Calculus Formulas Flashcards

Terms Definitions
∫cscx
-ln|cscx+cotx|
∫cotx
ln|sinx|+C
position
x(t)
1-tanh^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^2-1)]
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) cosx-1/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
b-a/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) / b-a
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/b-a 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
ln|u|+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/√(1-x^2)
d/dx(sec^-1 x)
1/[x*√(x^2-1)]
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
b-a/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 x-value more than once, it is not a function.
Mean value theorem for integrals
f(c) = 1/b-a ∫[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 y-intercept
so
f(x) = 4x - 1 the slope is 4/1 (rise over run) and the y-intercept is -1
/ 91
Term:
Definition:
Definition:

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