Mathematics: Analysis and Approaches Formula Booklet - Comprehensive Notes

Prior Learning – SL and HL
Area of a Parallelogram: $A = bh$ , where $b$ is the base and $h$ is the height.
Area of a Triangle: $A = {1 \over 2}bh$ , where $b$ is the base and $h$ is the height.
Area of a Trapezoid: $A = {1 \over 2}(a + b)h$ , where $a$ and $b$ are the parallel sides and $h$ is the height.
Area of a Circle: $A = \pi r^2$ , where $r$ is the radius.
Circumference of a Circle: $C = 2\pi r$ , where $r$ is the radius.
Volume of a Cuboid: $V = lwh$ , where $l$ is the length, $w$ is the width, and $h$ is the height.
Volume of a Cylinder: $V = \pi r^2h$ , where $r$ is the radius and $h$ is the height.
Volume of a Prism: $V = Ah$ , where $A$ is the area of cross-section and $h$ is the height.
Area of the Curved Surface of a Cylinder: $A = 2\pi rh$ , where $r$ is the radius and $h$ is the height.
Distance Between Two Points: Given points $(x1, y1)$ and $(x2, y2)$ , the distance $d$ between them is $d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$ .
Coordinates of the Midpoint of a Line Segment: Given endpoints $(x1, y1)$ and $(x2, y2)$ , the midpoint is $\left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)$ .

Topic 1: Number and Algebra – SL and HL

SL 1.2 The nth Term of an Arithmetic Sequence: $un = u1 + (n - 1)d$ , where $u_1$ is the first term, $n$ is the term number, and $d$ is the common difference.
The Sum of n Terms of an Arithmetic Sequence: $Sn = \frac{n}{2}(2u1 + (n - 1)d) = \frac{n}{2}(u1 + un)$
SL 1.3 The nth Term of a Geometric Sequence: $un = u1r^{n-1}$ , where $u_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.
The Sum of n Terms of a Finite Geometric Sequence: $Sn = \frac{u1(1 - r^n)}{1 - r} = \frac{u_1(r^n - 1)}{r - 1}$ , $r \neq 1$
SL 1.4 Compound Interest: $FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$ , where:
- $FV$ is the future value.
- $PV$ is the present value.
- $n$ is the number of years.
- $k$ is the number of compounding periods per year.
- $r\%$ is the nominal annual rate of interest.
SL 1.5 Exponents and Logarithms: $\log_a x = b \Leftrightarrow x = a^b$ , where a > 0, b > 0, $a \neq 1$
SL 1.7 Exponents and Logarithms:
- $\loga (xy) = \loga x + \log_a y$
- $\loga \frac{x}{y} = \loga x - \log_a y$
- $\loga x^m = m \loga x$
- $\logb x = \frac{\loga x}{\log_a b}$
SL 1.8 The Sum of an Infinite Geometric Sequence: $S\infty = \frac{u1}{1 - r}$ , where |r| < 1
SL 1.9 Binomial Theorem: $(a + b)^n = a^n + C1 a^{n-1}b + … + Cr a^{n-r}b^r + … + b^n$
- $C_n = \frac{n!}{r!(n - r)!}$

Topic 1: Number and Algebra – HL Only

AHL 1.10 Combinations: $C_r^n = \frac{n!}{(n - r)!r!}$
Permutations: $P_r^n = \frac{n!}{(n - r)!}$
AHL 1.12 Complex Numbers: $z = a + bi$
AHL 1.13 Modulus-Argument (Polar) and Exponential (Euler) Form: $z = r(\cos \theta + i \sin \theta) = re^{i\theta} = r \text{ cis } \theta$
AHL 1.14 De Moivre’s Theorem: $[r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta) = r^n e^{in\theta} = r^n \text{ cis } n\theta$

Topic 2: Functions – SL and HL

SL 2.1 Equations of a Straight Line: $y = mx + c$ ; $ax + by + d = 0$ ; $y - y1 = m(x - x1)$
Gradient Formula: $m = \frac{y2 - y1}{x2 - x1}$
SL 2.6 Axis of Symmetry of the Graph of a Quadratic Function: For $f(x) = ax^2 + bx + c$ , the axis of symmetry is $x = -\frac{b}{2a}$ .
SL 2.7 Solutions of a Quadratic Equation: For $ax^2 + bx + c = 0$ , $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a \neq 0$ .
- Discriminant: $\Delta = b^2 - 4ac$
SL 2.9 Exponential and Logarithmic Functions: $\lne x = x$ ; $a^{\loga x} = x$ ; $\log_a a^x = x$ , where a > 0, a \neq 1

Topic 2: Functions – HL Only

AHL 2.12 Sum and Product of the Roots of Polynomial Equations of the Form $a_r x^r = 0$ :
- Sum is $\frac{-a{n-1}}{an}$ ;
- Product is $(-1)^n \frac{a0}{an}$ .

Topic 3: Geometry and Trigonometry – SL and HL

SL 3.1 Distance Between Two Points: Given points $(x1, y1, z1)$ and $(x2, y2, z2)$ , the distance $d$ between them is $d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}$ .
Coordinates of the Midpoint of a Line Segment: Given endpoints $(x1, y1, z1)$ and $(x2, y2, z2)$ , the midpoint is $\left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}, \frac{z1 + z2}{2}\right)$ .
Volume of a Right-Pyramid: $V = \frac{1}{3}Ah$ , where $A$ is the area of the base and $h$ is the height.
Volume of a Right Cone: $V = \frac{1}{3}\pi r^2 h$ , where $r$ is the radius and $h$ is the height.
Area of the Curved Surface of a Cone: $A = \pi rl$ , where $r$ is the radius and $l$ is the slant height.
Volume of a Sphere: $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius.
Surface Area of a Sphere: $A = 4\pi r^2$ , where $r$ is the radius.
SL 3.2 Sine Rule: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$
Cosine Rule: $c^2 = a^2 + b^2 - 2ab \cos C$ ; $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$
Area of a Triangle: $A = \frac{1}{2}ab \sin C$
SL 3.4 Length of an Arc: $l = r\theta$ , where $r$ is the radius and $\theta$ is the angle measured in radians.
Area of a Sector: $A = \frac{1}{2}r^2 \theta$ , where $r$ is the radius and $\theta$ is the angle measured in radians.
SL 3.5 Identity for tan θ: $\tan \theta = \frac{\sin \theta}{\cos \theta}$
SL 3.6 Pythagorean Identity: $\cos^2 \theta + \sin^2 \theta = 1$
Double Angle Identities:
- $\sin 2\theta = 2\sin \theta \cos \theta$
- $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$

Topic 3: Geometry and Trigonometry – HL Only

AHL 3.9 Reciprocal Trigonometric Identities:
- $\sec \theta = \frac{1}{\cos \theta}$
- $\csc \theta = \frac{1}{\sin \theta}$
Pythagorean Identities:
- $1 + \tan^2 \theta = \sec^2 \theta$
- $1 + \cot^2 \theta = \csc^2 \theta$
AHL 3.10 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: $\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$
AHL 3.12 Magnitude of a Vector: $|v| = \sqrt{v1^2 + v2^2 + v3^2}$ , where $v = \begin{pmatrix} v1 \ v2 \ v3 \end{pmatrix}$ .
AHL 3.13 Scalar Product: $\vec{v} \cdot \vec{w} = v1w1 + v2w2 + v3w3$ , where $\vec{v} = \begin{pmatrix} v1 \ v2 \ v3 \end{pmatrix}$ and $\vec{w} = \begin{pmatrix} w1 \ w2 \ w3 \end{pmatrix}$ .
- $\vec{v} \cdot \vec{w} = |v||w| \cos \theta$ , where $\theta$ is the angle between $\vec{v}$ and $\vec{w}$ .
Angle Between Two Vectors: $\cos \theta = \frac{v1w1 + v2w2 + v3w3}{|v||w|}$
AHL 3.14 Vector Equation of a Line: $\vec{r} = \vec{a} + \lambda \vec{b}$ , where $\vec{a}$ is a point on the line and $\vec{b}$ is the direction vector.
Parametric Form of the Equation of a Line:
- $x = x_0 + l\lambda$
- $y = y_0 + m\lambda$
- $z = z_0 + n\lambda$
Cartesian Equations of a Line: $\frac{x - x0}{l} = \frac{y - y0}{m} = \frac{z - z_0}{n}$
AHL 3.16 Vector Product: $\vec{v} \times \vec{w} = \begin{pmatrix} v2w3 - v3w2 \ v3w1 - v1w3 \ v1w2 - v2w1 \end{pmatrix}$ , where $\vec{v} = \begin{pmatrix} v1 \ v2 \ v3 \end{pmatrix}$ and $\vec{w} = \begin{pmatrix} w1 \ w2 \ w3 \end{pmatrix}$ .
- $|\vec{v} \times \vec{w}| = |v||w| \sin \theta$ , where $\theta$ is the angle between $\vec{v}$ and $\vec{w}$ .
Area of a Parallelogram: $A = |\vec{v} \times \vec{w}|$ , where $\vec{v}$ and $\vec{w}$ form two adjacent sides of a parallelogram.
AHL 3.17 Vector Equation of a Plane: $\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}$
Equation of a Plane (Using the Normal Vector): $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$
Cartesian Equation of a Plane: $ax + by + cz = d$

Topic 4: Statistics and Probability – SL and HL

SL 4.2 Interquartile Range: $IQR = Q3 - Q1$
SL 4.3 Mean: $\bar{x} = \frac{\sum{i=1}^{k} fi xi}{n}$ , where $n = \sum{i=1}^{k} f_i$
SL 4.5 Probability of an Event A: $P(A) = \frac{n(A)}{n(U)}$
Complementary Events: $P(A) + P(A') = 1$
SL 4.6 Combined Events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Mutually Exclusive Events: $P(A \cup B) = P(A) + P(B)$
Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
Independent Events: $P(A \cap B) = P(A)P(B)$
SL 4.7 Expected Value of a Discrete Random Variable X: $E(X) = \sum x P(X = x)$
SL 4.8 Binomial Distribution: $X \sim B(n, p)$
- Mean: $E(X) = np$
- Variance: $Var(X) = np(1 - p)$
SL 4.12 Standardized Normal Variable: $z = \frac{x - \mu}{\sigma}$

Topic 4: Statistics and Probability – HL Only

AHL 4.13 Bayes’ Theorem: $P(Bi|A) = \frac{P(Bi)P(A|Bi)}{\sum{i=1}^{k} P(Bi)P(A|Bi)}$
AHL 4.14 Variance: $\sigma^2 = \frac{\sum{i=1}^{k} fi(xi - \mu)^2}{n} = \frac{\sum{i=1}^{k} fi xi^2}{n} - \mu^2$
Standard Deviation: $\sigma = \sqrt{\frac{\sum{i=1}^{k} fi(x_i - \mu)^2}{n}}$
Linear Transformation of a Single Random Variable:
- $E(aX + b) = aE(X) + b$
- $Var(aX + b) = a^2 Var(X)$
Expected Value of a Continuous Random Variable X: $E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$
Variance: $Var(X) = E(X^2) - [E(X)]^2 = E(X^2) - \mu^2$
Variance of a Discrete Random Variable X: $Var(X) = \sum (x - \mu)^2 P(X = x) = \sum x^2 P(X=x) - \mu^2$
Variance of a Continuous Random Variable X: $Var(X) = \int{-\infty}^{\infty} (x - \mu)^2 f(x) dx = \int{-\infty}^{\infty} x^2 f(x) dx - \mu^2$

Topic 5: Calculus – SL and HL

SL 5.3 Derivative of x^n: If $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ .
SL 5.5 Integral of x^n: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , $n \neq -1$
Area Between a Curve and the x-axis: If $y = f(x)$ and f(x) > 0, then $A = \int_a^b y dx$
SL 5.6 Derivatives:
- If $f(x) = \sin x$ , then $f'(x) = \cos x$
- If $f(x) = \cos x$ , then $f'(x) = -\sin x$
- If $f(x) = e^x$ , then $f'(x) = e^x$
- If $f(x) = \ln x$ , then $f'(x) = \frac{1}{x}$
Chain Rule: If $y = g(u)$ and $u = f(x)$ , then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$
Product Rule: If $y = uv$ , then $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$
Quotient Rule: If $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$
SL 5.9 Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$
Distance Travelled: $distance = \int{t1}^{t_2} |v(t)| dt$
Displacement: $displacement = \int{t1}^{t_2} v(t) dt$
SL 5.10 Standard Integrals:
- $\int \frac{1}{x} dx = \ln |x| + C$
- $\int \sin x dx = -\cos x + C$
- $\int \cos x dx = \sin x + C$
- $\int e^x dx = e^x + C$
SL 5.11 Area of Region Enclosed by a Curve and x-axis: $A = \int_a^b y dx$

Topic 5: Calculus – HL Only

AHL 5.12 Derivative of f(x) from First Principles: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
AHL 5.15 Standard Derivatives:
- If $f(x) = \tan x$ , then $f'(x) = \sec^2 x$
- If $f(x) = \sec x$ , then $f'(x) = \sec x \tan x$
- If $f(x) = \csc x$ , then $f'(x) = -\csc x \cot x$
- If $f(x) = \cot x$ , then $f'(x) = -\csc^2 x$
- If $f(x) = a^x$ , then $f'(x) = a^x (\ln a)$
- If $f(x) = \log_a x$ , then $f'(x) = \frac{1}{x \ln a}$
- If $f(x) = \arcsin x$ , then $f'(x) = \frac{1}{\sqrt{1 - x^2}}$
- If $f(x) = \arccos x$ , then $f'(x) = \frac{-1}{\sqrt{1 - x^2}}$
- If $f(x) = \arctan x$ , then $f'(x) = \frac{1}{1 + x^2}$
AHL 5.15 Standard Integrals:
- $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \arctan \frac{x}{a} + C$
- $\int \frac{1}{x \sqrt{x^2 - a^2}} dx = \frac{1}{a} \ln | \frac{x}{a} | + C$
- \int \frac{1}{\sqrt{a^2 - x^2}} dx = \arcsin \frac{x}{a} + C, |x| < a
AHL 5.16 Integration by Parts: $\int u dv = uv - \int v du$
AHL 5.17 Area of Region Enclosed by a Curve and y-axis: $A = \int_a^b x dy$
Volume of Revolution About the x or y-axes:
- About x-axis: $V = \pi \int_a^b y^2 dx$
- About y-axis: $V = \pi \int_a^b x^2 dy$
AHL 5.18 Euler’s Method: $y{n+1} = yn + h f(xn, yn)$ , where $h$ is a constant (step length) and $x{n+1} = xn + h$
Integrating Factor for y' + P(x)y = Q(x): $e^{\int P(x) dx}$
AHL 5.19 Maclaurin Series: $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + …$
Maclaurin Series for Special Functions:
- $e^x = 1 + x + \frac{x^2}{2!} + …$
- $\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - …$
- $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - …$
- $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - …$
- $\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - …$