Calculus

Elementary Differential and Integral Calculus FORMULA SHEET Exponents xa · xb = xa+b , ax · bx = (ab)x , (xa )b = xab , x0 = 1.

Logarithms ln xy = ln x + ln y, ax = ex ln a .

ln xa = a ln x,

ln 1 = 0,

eln x = x,

ln ey = y,

Trigonometry sin 0 = cos π = 0, cos 0 = sin π = 1, 2 2 2 2 cos θ + sin θ = 1, cos(−θ) = cos θ, sin(−θ) = − sin θ, cos(A + B) = cos A cos B − sin A sin B, cos 2θ = cos2 θ − sin2 θ, sin(A + B) = sin A cos B + cos A sin B, sin 2θ = 2 sin θ cos θ, sin θ 1 , sec θ = , 1 + tan2 θ = sec2 θ. tan θ = cos θ cos θ Inverse Functions y = sin−1 x means x = sin y and − π y π . 2 2 y = cos−1 x means x = cos y and 0 y π. y = tan−1 x means x = tan y and − π < y < π . 2 2 1/n n y=x means x = y . y = ln x means x = ey . Alternative Notation arcsin x = sin−1 x, arccos x = cos−1 x, arctan x = tan−1 x, loge x = ln x. Note: sin−1 x = (sin x)−1 , cos−1 x = (cos x)−1 , tan−1 x = (tan x)−1 . However: sin2 x = (sin x)2 , cos2 x = (cos x)2 , tan2 x = (tan x)2 . Lines The line y = mx + c has slope m. The line through (x1 , y1 ) with slope m has equation y − y1 = m(x − x1 ). y − y1 y2 − y1 y2 − y1 and equation = . The line through (x1 , y1 ) and (x2 , y2 ) has slope m = x2 − x1 x − x1 x2 − x1 The line y = mx + c is perpendicular to the line y = m x + c if mm = −1. Circles √ The distance between (x1 , y1 ) and (x2 , y2 ) is (x1 − x2 )2 + (y1 − y2 )2 . The circle with centre (a, b) and radius r is given by (x − a)2 + (y − b)2 = r2 . Triangles In a triangle ABC:

(Sine Rule)

b c a = = ; sin A sin B sin C

(Cosine Rule) a2 = b2 + c2 − 2bc cos A.

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Pascal’s Triangle (x + y)2 = x2 + 2xy + y2 , (x + y)3 = x3 + 3x2 y + 3xy2 + y3 and so on. The coefficients in (x + y)n form the nth row of Pascal’s triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ............. and so on.

Quadratics If ax + bx + c = 0, with a = 0, then x =

2

−b ±

√

b2 − 4ac . 2a

Calculus dy du dv dy du dv If y = u + v then = + . If y = uv then = v+u . dx{ dx dx } / dx dx dx du u dy dv...