CALCULAS

Integration Formulas
∫ x n dx = xn+1 /(n+1)     if n+1 ≠ 0
∫1 / x dx = ln |x|
∫ e nx dx = e nx/n     if n ≠ 0
Derivative Formulas
d/dx (xn) = nxn-1
d/dx (ln x) = 1/ x
d/dx (e mx) = me mx
Product and Quotient Rules
The Product Rule: d/dx (f(x)g(x)) = f '(x)g(x) + f(x)g '(x)
The Quotient Rule: d/dx (f(x)/g(x)) = (f '(x)g(x) - f(x)g '(x))/(g(x)2)
Chain Rules
d/dx (f(u(x))) = d/dx (f(u)) d/dx (u(x)) = f'(u)u'(x)
d/dx (u(x)n) = n u(x)n-1 u'(x)
d/dx (ln (u(x)) = u'(x)/ u(x)
d/dx (e u(x) ) = e u(x) u'(x)
Change of Variables
du =d/dx (u) dx = u'(x)dx
Integration by Parts
∫u dv = uv - ∫v du
Numerical Integration
∆x = (b-a)/n
x0 = a, x1 = x1 + ∆ x , x3 = x2 + ∆x, ... , xn= b.
Trapizoidal Approximation for ∫ ab f(x) dx:
Tn = 0.5∆x [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]
Simpson's Rule (Parabolic Approximation) for ∫ ab f(x) dx:
Pn = ∆x [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(xn-1) + f(xn)]/3
Limit
For n positive : lim x -- > ∞ 1/xn = 0.
For n positive : lim x -- > ∞ xn = ∞.
For n positive : lim x -- > ∞ 1/enx = 0.
For n positive : lim x -- > ∞ enx = ∞.
For n positive : lim x -- > - ∞ 1/xn = 0.
For n positive : limx -- > - ∞ xn = ±∞.
Maximum and Minimum : 2 Variables
Given a function f(x,y) :
The discriminant : D = fxx fyy - fxy2
Decision : For a critical point P= (a,b)
If D(a,b) > 0 and fxx(a,b) < 0 then f has a rel-Maximum at P.
If D(a,b) > 0 and fxx(a,b) > 0 then f has a rel-Minimum at P.
If D(a,b) < 0 then f has a saddle point at P.
If D(a,b) = 0 then the test is inconclusive.
Volume and Averager Value
(2 variables case.)
Suppose f(x,y) is a function and R is a region on the xy-plane.
Assume that f(x,y) is a nonnegative on R. Then the volume under the graph of z = f(x,y) above R is given by
Volume = ∫ ∫ R f(x,y) dA

Suppose f(x,y) is a function and R is a region on the xy-plane. Then the AVERAGE VALUE of z = f(x,y) over the region R is given by
Average Value = ( ∫ ∫ R f(x,y) dA) / (Area of A).

Taylor Polynomial
Given a function f(x) the Taylor Polynomial P n (x) of f(x) around x = a is given by
P n (x) =
f(a) + f '(a)(x -a) + f ''(a)(x-a)2/2! + f (3)(a)(x-a)3/3! + f (4)(a)(x-a)4/4! + ... + f (n)(a)(x-a)n/n!

Infinite Series
The sum of the Geometric Series
a + ar + ar 2 + ... + ar n + ... =
a/(1 - r)     if -1 < r < 1  
Does Not Converge       Otherwise
Derivative Formulas : Trigonometric Functions
d/dx (sin u) = cos u u'(x)
d/dx (cos u) = - sin u u'(x)
d/dx (tan u) = sec 2 u u'(x)
d/dx (csc u) =- csc (u)cot u u'(x)
d/dx (sec u) = sec (u) tan u u'(x)
d/dx (cot u) = - csc 2 u u'(x)
Integration Formulas : Trigonometric Functions
∫ sin x dx = -cos x
∫ cos x dx = sin x
∫ tan x dx = - ln |cos x|
∫ sec x dx = ln |sec x + tan x|
∫ csc x dx = ln |csc x -cot x|
∫ cot x = ln |sin x|