Comprehensive University-Level Advanced Mathematics Study Guide

Algebraic Laws and Foundations

Algebraic manipulations are governed by specific index and logarithmic laws that facilitate the simplification of complex expressions. The index laws state that for any base $a$, the following operations apply: $a^n imes a^m = a^{n+m}$, $a^n \div a^m = a^{n-m}$, and $(a^n)^m = a^{nm}$. Furthermore, negative indices represent reciprocals such that $a^{-n} = \frac{1}{a^n}$, and fractional indices denote roots where $a^{\frac{1}{n}} = \sqrt[n]{a}$. By definition, any non-zero base raised to the power of zero is equal to one, represented as $a^0 = 1$.

Logarithmic laws are the inverse of index laws. The relationship between an exponent and a logarithm is defined as $n = a^x \iff \log_a(n) = x$. For natural logarithms, which use the base $e$, several specific rules apply: $\ln(xy) = \ln(x) + \ln(y)$, $\ln(\frac{x}{y}) = \ln(x) - \ln(y)$, and for powers, $\ln(x^k) = k \ln(x)$. Special cases for natural logs include $\ln(\frac{1}{x}) = -\ln(x)$, $\ln(e) = 1$, and $\ln(1) = 0$.

The Factor Theorem is a critical tool for polynomial division and root finding. It states that if a function $f(x)$ is evaluated at a constant $a$ such that $f(a) = 0$, then $(x - a)$ is a factor of the polynomial, and vice versa. Regarding inequalities, a fundamental rule is that multiplying or dividing both sides by a negative number results in the negation and flipping of the inequality sign (e.g., < becomes >).

Partial fractions are used to decompose complex rational expressions into simpler components. For a denominator with distinct linear factors like $\frac{1}{(x + a)(x + b)}$, the form is $\frac{A}{x+a} + \frac{B}{x+b}$. If a linear factor is repeated, such as in $\frac{1}{(x + a)^2(x + b)}$, the decomposition must account for both powers of the repeated factor: $\frac{A}{x+a} + \frac{B}{(x+a)^2} + \frac{C}{x+b}$.

Geometry and Circle Theorems

In trigonometry and geometry, relationships within triangles are explored through the Cosine Rule, $a^2 = b^2 + c^2 - 2bc \cos(A)$, and the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$. The area of any triangle can be calculated using the formula $\text{Area} = \frac{1}{2} ab \sin(C)$.

Circular geometry involves calculations for arcs and sectors. The length of an arc is given by $l = r\theta$, and the area of a sector is $\frac{1}{2} r^2 \theta$. To find the area of a segment (the region between a chord and an arc), the formula used is $\frac{1}{2} r^2 (\theta - \sin(\theta))$. These formulas are generally provided in formula booklets but are recommended to be memorized for efficiency.

Functions and Graphing

Functions are defined as mappings where each input has exactly one output; "1-to-many" mappings are not considered functions. The term "domain" refers to the input values or the $x$-coordinates, while the "range" refers to the output values or the $y$-coordinates. Composite functions, denoted as $fg(x)$, are evaluated by applying the function $g$ first, followed by applying $f$ to the result. Inverse functions ($f^{-1}$) involve swapping the domain and range, reflecting the original function in the line $y = x$, and are found algebraically by setting $x = f(y)$ and rearranging to solve for $y$.

Graphing properties for lines include the midpoint formula $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$ and the gradient formula $m = \frac{\Delta y}{\Delta x}$. If two lines are perpendicular with gradients $m_1$ and $m_2$, then $m_1 = -\frac{1}{m_2}$. The equation of a straight line is expressed as $y - y_1 = m(x - x_1)$. For quadratic equations, the discriminant $b^2 - 4ac$ determines the nature of the roots: if > 0, there are two real roots; if $= 0$, there is one repeated real root; and if < 0, there are no real roots. Reciprocal graphs follow the structures $y = \frac{1}{x}$ or $y = \frac{1}{x^2}$. For circles, the standard equation is $(x - a)^2 + (y - b)^2 = r^2$, where the center is $(a, b)$ and the radius is $r$.

Graph transformations for a function $f(x)$ include shifts and stretches. Horizontal shifts are given by $f(x + a)$, moving the graph $a$ units to the left, while vertical shifts are $f(x) + a$, moving it $a$ units up. Vertical stretches are $a f(x)$ with a scale factor of $a$, and horizontal stretches are $f(ax)$ with a scale factor of $\frac{1}{a}$. Reflections include $-f(x)$ across the $x$-axis and $f(-x)$ across the $y$-axis. Absolute value transformations include $|f(x)|$, which reflects any part below the $x$-axis upward, and $f(|x|)$, which reflects the portion to the right of the $y$-axis onto the left.

Vectors

Vectors represent quantities with both magnitude and direction. A position vector $AB$ can be found by the difference in position vectors of its endpoints: $AB = b - a$. Two vectors $a$ and $b$ are parallel if one is a scalar multiple of the other: $a = \lambda b$, where $\lambda$ is a constant. The magnitude of a 3D vector $a = xi + yj + zk$ is calculated using the Pythagorean extension: $|a| = \sqrt{x^2 + y^2 + z^2}$. A unit vector is any vector with a magnitude of $1$. The angles a vector makes with the coordinate axes are found via direction cosines: $\cos(\theta_x) = \frac{x}{|a|}$, $\cos(\theta_y) = \frac{y}{|a|}$, and $\cos(\theta_z) = \frac{z}{|a|}$.

Binomial Expansion, Series, and Sequences

Binomial expansion uses the binomial coefficient ${}^n C_r = \binom{n}{r} = \frac{n!}{r!(n - r)!}$. The general formula for $(1 + x)^n$ is $1 + nx + \frac{n(n - 1)}{2!} x^2 + \frac{n(n - 1)(n - 2)}{3!} x^3 + \dots$, which is valid for |x| < 1. For expansions of the form $(a + bx)^n$, it is rearranged as $a^n (1 + \frac{bx}{a})^n$, valid for |\frac{bx}{a}| < 1.

Arithmetic series have an $n^{th}$ term $U_n = a + (n - 1)d$ and a sum $S_n = \frac{n}{2}(2a + (n - 1)d)$ or $S_n = \frac{n}{2}(a + L)$, where $L$ is the last term. Geometric series have an $n^{th}$ term $U_n = ar^{n-1}$ and a sum $S_n = \frac{a(1 - r^n)}{1 - r}$. The sum to infinity is $S_{\infty} = \frac{a}{1 - r}$, provided |r| < 1. Sequences are classified as increasing if U_{n+1} > U_n, decreasing if U_{n+1} < U_n, and periodic if $U_{n+k} = U_n$ for a period or order $k$.

Trigonometry: Identities and Equations

Radians are a standard unit of angular measure, where $2\pi = 360^{\circ}$, $\pi = 180^{\circ}$, $\frac{\pi}{2} = 90^{\circ}$, $\frac{\pi}{3} = 60^{\circ}$, $\frac{\pi}{4} = 45^{\circ}$, and $\frac{\pi}{6} = 30^{\circ}$. Small angle approximations for $\theta$ (in radians) include $\sin(\theta) \approx \theta$, $\cos(\theta) \approx 1 - \frac{\theta^2}{2}$, and $\tan(\theta) \approx \theta$. Fundamental identities include $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and the Pythagorean identities: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, and $1 + \cot^2(\theta) = \csc^2(\theta)$.

Reciprocal functions are defined as $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, and $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$. Co-function identities state $\sin(\theta) = \cos(90 - \theta)$ and vice versa. Addition formulae allow for the expansion of sums: $\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)$ and $\cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B)$. Double angle formulae include $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$ and $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2 \cos^2(\theta) - 1 = 1 - 2 \sin^2(\theta)$. In integration, a useful rearrangement is $\cos^2(\theta) = \frac{1}{2} + \frac{1}{2} \cos(2\theta)$.

Harmonic identities take the form $R \cos(\theta \pm \alpha)$ or $R \sin(\theta \pm \alpha)$. If $R \cos(\alpha) = a$ and $R \sin(\alpha) = b$, then $R = \sqrt{a^2 + b^2}$ and $\tan(\alpha) = \frac{b}{a}$. When solving trigonometric equations, the principal solutions are modified by periodic properties: $\sin(\theta)$ alternates via $180 - \theta \pm 360$, $\cos(\theta)$ via $360 - \theta \pm 360$, and $\tan(\theta)$ via $\pm 180$.

Calculus: Differentiation

Differentiation from first principles is calculated as $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$. Derivatives indicate the behavior of a function: if f'(x) < 0 it is decreasing, if $f'(x) = 0$ it is stationary, and if f'(x) > 0 it is increasing. Second derivatives describe curvature: if f''(x) < 0 the function is concave (maximum), if $f''(x) = 0$ it is a point of inflection, and if f''(x) > 0 it is convex (minimum).

Standard derivative forms include:

• $ax^n \rightarrow anx^{n-1}$

• $e^{ax} \rightarrow ae^{ax}$

• $a^x \rightarrow \ln(a) a^x$

• $\ln(x) \rightarrow \frac{1}{x}$

• $\sin(x) \rightarrow \cos(x)$

• $\cos(x) \rightarrow -\sin(x)$

• $\tan(x) \rightarrow \sec^2(x)$

• $\sec(x) \rightarrow \sec(x) \tan(x)$

• $\cot(x) \rightarrow -\csc^2(x)$

• $\csc(x) \rightarrow -\csc(x) \cot(x)$

Advanced calculus techniques include the Chain Rule ($\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$), the Product Rule ($uv' + u'v$), and the Quotient Rule ($\frac{vu' - uv'}{v^2}$). Implicit differentiation is used for equations involving both $x$ and $y$ by applying $\frac{d}{dx} f(y) = f'(y) \frac{dy}{dx}$. Parametric differentiation utilizes $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. Connected rates of change use the chain rule, such as $\frac{dv}{dt} = \frac{dv}{dr} \times \frac{dr}{dt}$, where "rate" implies a derivative with respect to time ($t$).

Calculus: Integration and Numerical Methods

Integration serves as the anti-derivative. Standard integrals (including a constant $c$) are:

• $x^n \rightarrow \frac{x^{n+1}}{n+1}$

• $e^x \rightarrow e^x$

• $\frac{1}{x} \rightarrow \ln|x|$

• $\sin(x) \rightarrow -\cos(x)$

• $\cos(x) \rightarrow \sin(x)$

• $\sec^2(x) \rightarrow \tan(x)$

• $\sec(x) \tan(x) \rightarrow \sec(x)$

• $\csc^2(x) \rightarrow -\cot(x)$

• $\csc(x) \cot(x) \rightarrow -\csc(x)$

Integration techniques include the Reverse Chain Rule (considering then scaling), Substitution (matching a $\sin$ variable to a derivative), and Integration by Parts: $\int u v' \,dx = uv - \int u' v \,dx$ (priority for choosing $u$ is given to $\ln(x)$). Parametric integration is performed using $\int y \frac{dx}{dt} \,dt$ while ensuring limits are adjusted. Integration tips include splitting numerators, algebraic division, and partial fractions. The Trapezium Rule provides an approximation for the area under a curve: $\int y \,dx \approx \frac{1}{2}h(y_{\text{first}} + y_{\text{last}} + 2(\text{middle values}))$.

Numerical methods for solving equations include looking for a change in sign in a continuous function over an interval to identify roots. Iterative methods use staircase or cobweb diagrams; Newton-Raphson finds roots using $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, though this method fails if $f'(x) = 0$.