pure maths bits

differentiate ln(x)

1/x

differentiate ln(f(x))

(f’(x))/(f(x))

differentiate y = a^x

ln(a) x a^x

differentiate y= a^(f(x))

f’(x) x ln(a) x a^(f(x))

differentiate y= e^x

e^x

differentiate y= e^(f(x))

f’(x) x e^(f(x))

arithmetic sequence formula

a + (n-1)d

sum of arithmetic sequence

(n/2)(2a + (n-1)d) or (n/2)(a+l)

proof of sum of arithmetic sequence

(1) sn = a + (a+d) + (a+2d) + … + (a+(n-2)d) + (a+(n-1)d)

• reverse

(2) sn = (a+(n-1)d) + (a+(n-2)d) + … + (a+2d) + (a+d)

(1) + (2) = 2sn = n(2a + (n-1)d)

so, sn = (n/2)(2a + (n-1)d)

geometric series formula

ar^(n-1)

sum of geometric series

(a(1-r^n))/(1-r) = (a(r^n - 1))/(r-1)

proof of sum of geometric series

(1) sn = a + ar + ar² + ar³ + … + ar^(n-1)

(2) rsn = ar + ar² + ar³ + … + ar^n

(1) - (2) = sn - rsn = a-ar^n

• factorise each side

sn(1-r) = a(1-r^n)

sn= (a(1-r^n))/(1-r)

divergent series meaning

each term gets bigger

convergent series meaning

each term gets bigger

when is a geometric sequence convergent

l r l < 1

sum to infinity for a convergent series

s(infinity) = a/(1-r)

arc length in radians

r(theta)

sector area in radians

(1/2)(theta)(r²)

arc length in degrees

(theta/360) x 2(pi)r

sector area in degrees

(theta/360) x (pi)r²

degrees to radians

x(pi/180)

cosine rule

a²=b² + c² - 2bccos(A)

sine rule

a/sinA = b/sinB = c/sinC

integrate y= a^x dx

(a^x)/ln(a)

integrate y= e^(ax+b)

(1/a)e^(ax+b)

sin cos diff int (clockwise)

sin —> cos —> -sin —> - cos

newton-raphson formula

x_n+1 = x_n - ((f(x_n))/(f’(x_n)))

parametric integration formula

integrate —> y(dx/dt)dt

for what values of x is the expansion of (1-9x)^1/2 valid

l 9x l < 1 so -1/9 < x < 1/9

how to find range of composite function

• find (for example) fg(x)

• domain of g(x) and function g(x) should satisfy domain of f(x)

• sketch fg(x)

order of graph transformations

1: inside bracket

• translation

• stretch/compress

• reflection

2: outside stretch/compression

3: outside translation

asymptotes of reciprocal graphs

vert asymptote: denominator= 0

horizontal asymptote: do long division ie

(3x+1)/(x-1) —> 3 + 4/(x+1) where y=3 is the horizontal asymptote

transform f(x) to f(lxl)

reflect all graphs that have +ve x value in the y-axis (so they’re on both sides)

for parametrics - what does show that all points on c satisfy “equation” mean

turn parametric equations into a cartesian equation