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Arithmetic Sequence formula
Un= a + (n-1)d
nth term of a geometric sequence
Un = ar^n-1
Inverse of 2×2 matrix
Maclaurin expansion for e^x
1 + x + x² / 2! + x³ / 3! + x^4 / 4!
Maclaurin expansion for sinx
x - x³ / 3! + x^5 / 5! - x^7 / 7! …
Maclaurin expansion for cosx
1 - x² / 2! + x^4 / 4! - x^6 / 6! …
(AB)^-1
A^-1 • B^-1
Sum of infinite geometric series
S=a/1-r
Angle between things
Odd function
f(-x)=-f(x)
Even function
f(-x)=f(x)
2nd derivative of parametric equations
d2y/d2x = d/dt • (dy/dx) • dt/dx
d/dx(f(y)) =
d/dy (f(y)) • dy/dx
Complex numbers - If coefficients of polynomials are real then
Any complex roots will occur as a conjugate pair
Polynomial degree 2 will either have 2 real roots or 2 complex roots (a conjugate pair).
a polynomial of degree 3 (i.e. cubic) will either have 3 real roots or 1 real root and 2 complex roots (a conjugate pair).
If the coefficients are complex there are still n roots but they are not in any fixed pattern.
Any real n-degree polynomial, with n odd, must have at least one real root.
Trig identity to know
Cos^2x + sin^2x = 1
Cos^2x = 1/2 ( 1 + cos 2x)
Cos^2x = 1/2 ( 1 - cos2x)
Cos(2A) = cos^2A - sin^2A
Sin (2A) = 2sinAcosA
Polar form formula
Z = r(cosx + isinx)
First Order Differential Equation
dy/dx + P(x)y = Q(x)
∫ tanx dx
ln |secx| + C
∫cotx dx
ln|sinx| + C
∫-tanx dx
ln |cosx| + c
Integrating Factor
e^(∫P(x) dx)
General solution first order differential equations from equation given
Form of first order differentiation
dy/dx + P(x)y = Q(x)
Homogenous equation general solution for real and distinct
Ae^ px + Be^qx
Homogenous equation general solution for real and equal
y = (Ax + B)e^px
Homogenous equation general solution for complex conjugates for p + iq
Y= e^px(Asinqx + Bcosqx)
General solution (GS) for non homogenous second order differential equation
GS = Complimentary function + Particular integral
Particular integral from f(x) = 2x + 1
y = px + q
Particular integral from f(x) = x^2 - 1
px^2 + qx + r
Particular integral from f(x) = 4e^2x
pe^2x
Particular integral from f(x) = 2sinx + cosx
psinx + qcosx
Particular integral from f(x) = 3sin2x
psin2x + qcos2x
General formula for first order differential equation
d/dx IFy = IFQ
What differentiates to ln(f(x)
f'(x) / f(x)
Reflection in the x-axis matrix
1 0
0 -1
Reflection in the y-axis matrix
-1 0
0 1
Enlargement by scale factor k
k 0
0 k
Reflection in the line y = x matrix
0 1
1 0
rotation 90° about the origin matrix
0 -1
1 0
General plane equation
a(x - x0) - b(y-y0) + c(z-z0)
Where (a,b,c) are components of normal vector n
And (x0, y0, z0) are the coordinates of a known point on the plane
Intersection of 2 or 3 planes method
Solve the plane equations simultaneously (e.g. Gaussian elimination)
• Three possibilities:
One solution → point of intersection
Only two unique equations → set z = t and solve in terms of t to give the equation of a line
• No solution → no point of intersection.
To find out why, check the normal vectors to see how many are parallel.
Line and plane intersection method
Write line in parametric form
Substitute for x, y and z into plane equation and solve to find lamda.
Substitute lamda into line equation to find point.
Distance formula
d = SQUARE ROOT OF ( (x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2 )
Vector equation different forms
Image
Plane equations different forms
Image
What does 1 equal in terms of trig identities
Cos²x + sin²x = 1
2cos²x - cos2x = 1
2sin²x - sin2x = 1
Double angle formula
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos²x - sin²x
Compound angles trigonometric identities
Sin(A ± B) = sinAcosB ± cosAsinB
Cos(A±B) = cosAcosB ± sinAsinB
Trig identities with sec, tan, cosec and cot
1 + tan²x = sec²x
cot²x + 1 = cosec²x
TanA (A±B)
(TanA ± TanB) / 1 ± tanA tanB
tan2A
2tanA / 1 - tan²A
Differentiate ln f(x)
f’(x) / f(x)
Tan x integral
ln (secx) + c
(Sigma with n on top and k=1 on bottom) of 1
n
Partial fractions of px + q / (x - a) (x - b)
A / (x - a) + B / (x-b)
Partial fractions of px + q / (x - a)²
A / (x - a) + B / (x - a)²
Partial fractions of px² + qx + r / (x-a) (x-b) (x-c)
A / (x - a) + B / (x - b) + C / (x - c)
Partial fractions for px² + qx + r / (x-a)² (x-b)
A / (x - a) + B / (x - a)² + C / (x - b)
Partial fractions for px² + qx + r / (x - a) (x² + bx + c)
A / (x - a)
+
Bx + C / (x² + bx + c)