Basic Formulae Flashcards
Geometric sum
Sn = a + ar + ar^2 +...+ ar^(n-1) Sn = a(1-r^n)/(1-r)
Geometric sum to infinity
S = a/(1-r)
when |r| > 1
Binomial Expansion (1+x)^n
(1+x)^n = 1 + nx + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3 +…
Binomial Expansion (a+b)^n
(a+b)^n = a^n + nC1 a^(n-1) b + nC2 a^(n-2) b^2 +…+ b^n
Maclaurin Series
f(x) = f(0) + xf’(0) + x^2 f’‘(0)/2! + x^3 f’’‘(0)/3! +…
Taylor Series f(a+x)
f(a+x) = f(a) + xf’(a) + x^2 f’‘(a)/2! + x^3 f’’‘(a)/3! +…
Taylor Series f(x)
f(x) = f(a) + (x-a)f’(a) + (x-a)^2 f’‘(a)/2! + (x-a)^3 f’’‘(a)/3! +…
Power Series e^x
e^x = 1 + x + x^2 /2! + x^3 /3! +…+ x^n /n! +…
Power Series ln(1+x)
ln(1+x) = x - x^2 /2 + x^3 /3 -…+ (-1)^(n+1) x^n /n +…
Power Series sin(x)
sin(x) = x - x^3 /3! + x^5 /5! -…
Power Series cos(x)
cos(x) = 1 - x^2 /2! + x^4 / 4!
De Moivre’s Theorem
(cos(θ) + isin(θ))^n = cos(nθ) + isin(nθ)
Trig in a circle
cos^2 (A) + sin^2 (A) = 1
sec^2(A) = tan^2 (A) + 1
cosec^2 (A) = cot^2 (A) + 1
Double angle formulae
sin(2A) = 2sin(A)cos(A)
cos(2A) = cos^2 (A) - sin^2 (A)
= 2 cos^2 (A) - 1
= 1 - 2sin^2 (A)
sin(A + B)
sin(A - B)
sin(A)cos(B) + cos(A)sin(B)
sin(A)cos(B) - cos(A)sin(B)
cos(A + B)
cos(A - B)
cos(A)cos(B) - sin(A)sin(B)
cos(A)cos(B) + sin(A)sin(B)
tan(A + B)
tan(A - B)
(tan(A) + tan(B))/(1 - tan(A)tan(B))
tan(A) - tan(B))/(1 + tan(A)tan(B)
cos(A)cos(B)
[cos(A + B) + cos(A - B)]/2
sin(A)sin(B)
[cos(A - B) - cos(A + B)]/2
sin(A)cos(B)
[sin(A + B) + sin(A - B)]/2
sin(A) + sin(B)
2sin((A + B)/2)cos((A - B)/2)
sin(A) - sin(B)
2sin((A - B)/2)cos((A + B)/2)
cos(A) + cos(B)
2cos((A + B)/2)cos((A - B)/2)
cos(A) - cos(B)
-2sin((A + B)/2)sin((A - B)/2)
Sine Rule
a/sin(A) = b/sin(B) = c/sin(C) sin(A)/a = sin(B)/b = sin(C)/c
Cosine Rule
a^2 = b^2 + c^2 - 2bccos(A) cos(A) = (b^2 + c^2 - a^2)/2bc
Product Rule
d(uv)/dx = vu’ + uv’
Quotient Rule
d(u/v)/dx = (vu’ - uv’)/v^2
Differential of x^n
nx^(n-1)
Differential of e^(kx)
ke^(kx)
Differential of ln(x)
1/x
Differential of e^(f(x))
f’(x)e^(f(x))
Differential of ln(f(x))
f’(x)/f(x)
Differential of sin(x)
cos(x)
Differential of cos(x)
-sin(x)
Differential of tan(x)
sec^2 (x)
Differential of sec(x)
sec(x)tan(x)
Differential of cot(x)
-cosec^2 (x)
Differential of cosec(x)
-cosec(x)cot(x)
Integral of sec^2(x)
tan(x)
Integral of sec(x)tan(x)
sec(x)
Integral of -cosec^2 (x)
cot(x)
Integral of -cosec(x)cot(x)
cosec(x)
Quadratic Formula roots
x = (-b +/- sqrt(b^2 - 4ac))/2a
Logarithm Rules
log(xy) = log(x) + log(y) log(x/y) = log(x) - log(y) log(x^k) = klog(x)
Surface area of a sphere
4πr^2
Volume of a sphere
4/3 πr^3
