Math 3 (Grade 12)
The heaviest maths book: analysing functions, exponentials and logs, trig identities, conics, vectors, polar coordinates, statistics, and limits, derivatives…
The heaviest maths book: analysing functions, exponentials and logs, trig identities, conics, vectors, polar coordinates, statistics, and limits, derivatives and integrals.
Units of the book
| Unit | What to know |
|---|---|
| Analysing functions | domain, continuity, extrema, average rate of change, transformations |
| Exponential and logarithmic | exponential equations, log properties, natural logs |
| Trig identities | basic identities, sum and difference, double angle, solving equations |
| Conics | circles, parabolas, ellipses, hyperbolas |
| Vectors | magnitude, dot product, perpendicularity, 3-D vectors |
| Polar and complex | converting forms, De Moivre’s theorem |
| Probability and statistics | the normal distribution, z-scores, conditional probability |
| Limits and derivatives | limits, derivatives, tangents and velocity, integrals and area |
Formulas to remember
- log b x = y ⇔ bʸ = x: definition of a log
- log ab = log a + log b: product rule
- sin²θ + cos²θ = 1: Pythagorean identity
- sin 2θ = 2 sinθ cosθ: double angle
- (x−h)² + (y−k)² = r²: circle
- y² = 4px: parabola
- c² = a² − b²: ellipse
- c² = a² + b²: hyperbola
- a · b = a₁b₁ + a₂b₂: dot product
- x = r cosθ, y = r sinθ: polar coordinates
- z = (x − μ)/σ: z-score
- (xⁿ)′ = n xⁿ⁻¹: power rule
🧠 Memory kit: functions, logarithms and identities
| Formula | In plain words | How to remember it |
|---|---|---|
| (f(b) − f(a)) / (b − a) | Average rate of change = slope of the line through two points. | It is the slope formula used on a curve. |
| f(x − h) + k | Shift h right and k up. | Inside the bracket does the opposite: (x − 3) → 3 right. Outside does what it says. |
| −f(x) | f(−x) | Minus outside → reflect in the x-axis; minus inside → reflect in the y-axis. | Outside controls y; inside controls x. |
| log b x = y ⇔ bʸ = x | A log is a power: what power of b gives x? | “The base stays the base, and the answer becomes the power.” |
| log(ab) = log a + log b | Log of a product = sum of logs. | Like powers: xᵃ · xᵇ = xᵃ⁺ᵇ; multiplying becomes adding. |
| log(a/b) = log a − log b | Log of a quotient = difference of logs. | Dividing becomes subtracting. |
| log(aⁿ) = n log a | The power comes down in front. | “The power jumps to the front.” |
| log b x = log x / log b | Change of base. | The base goes to the bottom. |
| log b 1 = 0, log b b = 1, b^(log b x) = x | Quick facts. | Because b⁰ = 1 and b¹ = b. |
| ln x = logₑ x, e ≈ 2.718 | The natural log has base e. | ln e = 1 and e^(ln x) = x. |
| A = P(1 + r/n)ⁿᵗ | A = Pe^(rt) | Interest compounded n times a year, and continuously. | Continuous interest spells “Pert”. |
| sin²θ + cos²θ = 1 | The Pythagorean identity. | It is Pythagoras on the unit circle (hypotenuse = 1). |
| 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ | Two derived identities. | Divide the main identity by cos²θ, then by sin²θ. |
| csc = 1/sin, sec = 1/cos, cot = 1/tan | The reciprocals. | Each pairs with the opposite letter: sec ↔ cos, csc ↔ sin (not what you’d guess). |
| sin(A ± B) = sinA cosB ± cosA sinB | Sine of a sum or difference. | Sine mixes (sin cos + cos sin) and keeps the sign. |
| cos(A ± B) = cosA cosB ∓ sinA sinB | Cosine of a sum or difference. | Cosine keeps like with like (cos cos, sin sin) and flips the sign. |
| sin 2θ = 2 sinθ cosθ | Double angle for sine. | Put A = B = θ in the sum formula. |
| cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ | Double angle for cosine, three forms. | The last two come from the first using sin² + cos² = 1. |
| tan 2θ = 2tanθ / (1 − tan²θ) | Double angle for tangent. | “Two tan over one minus tan squared.” |
🧠 Memory kit: conics, vectors, statistics and calculus
| Formula | In plain words | How to remember it |
|---|---|---|
| (x − h)² + (y − k)² = r² | Circle: centre (h, k), radius r. | It is the distance formula from any point to the centre. |
| y² = 4px | x² = 4py | Parabola: the focus is p from the vertex, the directrix p on the other side. | The unsquared letter tells the direction it opens. |
| x²/a² + y²/b² = 1, c² = a² − b² | Ellipse: a is the semi-major axis; the foci are inside. | Ellipse uses minus: c² = a² − b²; eccentricity e = c/a is below 1. |
| x²/a² − y²/b² = 1, c² = a² + b² | Hyperbola: the foci lie beyond the vertices; asymptotes y = ±(b/a)x. | Hyperbola uses plus: c² = a² + b²; e is above 1. |
| Ax² + Cy² + …: A = C, AC > 0, AC | Recognise the conic: circle, ellipse, hyperbola, parabola. | Same signs → ellipse (equal → circle), different signs → hyperbola, a square missing → parabola. |
| |v| = √(x² + y²), u = v / |v| | Length of a vector and the unit vector along it. | Unit vector = the vector divided by its length, so its length is 1. |
| a · b = a₁b₁ + a₂b₂ = |a||b| cos θ | Dot product, which gives the angle between vectors. | A dot product of 0 means the vectors are perpendicular. |
| x = r cosθ, y = r sinθ, r² = x² + y² | Between polar and rectangular coordinates. | They are the trig ratios in a triangle with hypotenuse r. |
| [r(cosθ + i sinθ)]ⁿ = rⁿ(cos nθ + i sin nθ) | De Moivre’s theorem for powers of complex numbers. | “Raise the length, multiply the angle.” |
| z = (x − μ) / σ | z-score: how many standard deviations a value is from the mean. | Positive → above the mean, negative → below. |
| 68% – 95% – 99.7% | Normal distribution: within 1, 2 and 3 standard deviations. | Learn it as one chant: “68-95-99.7”. |
| P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ | Probability of k successes in n trials (binomial). | Choose where the successes go, then success k times and failure the rest. |
| (xⁿ)′ = n xⁿ⁻¹ | Power rule for derivatives. | “Bring the power down, then drop it by one.” |
| (fg)′ = f′g + fg′ | Product rule. | “Derive the first, keep the second, plus keep the first, derive the second.” |
| (f/g)′ = (f′g − fg′) / g² | Quotient rule. | Like the product rule but with minus, over the bottom squared. |
| f′(x) = 0 | Critical points: candidates for maxima and minima. | At a peak or valley the tangent is flat (slope 0). |
| v = s′(t), a = v′(t) | Velocity is the derivative of position; acceleration of velocity. | Each derivative is one step: position → velocity → acceleration. |
| ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C | Antiderivative of xⁿ (n ≠ −1). | The reverse of differentiating: “raise the power by one, divide by it”, and add + C. |
| ∫ₐᵇ f(x) dx = F(b) − F(a) | The definite integral = area under the curve from a to b. | “Top limit minus bottom limit.” |
| lim (x→∞) لكسر كثيري حدود | Top degree smaller → 0, equal → ratio of leading coefficients, top bigger → no limit (∞). | At infinity the highest power swallows everything else. |
- Ellipse vs hyperbola: ellipse → minus c² = a² − b², hyperbola → plus c² = a² + b².
- Derivative and integral undo each other: the derivative of x³ is 3x², and the integral of 3x² is x³ + C. Check an integral by differentiating it.
- Log rules are power rules in disguise: product → sum, quotient → difference, power → multiple. If you forget one, write the matching power rule.
A way of memorizing that sticks
- Understand before you memorize: read “in plain words” for each formula and say out loud what it measures and why each quantity is multiplied or divided. A formula you understand takes a minute to learn.
- Active recall: cover the formula column and write it from memory, then check. Writing from memory beats rereading by far.
- Spaced repetition: review the same sheet after 1 day, 3 days, a week and two weeks. Four short reviews lock it in until test day.
- Your own formula page: write every formula of a book on one page by hand. If it does not fit, you are writing things you can derive, so cut them.
- Teach it: explain the formula to a friend, or to yourself in a mirror, in two sentences. If you cannot explain it, you do not know it yet.
- Use it at once: after each group, press “Try one” in the patterns below. Memory sets when you use it.
Question patterns
Limits and rate of change
If you get 0/0, factor and cancel. At infinity, compare degrees.
Exponents and logarithms
Make the bases equal, then match the exponents. A logarithm is an exponent.
Identities and trig equations
Use 3–4–5 triangles for ratios; choose quadrants by sign.
Conic sections
Read the centre and radius from standard form (flip the signs). A parabola’s focus is (p, 0).
Vectors
Magnitude by Pythagoras; perpendicular when the dot product is zero.
Polar and complex
De Moivre: raise the modulus, multiply the angle.
Statistics and probability
68–95–99.7 for 1, 2, 3 standard deviations. P(A|B) = P(A∩B)/P(B).
Derivatives and integrals
Derivative = tangent slope = instantaneous velocity. Integral = area under the curve.
- In a circle, (x + 2)² means h = −2.
- Ellipse c² = a² − b²; hyperbola c² = a² + b².
- log(a + b) is not log a + log b.
More patterns (figures and calculations)
Unit vectors
Divide the vector by its length.
Extreme values on an interval
Evaluate at the endpoints and critical points, then compare.
Area by integration
∫ₐᵇ f(x) dx = F(b) − F(a)
Antiderivatives
Raise the power by one, divide by it, and add C.
Logarithmic equations
Rewrite in exponential form.
Reading a graph
Value = the y-coordinate, maximum = highest point, range = all y-values.