x)

L'Hopital's Rule 
if the derivative is in intederminate form then use which is 0/0 and infinity/infinity
lim x>a f(x)/g(x) = lim x>a f'(x)/g'(x) 
point discontinuity 
curve has a "hole"
(limit as x approaches a ≠ f at a) 
simpson's rule 
ba/3n [f(xo) + 4f(x1) + 2f(x2) + 4f(x3) .... f(xn)]

global max and min 
overall maximum and minimum highest and lowest y values substitute the cp into the f(x) set equal to zero to find y location on the graph to see which is the highest

Mean Value theorem 
f'(c) = f(b)  f(a) / ba

The distance between two points of a line can be found using the ? 
Distance formula

The distance between two points of a line can be found using the ? 
d = square root of ( X2  X1 )2 + ( Y2 Y1)2

average value 
1/ba integral from b to a f(x) dx

x is what on the graph 
x is right or left
horizontal 
hyperbolic functions to know 
cosh(0) = 1
sinh(0) = 0 cosh(x) = cosh x sinh (x) = sinh x cosh^2 x  sinh^2 x = 1 
definite integral 
F(b)  F(a) = integral from b to a F'(t) dt

definite integral 
F(b)  F(a) = integral from b to a F'(t) dt

y is what on the graph 
y is vertical or up and down

slope of a line is positive if 
line is increasing from left to right

acceleration 
x''(t)

∫du/u 
lnu+C

velocity 
x'(t)

d/dx(a^x) 
a^x*ln(a)

d/dx sinx 
cosx

Marginal Revenue 
R'(q)

d/dx(sin^1 x) 
1/√(1x^2)

d/dx(sec^1 x) 
1/[x*√(x^21)]

cosh(x) 
cosh x

Marginal Revenue 
R'(q)

d/dx e^u 
e^u du/dx

Volume of cylinder 
πr^2h

Revenue 
price x quantity

sinh(x+y) 
sinh(x)*cosh(y) + cosh(x)*sinh(y)

∫(a^x)dx 
a^x/ln(a) + C

surface area of cyclinder 
2πrh

d/dx cosh x 
sinh x

d/dx(log a X) 
1/[x ln(a)]

d/dx(sech^1 x) 
1/[x*√(1  x^2)]

d/dx(sinh^1 x) 
1/√(x^2 + 1)

d/dx cosh x 
sinh x

semicircle 
y = root(r^2  x^2)

antiderivative 
working backwards from the derivative

antiderivative 
working backwards from the derivative

Volume of disk 
π (radius) ^2 (thickness)

cosh x 
e^x + e^x / 2

removable discontinuity 
rational expression w/ common factors that cancel out

maximum profit 
Marginal Cost = Marginal revenueR'(q) = c'(q)on graph where space between is widest

maximum profit 
Marginal Cost = Marginal revenue
R'(q) = c'(q) on graph where space between is widest 
Modeling Optimization problems 
need two equationsthe quantity needed to be optimized is the thing you want the derivative of sketchesfind the cp and ep to evaluate global max and min

Modeling Optimization problems 
need two equations
the quantity needed to be optimized is the thing you want the derivative of sketches find the cp and ep to evaluate global max and min 
Second fundamental theorem 
dF/dx=d/dx ∫[a to x] f(t)dt = f(x)

delta t 
ba/n n is the number of intervals and b and a is the interval

global max and min 
overall maximum and minimum
highest and lowest y values substitute the cp into the f(x) set equal to zero to find y location on the graph to see which is the highest 
A function is 
a relation in which each element of the domain (x value  independent variable ) is paired with only one element of the range ( y value  dependent variable )

A function is 
A relation can be tested to see if it is a function by the vertical line test. Draw a vertical line through any graph, and if it hits an xvalue more than once, it is not a function.

Mean value theorem for integrals 
f(c) = 1/ba ∫[a to b] f(x)dx

Extreme Value theorem 
if f is continuous then must have a global max and global min on the closed interval

left and right increments 
in a chart begin with the first value but don't take the last for the right hand value begin with the second value and end with the last value for the left hand sum average the two to find the most acurate measurement of the integral

left and right increments 
in a chart begin with the first value but don't take the last for the right hand value
begin with the second value and end with the last value for the left hand sum average the two to find the most acurate measurement of the integral 
Linear functions take the form of 
f(x) = mx + b
or y = mx + b 
Linear functions take the form of 
Where m = the slope
b = the yintercept so f(x) = 4x  1 the slope is 4/1 (rise over run) and the yintercept is 1 
Leave a Comment ({[ getComments().length ]})
Comments ({[ getComments().length ]})
{[ comment.comment ]}