FAC1004 + FAD1014 — Must-Memorize Cheat Sheet
Everything you need cold. No derivations. No explanations. Just the formulas and procedures that must be instant.
MUST KNOW — ALL
Maclaurin Series (FAC1004 + FAD1014)
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$ $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$ $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$ $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (-1 < x \leq 1)$$ $$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \quad (|x| < 1)$$ $$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$ $$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$$ $$\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$$ $$\tan x \approx x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots \quad \text{(leak)}$$
Integration Standard Forms (FAD1014)
$$\int x^n,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$ $$\int \frac{1}{x},dx = \ln|x| + C$$ $$\int e^{ax},dx = \frac{1}{a}e^{ax} + C$$ $$\int \sin(ax),dx = -\frac{1}{a}\cos(ax) + C$$ $$\int \cos(ax),dx = \frac{1}{a}\sin(ax) + C$$ $$\int \sec^2 x,dx = \tan x + C$$ $$\int \frac{1}{\sqrt{a^2 - x^2}},dx = \sin^{-1}\left(\frac{x}{a}\right) + C$$ $$\int \frac{1}{\sqrt{x^2 \pm a^2}},dx = \ln\left|x + \sqrt{x^2 \pm a^2}\right| + C$$ $$\int \frac{1}{a^2 + x^2},dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$$
Inverse Hyperbolic Derivatives (FAC1004)
$$\frac{d}{dx}\sinh^{-1}u = \frac{1}{\sqrt{1+u^2}}\frac{du}{dx}$$ $$\frac{d}{dx}\cosh^{-1}u = \frac{1}{\sqrt{u^2-1}}\frac{du}{dx}$$ $$\frac{d}{dx}\tanh^{-1}u = \frac{1}{1-u^2}\frac{du}{dx}$$ $$\frac{d}{dx}\operatorname{sech}^{-1}u = -\frac{1}{u\sqrt{1-u^2}}\frac{du}{dx}$$
Inverse Hyperbolic Integrals (FAC1004)
$$\int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1}\left(\frac{x}{a}\right) + C = \ln\left|x + \sqrt{x^2 + a^2}\right| + C$$ $$\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}\left(\frac{x}{a}\right) + C = \ln\left|x + \sqrt{x^2 - a^2}\right| + C$$ $$\int \frac{dx}{a^2 - x^2} = \frac{1}{a}\tanh^{-1}\left(\frac{x}{a}\right) + C = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$$
Summation Formulas (FAD1014)
$$\sum_{r=1}^n r = \frac{n(n+1)}{2}$$ $$\sum_{r=1}^n r^2 = \frac{n(n+1)(2n+1)}{6}$$ $$\sum_{r=1}^n r^3 = \frac{n^2(n+1)^2}{4}$$
Key Identities (FAC1004)
$$z + \frac{1}{z} = 2\cos x, \quad z - \frac{1}{z} = 2i\sin x$$ $$z^n + z^{-n} = 2\cos(nx), \quad z^n - z^{-n} = 2i\sin(nx)$$ $$\cos x = \frac{e^{ix} + e^{-ix}}{2}, \quad \sin x = \frac{e^{ix} - e^{-ix}}{2i}$$
Integration by Parts (FAD1014)
$$\int u,dv = uv - \int v,du$$
LIATE (order for $u$): Log → Inverse trig → Algebraic → Trig → Exponential
Trig Substitution (FAD1014)
- $\sqrt{a^2 - x^2}$ → $x = a\sin\theta$
- $\sqrt{a^2 + x^2}$ → $x = a\tan\theta$
- $\sqrt{x^2 - a^2}$ → $x = a\sec\theta$
Method of Differences (FAD1014)
$$\frac{1}{r(r+1)} = \frac{1}{r} - \frac{1}{r+1}$$ $$\frac{1}{r(r+2)} = \frac{1}{2}\left(\frac{1}{r} - \frac{1}{r+2}\right)$$
Binomial Expansion I (FAD1014)
$$(a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r}b^r, \quad \binom{n}{r} = \frac{n!}{r!(n-r)!}$$
REFERENCE — Procedures & Details
FAC1004 — Advanced Mathematics II
Complex Numbers
Modulus, Argument, Polar Form
- $|z| = \sqrt{a^2 + b^2}$ where $z = a + bi$
- $\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)$, quadrant-adjusted
- Polar: $z = r(\cos\theta + i\sin\theta)$
- Euler: $e^{i\theta} = \cos\theta + i\sin\theta$
- De Moivre: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$
Loci — Boundary Types
- $|z - z_0| = r$ → Circle (centre $z_0$, radius $r$)
- $|z - z_1| = |z - z_2|$ → Perp bisector (midpoint of $(z_1,z_2)$, slope = $-1/$seg slope)
- $\arg(z - z_0) = \theta$ → Ray (from $z_0$ at angle $\theta$)
- $<$ or $>$ → Dotted boundary
- $\leq$ or $\geq$ → Solid boundary
Tangent Shortcut
$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}$$ For $\tan\left(\frac{\pi}{4} - i\right)$: use compound angle, NOT $e^{ix}$
Differential Equations
Bernoulli
$$\frac{dy}{dx} + P(x)y = Q(x)y^n \quad (n \neq 0,1)$$
Steps:
- If coefficient of $\frac{dy}{dx} \neq 1$, divide through
- Identify $n$, $P$, $Q$
- $v = y^{1-n}$, so $\frac{dv}{dx} = (1-n)y^{-n}\frac{dy}{dx}$
- Multiply original by $(1-n)y^{-n}$ → linear DE in $v$
- Solve using integrating factor $\mu = e^{\int P,dx}$
- Back-substitute $y^{1-n} = v$
Warning: $\frac{C}{x^{-2}} = Cx^2$ — $x$ is a variable, NOT a constant. Don't simplify to $A$.
Non-Homogeneous DEs
$$(a_1x + b_1y + c_1),dx + (a_2x + b_2y + c_2),dy = 0$$
Ratio test: $\frac{a_1}{a_2} = \frac{b_1}{b_2}$?
- YES → dependent → substitute $z = a_2x + b_2y$ → separable
- NO → independent → solve simultaneous eqns for $x,y$ → $X = x-h$, $Y = y-k$ → homogeneous
Ignore constants $C_1, C_2$ when testing linear dependence.
Exact DEs
$$M,dx + N,dy = 0$$
- Test: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
- Solve: $F = \int M,dx + g(y)$, $F_y = N$ → find $g(y)$. Solution: $F(x,y) = C$.
Mixing Tank
$$\frac{dQ}{dt} = R_{in}C_{in} - R_{out}\frac{Q}{V(t)} \quad \text{where} \quad V(t) = V_0 + (R_{in} - R_{out})t$$
Maclaurin Series Operations
- Substitution: Replace $x$ with argument (e.g., $e^{-x^2}$)
- Multiplication: Multiply two series term-by-term
- Integration: Integrate term-by-term
- Differentiation: Differentiate term-by-term
FAD1014 — Mathematics II
Integration
Area Under Curve
$$A = \int_a^b f(x),dx \quad \text{(above $x$-axis only)}$$
Differential Equations
Separable DE
$$\frac{dy}{dx} = f(x)g(y)$$
Steps: $\frac{dy}{g(y)} = f(x),dx$ → Integrate both sides → Apply IC.
Homogeneous DE
Test: Every term same total degree (e.g., $x^2$, $y^2$, $xy$ all degree 2)
Steps: $y = vx$, $\frac{dy}{dx} = v + x\frac{dv}{dx}$ → substitute → separate → integrate → back-sub $v = y/x$
Series & Summation
Convergence Tests
- nth term test: If $\lim_{n\to\infty} a_n \neq 0$, series diverges
- Geometric series: $\sum ar^{n-1}$ converges if $|r| < 1$
- p-series: $\sum \frac{1}{n^p}$ converges if $p > 1$
Geometry — Conic Sections
Parabola
- $(x-h)^2 = 4a(y-k)$ → opens up: focus $(h, k+a)$, directrix $y = k-a$
- $(x-h)^2 = -4a(y-k)$ → opens down: focus $(h, k-a)$, directrix $y = k+a$
- $(y-k)^2 = 4a(x-h)$ → opens right: focus $(h+a, k)$, directrix $x = h-a$
- $(y-k)^2 = -4a(x-h)$ → opens left: focus $(h-a, k)$, directrix $x = h+a$
- Vertex always $(h,k)$
Ellipse vs Hyperbola
- Ellipse: $c^2 = a^2 - b^2$ (subtract!)
- Hyperbola: $c^2 = a^2 + b^2$ (add!)
Ellipse
$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$
- Horizontal $(a > b)$: vertices $(h \pm a, k)$, foci $(h \pm c, k)$
- Vertical $(b > a)$: vertices $(h, k \pm b)$, foci $(h, k \pm c)$
Hyperbola
- Horizontal: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ → vertices $(h \pm a, k)$, asymptotes $y-k = \pm \frac{b}{a}(x-h)$
- Vertical: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ → vertices $(h, k \pm a)$, asymptotes $y-k = \pm \frac{a}{b}(x-h)$
Completing the Square
$$x^2 + bx \to \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4}$$
Parametric Equations
Derivatives
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad \frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \Big/ \frac{dx}{dt}$$
Converting to Cartesian
- Express $t$ in terms of $x$ and substitute into $y$
- Use identities ($\cos^2 t + \sin^2 t = 1$)
QUICK REFERENCE: NOT TESTED (Skip)
FAD1014 Skip
- Integration of powers of trig functions
- Integration by partial fractions
- Area between curves
- Volume of revolution
- Linear DE (integrating factor method)
- Bernoulli DE
- Binomial II (non-positive $n$)
- Taylor series (about $a \neq 0$)
- Circle geometry
PART B STRATEGY — FAD1014
Choose 4 of 6. Leak recommends:
- Maclaurin Series ($\checkmark$ leaked) — substitution, multiplication, integration
- Ellipse ($\checkmark$ leaked) — completing the square is mechanical
- Differential Equations ($\checkmark$ leaked) — identify type first
- Integration by Parts or Trig Substitution — pick your stronger
Avoid if weak: Parametric, Homogeneous DE