Mathematics: Analysis & Approaches SL & HL
Interactive Formula Sheet – First Examinations 2021 – Updated Version 1.3
Prior Learning SL & HL
Area: Parallelogram
$$A = bh$$
b = base, h = height
Area: Triangle
$$A = \frac{1}{2}bh$$
b = base, h = height
Area: Trapezoid
$$A = \frac{1}{2}(a+b)h$$
a, b = parallel sides, h = height
Area: Circle
$$A = \pi r^2$$
r = radius
Circumference: Circle
$$C = 2\pi r$$
r = radius
Volume: Cuboid
$$V = lwh$$
l = length, w = width, h = height
Volume: Cylinder
$$V = \pi r^2 h$$
r = radius, h = height
Volume: Prism
$$V = Ah$$
A = cross-section area, h = height
Area: Cylinder curve (lateral surface)
$$A = 2\pi rh$$
r = radius, h = height
Distance between two points \((x_1, y_1), (x_2, y_2)\)
$$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$
Coordinates of midpoint
$$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$
for endpoints \((x_1, y_1), (x_2, y_2)\)
Topic 1: Number and algebra
SL & HL Content
The \(n^{th}\) term of an arithmetic sequence
$$u_n = u_1 + (n-1)d$$
Sum of \(n\) terms of an arithmetic sequence
$$S_n = \frac{n}{2}(2u_1 + (n-1)d) = \frac{n}{2}(u_1 + u_n)$$
The \(n^{th}\) term of a geometric sequence
$$u_n = u_1 r^{n-1}$$
Sum of \(n\) terms of a finite geometric sequence
$$S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1 - r^n)}{1-r}, \quad r \neq 1$$
Compound interest
$$FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$$
FV: future value, PV: present value, n: number of years, k: compounding periods per year, r%: nominal annual rate of interest
Exponents & logarithms (definition)
$$a^x = b \iff x = \log_a b$$
\(a, b > 0, a \neq 1\)
Exponents & logarithms (properties)
$$\log_a xy = \log_a x + \log_a y$$ $$\log_a \frac{x}{y} = \log_a x - \log_a y$$ $$\log_a x^m = m \log_a x$$ $$\log_a x = \frac{\log_b x}{\log_b a}$$
The sum of an infinite geometric sequence
$$S_\infty = \frac{u_1}{1-r}, \quad |r| < 1$$
Binomial Theorem for \(n \in \mathbb{N}\)
$$(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \dots + \binom{n}{r}a^{n-r}b^r + \dots + b^n$$
Also: \((a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r}b^r\)
Binomial coefficient
$$\binom{n}{r} = {}^nC_r = \frac{n!}{r!(n-r)!}$$
HL Only Content
Combinations; Permutations
$${}^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}; \quad {}^nP_r = \frac{n!}{(n-r)!}$$
Extension of Binomial Theorem, \(n \in \mathbb{Q}\)
$$(a+b)^n = a^n\left(1 + n\left(\frac{b}{a}\right) + \frac{n(n-1)}{2!}\left(\frac{b}{a}\right)^2 + \dots \right)$$
Valid for \( \left|\frac{b}{a}\right| < 1 \)
Complex numbers
$$z = a + bi$$
\(a, b \in \mathbb{R}\), \(i^2 = -1\)
Modulus-argument (polar) & Exponential (Euler) form
$$z = r(\cos\theta + i\sin\theta) = re^{i\theta} = r\text{cis}\theta$$
\(r = |z|\), \(\theta = \arg(z)\)
De Moivre's theorem
$$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta)) = r^n e^{in\theta} = r^n\text{cis}(n\theta)$$
\(n \in \mathbb{Z}\) (extends to \(n \in \mathbb{Q}\) for roots)
Topic 2: Functions
SL & HL Content
Equations of a straight line
$$y = mx+c$$ $$ax+by+d=0$$ $$y-y_1 = m(x-x_1)$$
Gradient formula
$$m = \frac{y_2-y_1}{x_2-x_1}$$
Axis of symmetry of a quadratic function
$$f(x) = ax^2+bx+c \implies x = -\frac{b}{2a}$$
Solutions of a quadratic equation \(ax^2+bx+c=0\)
$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}, \quad a \neq 0$$
Discriminant
$$\Delta = b^2-4ac$$
Exponential and logarithmic functions
$$a^x = e^{x\ln a}$$ $$\log_a a^x = x = a^{\log_a x}$$
where \(a, x > 0, a \neq 1\)
HL Only Content
Sum & product of roots of polynomial \( \sum_{r=0}^{n} a_r x^r = 0 \)
For \(a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0\):
$$\text{Sum of roots} = -\frac{a_{n-1}}{a_n}$$ $$\text{Product of roots} = (-1)^n \frac{a_0}{a_n}$$
Topic 3: Geometry and trigonometry
SL & HL Content
Distance between 2 points \((x_1,y_1,z_1), (x_2,y_2,z_2)\)
$$d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$$
Coordinates of midpoint of a line with endpoints \((x_1,y_1,z_1), (x_2,y_2,z_2)\)
$$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$$
Volume: Right-pyramid
$$V = \frac{1}{3}Ah$$
A = base area, h = height
Volume: Right cone
$$V = \frac{1}{3}\pi r^2 h$$
r = radius, h = height
Area: Cone curve (lateral surface)
$$A = \pi r l$$
r = radius, l = slant height
Volume: Sphere
$$V = \frac{4}{3}\pi r^3$$
r = radius
Surface area: Sphere
$$A = 4\pi r^2$$
r = radius
Sine rule
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Cosine rule
$$c^2 = a^2+b^2 - 2ab\cos C$$ $$\cos C = \frac{a^2+b^2-c^2}{2ab}$$
Area: Triangle
$$A = \frac{1}{2}ab\sin C$$
Length of an arc
$$l = r\theta$$
r = radius, \(\theta\) = angle in radians
Area of a sector
$$A = \frac{1}{2}r^2\theta$$
r = radius, \(\theta\) = angle in radians
Identity for \(\tan\theta\)
$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$
Pythagorean identity
$$\cos^2\theta + \sin^2\theta = 1$$
Double angle identities
$$\sin(2\theta) = 2\sin\theta\cos\theta$$ $$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$
HL Only Content
Reciprocal trigonometric identities
$$\sec\theta = \frac{1}{\cos\theta}; \quad \csc\theta = \frac{1}{\sin\theta}; \quad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$$
Pythagorean identities (extended)
$$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$
Compound angle identities
$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$ $$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$ $$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$
Double angle identity for \(\tan\)
$$\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}$$
Magnitude of a vector \(\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\) or \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k} \)
$$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$$
Scalar product (dot product)
$$\mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3$$ $$\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}||\mathbf{w}|\cos\theta$$
\(\theta\) is the angle between \(\mathbf{v}\) and \(\mathbf{w}\)
Angle between two vectors
$$\cos\theta = \frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|} = \frac{v_1w_1 + v_2w_2 + v_3w_3}{|\mathbf{v}||\mathbf{w}|}$$
Vector equation of a line
$$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$$
\(\mathbf{a}\) is position vector of a point on line, \(\mathbf{b}\) is direction vector
Parametric form of the equation of a line
$$x = x_0 + \lambda l, \quad y = y_0 + \lambda m, \quad z = z_0 + \lambda n$$
\((x_0, y_0, z_0)\) is a point, direction vector \(\mathbf{b} = \begin{pmatrix} l \\ m \\ n \end{pmatrix}\)
Cartesian equations of a line
$$\frac{x-x_0}{l} = \frac{y-y_0}{m} = \frac{z-z_0}{n}$$
Vector product (cross product)
$$\mathbf{v} \times \mathbf{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix} = (v_2w_3 - v_3w_2)\mathbf{i} + (v_3w_1 - v_1w_3)\mathbf{j} + (v_1w_2 - v_2w_1)\mathbf{k}$$ $$|\mathbf{v} \times \mathbf{w}| = |\mathbf{v}||\mathbf{w}|\sin\theta$$
\(\theta\) is the angle between \(\mathbf{v}\) and \(\mathbf{w}\)
Area of a parallelogram
$$A = |\mathbf{v} \times \mathbf{w}|$$
\(\mathbf{v}\) and \(\mathbf{w}\) form two adjacent sides. Area of triangle is \(\frac{1}{2}|\mathbf{v} \times \mathbf{w}|\).
Vector equation of a plane
$$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$$
\(\mathbf{a}\) is position vector of a point, \(\mathbf{b}, \mathbf{c}\) are non-parallel direction vectors in the plane
Equation of a plane (scalar product form)
$$(\mathbf{r} - \mathbf{a}) \cdot \mathbf{n} = 0 \quad \text{or} \quad \mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n} = d$$
\(\mathbf{n}\) is the normal vector to the plane, \(\mathbf{a}\) is position vector of a point on the plane.
Cartesian equation of a plane
$$ax + by + cz = d$$
where normal vector \(\mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\) and \(d = \mathbf{a} \cdot \mathbf{n}\)
Topic 4: Statistics and probability
SL & HL Content
Interquartile range
$$\text{IQR} = Q_3 - Q_1$$
Mean, \(\bar{x}\), of a set of data
$$\bar{x} = \frac{\sum_{i=1}^k f_i x_i}{n}$$
where \(n = \sum_{i=1}^k f_i\) is the total frequency.
Probability of an event A
$$P(A) = \frac{\text{number of outcomes in A}}{\text{total number of outcomes in sample space}} = \frac{n(A)}{n(U)}$$
Complementary events
$$P(A) + P(A') = 1$$
\(A'\) is the complement of A.
Combined events (Addition Rule)
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Probability of A or B or both.
Mutually exclusive events
$$P(A \cup B) = P(A) + P(B)$$
Events A and B cannot occur simultaneously, so \(P(A \cap B) = 0\).
Conditional probability
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
Probability of A given that B has occurred. Assumes \(P(B) \neq 0\).
Independent events
$$P(A \cap B) = P(A)P(B)$$
If A and B are independent, then \(P(A|B) = P(A)\) and \(P(B|A) = P(B)\).
Expected value of a discrete random variable X
$$E(X) = \mu = \sum x P(X=x)$$
Binomial distribution \(X \sim B(n,p)\)
$$P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x=0,1,\dots,n$$ $$E(X) = np$$ $$\text{Var}(X) = np(1-p)$$
n trials, probability of success p.
Standardized normal variable (Z-score)
$$Z = \frac{X-\mu}{\sigma}$$
If \(X \sim N(\mu, \sigma^2)\), then \(Z \sim N(0, 1)\) (standard normal distribution).
HL Only Content
Bayes' theorem
$$P(B|A) = \frac{P(B)P(A|B)}{P(A)}$$ $$P(B_i|A) = \frac{P(B_i)P(A|B_i)}{\sum_j P(B_j)P(A|B_j)}$$
Using Law of Total Probability for \(P(A)\): \(P(A) = P(A|B)P(B) + P(A|B')P(B')\). For partitions \(B_j\), \(P(A) = \sum_j P(A|B_j)P(B_j)\).
Variance \(\sigma^2\) (of a population for grouped data)
$$\sigma^2 = \frac{\sum_{i=1}^k f_i (x_i - \mu)^2}{N} = \frac{\sum_{i=1}^k f_i x_i^2}{N} - \mu^2$$
\(N = \sum f_i\) is total population size. For sample variance \(s^2\), divisor is often \(n-1\).
Standard Deviation \(\sigma\) (of a population for grouped data)
$$\sigma = \sqrt{\frac{\sum_{i=1}^k f_i (x_i - \mu)^2}{N}}$$
Linear transformation of a single random variable X
$$E(aX+b) = aE(X)+b$$ $$\text{Var}(aX+b) = a^2\text{Var}(X)$$
Expected value of a continuous random variable X
$$E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$$
where \(f(x)\) is the probability density function (PDF).
Variance (general definition for a random variable X)
$$\text{Var}(X) = E[(X-\mu)^2] = E(X^2) - [E(X)]^2$$
where \(\mu = E(X)\).
Variance of a discrete random variable X
$$\text{Var}(X) = \sum (x-\mu)^2 P(X=x) = \left(\sum x^2 P(X=x)\right) - \mu^2$$
Variance of a continuous random variable X
$$\text{Var}(X) = \int_{-\infty}^{\infty} (x-\mu)^2 f(x) dx = \left(\int_{-\infty}^{\infty} x^2 f(x) dx\right) - \mu^2$$
Topic 5: Calculus
SL & HL Content
Derivative of \(x^n\)
$$f(x) = x^n \implies f'(x) = nx^{n-1}$$
Integral of \(x^n\)
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$
Area between curve \(y=f(x)\) & x-axis from \(x=a\) to \(x=b\)
$$A = \int_a^b |f(x)| dx$$
If \(f(x) \ge 0\) on \([a,b]\), then \(A = \int_a^b f(x) dx\). If \(f(x) \le 0\), then \(A = -\int_a^b f(x) dx\). Split integral if \(f(x)\) changes sign.
Derivative of \(\sin x\)
$$f(x) = \sin x \implies f'(x) = \cos x$$
Derivative of \(\cos x\)
$$f(x) = \cos x \implies f'(x) = -\sin x$$
Derivative of \(e^x\)
$$f(x) = e^x \implies f'(x) = e^x$$
Derivative of \(\ln x\)
$$f(x) = \ln x \implies f'(x) = \frac{1}{x}, \quad x > 0$$
Chain rule
$$\text{If } y = g(u) \text{ and } u=f(x), \text{ then } \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$
Alternatively, if \(y = (g \circ f)(x) = g(f(x))\), then \(y' = g'(f(x))f'(x)\).
Product rule
$$\text{If } y = uv, \text{ then } \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$
Or \( (uv)' = u'v + uv' \).
Quotient rule
$$\text{If } y = \frac{u}{v}, \text{ then } \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$
Or \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \). Assumes \(v \neq 0\).
Acceleration
$$a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$$
\(s(t)\) is displacement, \(v(t)\) is velocity, \(a(t)\) is acceleration at time \(t\).
Distance & Displacement travelled from \(t_1\) to \(t_2\)
$$\text{Total Distance} = \int_{t_1}^{t_2} |v(t)| dt$$ $$\text{Displacement} = s(t_2) - s(t_1) = \int_{t_1}^{t_2} v(t) dt$$
Standard integrals
$$\int \frac{1}{x} dx = \ln|x| + C$$ $$\int \sin x dx = -\cos x + C$$ $$\int \cos x dx = \sin x + C$$ $$\int e^x dx = e^x + C$$
HL Only Content
Derivative of \(f(x)\) from first principles
$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
Standard derivatives (extended)
$$\frac{d}{dx}(\tan x) = \sec^2 x$$ $$\frac{d}{dx}(\sec x) = \sec x \tan x$$ $$\frac{d}{dx}(\csc x) = -\csc x \cot x$$ $$\frac{d}{dx}(\cot x) = -\csc^2 x$$ $$\frac{d}{dx}(a^x) = a^x \ln a$$ $$\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$$ $$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}, \quad |x|<1$$ $$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}, \quad |x|<1$$ $$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$$
Standard integrals (extended)
$$\int a^x dx = \frac{a^x}{\ln a} + C$$ $$\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$$ $$\int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin\left(\frac{x}{a}\right) + C, \quad |x| < a$$
Integration by parts
$$\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx \quad \text{or} \quad \int u \, dv = uv - \int v \, du$$
Often used for integrals of products, e.g., \(\int x e^x dx\), \(\int \ln x dx\).
Area enclosed by a curve \(x=g(y)\) and y-axis from \(y=c\) to \(y=d\)
$$A = \int_c^d |g(y)| dy$$
Volume of revolution about x-axis or y-axis
$$V_x = \pi \int_a^b y^2 dx \quad (\text{about x-axis})$$ $$V_y = \pi \int_c^d x^2 dy \quad (\text{about y-axis})$$
Solid generated by rotating region bounded by curve, axis, and lines.
Euler's method for approximating \(y(x)\) given \(\frac{dy}{dx} = f(x,y)\)
$$y_{n+1} = y_n + h \cdot f(x_n, y_n)$$ $$x_{n+1} = x_n + h$$
\(y_0 = y(x_0)\) is initial condition, \(h\) is step length.
Integrating factor for linear first-order ODE: \(y' + P(x)y = Q(x)\)
$$\text{Integrating Factor, } I(x) = e^{\int P(x)dx}$$
Multiply ODE by \(I(x)\) to get \((I(x)y)' = I(x)Q(x)\), then integrate.
Maclaurin series expansion of \(f(x)\)
$$f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \frac{x^3}{3!}f'''(0) + \dots + \frac{x^n}{n!}f^{(n)}(0) + \dots$$
Maclaurin series for special functions
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots = \sum_{n=0}^\infty \frac{x^n}{n!}, \quad \forall x$$ $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots = \sum_{n=1}^\infty (-1)^{n-1}\frac{x^n}{n}, \quad -1 < x \le 1$$ $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}, \quad \forall x$$ $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}, \quad \forall x$$ $$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}, \quad -1 \le x \le 1$$
Why Another Formula Guide?
Back when I wrestled with IB Math AA HL, I carried a dog-eared booklet splattered with latte stains and panic sweat. Problem was, memorising those neat little boxes of symbols never guaranteed I could use them under a ticking exam clock. After tutoring IB kids for five years—and collecting their "wish-I'd-known" confessions—I built this hybrid page:
- Official booklet (2025 update) in a single click.
- Bite-size, story-driven explanations so the algebra sticks.
Five Formulas Students Misuse—and How to Nail Them
| Syllabus Topic | The Classic Slip-Up | Fix in One Sentence |
|---|---|---|
| Functions | Mixing domain and range when defining inverse | Always state domain of f = range of f⁻¹—swap them, don't duplicate. |
| Calculus | Forgetting + C after integrating | That lone "+ C" has rescued more HL marks than any mnemonic—write it before you simplify. |
| Statistics | Using population σ for sample data | If the question says "sample," divide by n – 1, not n. |
| Vectors | Dropping the negative sign in angle formulas | Draw the vector tail-to-tail; if the angle exceeds 90°, your cos θ should be negative. |
| Complex Numbers | Converting polar→rectangular with degrees instead of radians | Radians live in math mode; hit the ° key only in geometry questions. |
Real-World Mini-Stories (Because Symbols Need Context)
The Drone Delivery Route (Vectors)
Zara's CAS-driven drone must drop parcels across Dubai Marina. By feeding waypoints into the calculator she spotted a 12% path overlap—saved the club's competition run.
Coffee-Shop Forecast (Poisson)
A barista used λ = 18 customers/hr to predict wait times before the morning rush. The calculator showed a 9% chance of zero orders in any three-minute window—just enough for a bathroom break.
These quick wins turn those sterile formula boxes into aha! moments you actually remember at 2 a.m.
Break-the-Template Study Hacks
✂️
Rip It, Colour It, Tape It. Cut the booklet into strands by topic, colour-code them, and tape the week's target sheet onto your laptop lid.
🗣️
Teach to an Empty Chair. After solving an integration by parts, explain it aloud to, well, nobody. Your brain hates holes in its own lecture—instant gap-filler.
🎲
Dice Drill. Roll a pair of dice: first die chooses topic (1 = Algebra ... 6 = Probability), second picks question number in the textbook. Zero decision fatigue.
FAQs (Gut-Level Answers)
Do I need to memorise the booklet?
No. Know where a formula lives and when to whip it out—speed matters more than rote recall.
HL only: What new formulas show up in 2025?
Look for Maclaurin series tweaks and a fresh Chi-squared contingency layout on page 9.
Final Word from a Former IB Survivor
Your booklet is a parachute, not a cheat sheet. Learn how each equation behaves, practise until your GDC muscle memory sings, and the Paper 3 curve won't scare you. If today's tool shaved even five minutes off tomorrow's study session, I've done my job—drop a comment and tell me which formula still haunts you. I read every note (yes, really).
