Combined pure formulas

Question 1

Simplifying partial fractions

Answer 1

An expression of the form
ax+b / (px + q)(rx + s)
can be split into partial fractions of the form
A(rx +s) + B(px + q)

An expression of the form
ax² + bx + c / (px + q)(rx + s)²
can be split into partial fractions of the form
|A / (px + q)| + |B / (rx + s)|+|C / (rx + s)²|
A(rx + s)² + B(rx + s)(px + q) + C(px + q)

ax² + bx + c / (px + q)(rx² + s)
can be split into partial fractions of the form
|A / (px + q)| + |B / (rx² + s)|
A(rx² + s) + (Bx + C)(px+q)

Question 2

Binomial expansion

Answer 2

(1 + x)ⁿ = 1 + nx + [n(n - 1) / 2! ]x² +
[n(n – 1)(n – 2) / 3! ]x³ + …

(a + x)ⁿ = aⁿ { 1 + n(x/a) + n(n - 1) / 2! ² +
n(n – 1)(n – 2) / 3! ³ + …

Question 3

Logarithmic notation

Answer 3

If y = bˣ ⇔ x = logᵦ(y)

Question 4

Properties of logarithmic functions

Answer 4

logᵦ(b) = 1
logᵦ(1) = 0
logᵦ(bⁿ) = n
logᵦ(xⁿ) = n * logᵦ(x)
logᵦ(ⁿ√x) = [1/n] * logᵦ(x)
logᵦ(uv) = logᵦ(u) + logᵦ(v)
logᵦ(u/v) = logᵦ(u) - logᵦ(v)

Question 5

Parallelogram rule of subtraction

Answer 5

AB> = OB> – OA>

Question 6

Differentiating products

Answer 6

y = u’v + v’u

Question 7

Differentiating quotients

Answer 7

y = [u’v - v’u] / v²

Question 8

Implicit differentiation

Answer 8

when differentiating implicitly

1.xy becomes y + x(dy/dx)
2. x²y becomes 2xy + x²(dy/dx)
3. xy² becomes y² + 2xy(dy/dx)

Rule of thumb:
1.the y gets taken out as its own term as is (with an x value if x remains when differentiated)
2. the second dy/dx term (implicit term) the x gets taken out as its own term as is (with a y value if y remains when differentiated)

Question 9

The trapezium rule

Answer 9

∫ᵇₐ 𝑓(𝑥) 𝑑𝑥 ≈ h/2 (y₀+2y₁+2y₂+ … +2yₙ₋₁+yₙ)

where h = b-a/n

Find y values by substituting x values into equation of line or curve

Question 10

Integration by parts

Answer 10

∫uv’ = uv - ∫vu’