12. Applications of Differentiation

Quadratic functions am-c1

Completed square form	$y = a {(x - h)}^{2} + k$ turning point (h, k); minimum if a > 0, maximum if a < 0
Always positive / always negative	a > 0 and b² − 4ac < 0 · a < 0 and b² − 4ac < 0the curve never touches the x-axis

Equations and inequalities am-c2

The discriminant	$b^{2} - 4 a c$ > 0: two real distinct roots · = 0: two real equal roots · < 0: no real roots · “real roots”: ≥ 0
A line and a curve	substitute the line into the curve, collect to … = 0, then test that quadratic’s discriminanttangent ⇔ = 0 · meets at two points ⇔ > 0 · does not meet ⇔ < 0
Quadratic inequality, roots α < β	(x − α)(x − β) < 0 ⇔ α < x < β · (x − α)(x − β) > 0 ⇔ x < α or x > βrearrange to … > 0 or … < 0 first

Surds am-c3

The three laws	$\sqrt{a b} = \sqrt{a} \times \sqrt{b}$ $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ $\sqrt{a} \times \sqrt{a} = a$ for a, b > 0
Conjugate surds	$a + b \sqrt{k}$ and $a - b \sqrt{k}$ multiply top and bottom by the conjugate to rationalise a denominator
Equal surds	if $a + b \sqrt{k} = c + d \sqrt{k}$ then a = c and b = da, b, c, d rational, √k irrational

Polynomials and partial fractions am-c4

Division algorithm	dividend = divisor × quotient + remainderthe remainder has a lower degree than the divisor
Remainder and Factor Theorems	dividing f(x) by x − a leaves f(a); x − a is a factor exactly when f(a) = 0for ax + b, substitute x = −b/a
Sum and difference of cubes	$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
Partial fractions — the three forms	distinct linear factors: $\frac{A}{a x + b} + \frac{B}{c x + d}$ a repeated factor: $\frac{A}{a x + b} + \frac{B}{c x + d} + \frac{C}{{(c x + d)}^{2}}$ a quadratic factor: $\frac{A}{a x + b} + \frac{B x + C}{x^{2} + c^{2}}$ an improper fraction is divided first

Binomial Theorem am-c5 G3

Factorial	$n! = n \times (n - 1) \times \dots \times 2 \times 1$ and 0! = 1the theorem, both forms of the coefficient and the general term are all on your paper
General term	$T_{r + 1} = (\binom{n}{r}) a^{n - r} b^{r}$ the middle term of the printed expansion — the kth term has r = k − 1

Exponential and logarithmic functions am-c6 G3

Equivalence	$y = a^{x}$ ⇔ $x = \log_{a} y$ for a > 0, a ≠ 1
Special values	$\log_{a} 1 = 0$ $\log_{a} a = 1$ $a^{\log_{a} b} = b$
The three laws	$\log_{a} x + \log_{a} y = \log_{a} x y$ $\log_{a} x - \log_{a} y = \log_{a} \frac{x}{y}$ $\log_{a} x^{r} = r \log_{a} x$
Change of base	$\log_{a} b = \frac{\log_{c} b}{\log_{c} a}$ $\log_{a} b = \frac{1}{\log_{b} a}$
Growth and decay models	$N = N_{0} a^{t}$ or $N = N_{0} e^{k t}$ N₀ is the value at t = 0; growth k > 0, decay k < 0

Coordinate geometry am-c7

Parallel and perpendicular lines	$m_{1} = m_{2}$ $m_{1} m_{2} = - 1$
Midpoint	$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
Line through a point	$y - y_{1} = m (x - x_{1})$
Circle	${(x - a)}^{2} + {(y - b)}^{2} = r^{2}$ $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ centre (−g, −f), radius √(g² + f² − c)
Area of a rectilinear figure	the shoelace array, vertices taken anticlockwise, halved

Linear law am-c8 G3

Straight-line form

Y = m X + c

rewrite the relationship so the unknown constants become the gradient and the Y-intercept; for y = axⁿ or y = kb^x, take logs first

Trigonometric functions and graphs am-c9

Radians	$π rad = 180 °$ $θ = \frac{s}{r}$ arc length rθ and sector area ½r²θ are on your paper
Exact values	sin 30° = ½, sin 45° = $\frac{\sqrt{2}}{2}$ , sin 60° = $\frac{\sqrt{3}}{2}$ tan 30° = $\frac{\sqrt{3}}{3}$ , tan 45° = 1, tan 60° = $\sqrt{3}$ cos runs the sine values backwards: cos 30° = sin 60°, and so on
Signs by quadrant	ASTC: All positive, then Sin, then Tan, then Cosquadrant and basic angle first, then the sign
Negative angles	$\sin (- θ) = - \sin θ$ $\cos (- θ) = \cos θ$ $\tan (- θ) = - \tan θ$
y = a sin(bx) + c	amplitude \|a\| · period $\frac{360 °}{b}$ or $\frac{2 π}{b}$ · axis y = cmax \|a\| + c, min −\|a\| + c; for y = a tan(bx) the period is 180°/b

Trigonometric equations and identities am-c10

Reciprocals and quotients	$cosec A = \frac{1}{\sin A}$ $\sec A = \frac{1}{\cos A}$ $\cot A = \frac{1}{\tan A}$ $\tan A = \frac{\sin A}{\cos A}$ cosec goes with sin, sec with cos
The R-formula — asked for by name, not given	$a \sin θ \pm b \cos θ = R \sin (θ \pm α)$ $a \cos θ \pm b \sin θ = R \cos (θ \mp α)$ with $R = \sqrt{a^{2} + b^{2}}$ and $\tan α = \frac{b}{a}$ , a and b positive; maximum R, minimum −R

Proofs in plane geometry am-c17 G3

Midpoint Theorem	M, N midpoints of AB, AC ⇒ MN ∥ BC and MN = ½BCand its converse: parallel and half ⇒ midpoints
Tangent-Chord Theorem	the angle between a tangent and a chord = the angle in the alternate segmentalso called the Alternate Segment Theorem
Tangent and secant	${P T}^{2} = P A \times P B$ worth recognising: tangent PT, and PAB cutting the circle at A and B
Every line of a proof	a statement and its reason in bracketsE-Math’s circle properties and congruence tests are assumed

Gradients and differentiation techniques am-c11

Power rule	$\frac{d}{d x} (x^{n}) = n x^{n - 1}$ any rational n; rewrite roots and reciprocals as powers first
Chain rule	$\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$
Product rule	$\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$
Quotient rule	$\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$ bottom × d(top) minus top × d(bottom), over bottom squared

Applications of differentiation am-c12

Tangent and normal at (x₁, y₁)	$y - y_{1} = m (x - x_{1})$ tangent gradient m = dy/dx at the point; normal gradient −1/m
Increasing and decreasing	$\frac{d y}{d x} > 0$ : increasing $\frac{d y}{d x} < 0$ : decreasing
Connected rates of change	$\frac{d y}{d t} = \frac{d y}{d x} \times \frac{d x}{d t}$
Stationary points	$\frac{d y}{d x} = 0$ ; then $\frac{d^{2} y}{d x^{2}} < 0$ maximum, $\frac{d^{2} y}{d x^{2}} > 0$ minimumif the second derivative is 0, use the first-derivative (sign) test

Differentiation of trigonometric, exponential and logarithmic functions am-c13 G3

The five derivatives	$\frac{d}{d x} \sin x = \cos x$ $\frac{d}{d x} \cos x = - \sin x$ $\frac{d}{d x} \tan x = \sec^{2} x$ $\frac{d}{d x} e^{x} = e^{x}$ $\frac{d}{d x} \ln x = \frac{1}{x}$ x in radians
With a linear inside	$\frac{d}{d x} \sin (a x + b) = a \cos (a x + b)$ $\frac{d}{d x} e^{a x + b} = a e^{a x + b}$ and so on: the derivative of the inside multiplies the front
Logarithm of a function	$\frac{d}{d x} \ln f (x) = \frac{f' (x)}{f (x)}$

Integration am-c14

Power rule	$\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c$ n ≠ −1; divide by the new index
Linear inside	$\int {(a x + b)}^{n} d x = \frac{{(a x + b)}^{n + 1}}{a (n + 1)} + c$ n ≠ −1
Trigonometric G3	$\int \cos (a x + b) d x = \frac{1}{a} \sin (a x + b) + c$ $\int \sin (a x + b) d x = - \frac{1}{a} \cos (a x + b) + c$ $\int \sec^{2} (a x + b) d x = \frac{1}{a} \tan (a x + b) + c$ watch the minus sign on sin
Exponential and n = −1 G3	$\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + c$ $\int \frac{1}{a x + b} d x = \frac{1}{a} \ln (a x + b) + c$

Applications of integration am-c15

Definite integral	$\int_{a}^{b} f (x) d x = F (b) - F (a)$ F is any antiderivative of f
Area above the x-axis	area = $\int_{a}^{b} f (x) d x$ when f(x) ≥ 0 on [a, b]
Area below the x-axis G3	area = $- \int_{a}^{b} f (x) d x$ split the region where the curve crosses the axis

Kinematics am-c16 G3

The ladder	$v = \frac{d s}{d t}$ $a = \frac{d v}{d t}$ $s = \int v d t$ $v = \int a d t$ find each “+ c” from the given values, usually at t = 0
Key conditions	instantaneously at rest: v = 0 · at O: s = 0 · greatest speed: a = 0distance travelled: add the distances between the times when v = 0

Quadratic equationfor ax² + bx + c = 0	$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ The E-Math paper does not give you this one.
Binomial expansion G3	${(a + b)}^{n} = a^{n} + (\binom{n}{1}) a^{n - 1} b + (\binom{n}{2}) a^{n - 2} b^{2} + \dots + (\binom{n}{r}) a^{n - r} b^{r} + \dots + b^{n}$ where n is a positive integer
and G3	$(\binom{n}{r}) = \frac{n!}{r! (n - r)!} = \frac{n (n - 1) \dots (n - r + 1)}{r!}$

Pythagorean	$\sin^{2} A + \cos^{2} A = 1$ $\sec^{2} A = 1 + \tan^{2} A$ ${cosec}^{2} A = 1 + \cot^{2} A$
Compound angle	$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Double angle	$\sin 2 A = 2 \sin A \cos A$ $\cos 2 A = \cos^{2} A - \sin^{2} A = 2 \cos^{2} A - 1 = 1 - 2 \sin^{2} A$ $\tan 2 A = \frac{2 \tan A}{1 - \tan^{2} A}$

Sine Rule	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine Rule	$a^{2} = b^{2} + c^{2} - 2 b c \cos A$
Area	$∆ = \frac{1}{2} b c \sin A$

Paper 1 · 2 h 15 min · 90 marks · 50%	12 to 14 questions of varying marks and lengths, up to 10 marks each. Answer all of them.
Paper 2 · 2 h 15 min · 90 marks · 50%	9 to 11 questions of varying marks and lengths, up to 12 marks each. Answer all of them.

Paper 1 · 1 h 45 min · 70 marks · 50%	13 to 15 questions of varying marks and lengths. Answer all of them.
Paper 2 · 1 h 45 min · 70 marks · 50%	8 to 10 questions of varying marks and lengths. Answer all of them.

Working	“Omission of essential working will result in loss of marks.” (A bare answer earns the answer mark at most; the method marks sit on the lines that lead to it.)
Accuracy	“Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. In questions which explicitly require an answer to be shown to be correct to a specific accuracy, the answer must be first shown to a higher degree of accuracy.” (Round once, at the end. Carry at least 4 s.f. through the working, or keep the value in the calculator.)
π	“Unless the question requires the answer in terms of π, the calculator value for π or π = 3.142 should be used.”
Units and time	SI units are used for mass and measures; compound units use the solidus (5 m/s, 13.6 g/cm³); the 12-hour and 24-hour clock may both be used (3.15 p.m. is 15 15).
Calculators	“An approved calculator may be used in both Paper 1 and Paper 2.” (A question that says without using a calculator or do not use a calculator is marked on the exact working it asks for.)
Formulae	“Relevant mathematical formulae will be provided for candidates.” (Exactly what is provided, and what is not, is the other two tabs of this window.)