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} - …