Mathematics: Coordinate Geometry & Logarithms

Coordinate Geometry

• The most useful version of the straight line equation is y - y1 = m(x - x1)
• Two lines are perpendicular when their gradients multiply together to make minus one.

Sequences and Series

• The sum to n terms for arithmetic and geometric sequences and the sum to infinity for geometric sequences are given in the formula booklet.
• The nth term for an arithmetic sequence is un = a + (n - 1)d
• The nth term for a geometric sequence is un = ar^(n-1)

Logarithms

• log base a of x + log base a of y = log base a of x * y
• log base a of x - log base a of y = log base a of x / y
• log base a of x^y = y * log base a of x
• log base a of a = 1
• log base a of 1 = 0
• b^x = y converts to x = log base b of y

Differentiation

• The derivative of e^(kx) is ke^(kx)
• The derivative of sin(kx) is k cos(kx)
• The derivative of cos(kx) is -k sin(kx)
• The derivative of tan(kx) is k sec^2(kx)
• The derivative of log(kx) is 1/x
• The derivative of a^(kx) is k log(a) a^(kx)

Integration

• The integral of e^(kx) is (1/k)e^(kx) + c
• The integral of sin(kx) is (-1/k)cos(kx) + c
• The integral of cos(kx) is (1/k)sin(kx) + c
• The integral of sec^2(kx) is (1/k)tan(kx) + c
• The integral of 1/x is ln(x) + c
• The integral of a^(kx) is (1/ k ln(a)) * a^(kx) + c

Vectors

• To find the vector from point A to point B, subtract the position vector of A from the position vector of B (AB = B - A)
• A unit vector is a vector with length 1, it is calculated by dividing a vector by its length.
• The length of a vector in 3D is found using the 3D Pythagorean theorem (e.g., for vector (3, 2, 5), length = sqrt(3^2 + 2^2 + 5^2)).

Trig Identities

• tan(theta) = sin(theta)/cos(theta)
• sin^2(theta) + cos^2(theta) = 1
• 1 + cot^2(theta) = cosec^2(theta)
• tan^2(theta) + 1 = sec^2(theta)
• sin(2theta) = 2sin(theta)cos(theta)
• cos(2theta) = cos^2(theta) - sin^2(theta) = 2cos^2(theta) - 1 = 1 - 2sin^2(theta)
• tan(2theta) = 2tan(theta) / (1 - tan^2(theta))

Indices

• a^m * a^n = a^(m + n)
• a^m / a^n = a^(m - n)
• (a^m)^n = a^(m*n)
• a^(1/m) = mth root of a
• a^(-m) = 1/a^m

Sectors

• Arc length = angle (in radians) * radius
• Area = (1/2) * angle (in radians) * radius^2

Forces

• Net force = mass * acceleration (Newton's Second Law)
• Weight = mass * gravity
• Friction (Fr) <= mu * R (mu = coefficient of friction, R = normal reaction force)
• If the block is moving, Fr = mu * R
• If the block is at the point of moving, Fr = mu * R

Statistics

• Mean (x̄) = (sum of all x values) / (number of values)
• The variance is the standard deviation squared.

Normal Distribution

• The standard normal distribution formula is: z = (x - mu) / sigma
• z is the z-score, x is the observed value, mu is the mean, and sigma is the standard deviation.

Parametric Equations

• The equations of an ellipse will have different coefficients for the cos(theta) and sin(theta) terms, allowing for differing x and y values.
• Parametric differentiation: dy/dx = (dy/dt) * (dt/dx)
• Parametric integration: ∫ y dx = ∫ y (dx/dt) dt

Other Useful Concepts

• Integration by parts formula is in the formula booklet.
• Integration by substitution: Step 1 is to find du/dx
• Changing the limits when doing substitution.
• Partial Fractions: For expressions like 1/(x+1)(x-1), it can be written as A/(x+1) + B/(x-1).
• For integrals of sin^2(x) or cos^2(x), use double angle formulas.
• The gradient of a line is the same as tan(angle) when the line makes an angle with the horizontal.