Pure Year 1 Overview

Proof

Logical connectives

The three logical connectives are:

• If P, then Q ($P⟹Q$)
• If and only if P, then Q ($P⟺Q$)
• P is implied by Q ($P⟸Q$) (not in spec)

Disproof by counter example

Disproof by counter example involves disproving an “if $P$ then $Q$” statement by finding an example where $P$ is true but $Q$ is not.

Identities

An identity is a relation that is true for all values of the variable. An identity uses the symbol $\equiv$, and the two sides of the identity are congruent expressions.

Sets of numbers

Integers

$\mathbb{Z}$ - whole numbers.

Real numbers

$\mathbb{R}$ - any number on the real number line, including all rational and irrational numbers.

Rational numbers

$\mathbb{Q}$ - any number than can be represented as $\frac{p}{q}$, where p and q are integers, and $q$ is nonzero.

Irrational numbers

Any real number that is not rational, e.g. $\pi$ or $\sqrt{2}$.

Inequality notation

Interval notation

A round bracket $($ or $)$ means the number is not included, e.g. $(3,5)$ is the interval from 3 to 5, non-inclusive. A square bracket $[$ or $]$ means the number is included, e.g. $[3,5]$ means the interval from 3 to 5, inclusive.

Equivalent interval notation

• $x>a$: $x\in (a,\infty )$
• $x<a$: $x\in (-\infty ,a)$
• $x\ge a$: $x\in [a,\infty )$
• $x\le a$: $x\in (-\infty ,a]$
• $a<x<b$: $x\in (a,b)$
• $a<x\le b$: $x\in (a,b]$
• $x<a$ or $x\ge b$: $x\in (-\infty ,a)\cup [b,\infty )$

Indices

Laws of indices

• Multiplication: $x^a\times x^b=x^{a+b}$
• Division: $x^a÷x^b=x^{a-b}$
• Exponentiation: $(x^a{)}^b=x^{ab}$
• Fractional exponents: $x^{1/a}=\text{root}a\text{of}x$
• Negative exponents$x^{-a}=\frac{1}{x^a}$

Rationalising the denominator

To rationalise the denominator of a fraction, multiply by the conjugate radical of the denominator (e.g. $a-b\sqrt{c}$ for $a+b\sqrt{c}$) to create a difference of two squares.

Quadratics

Quadratic formula

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Derivation

The quadratic formula can be derived by completing the square on a general quadratic $a{x}^2+bx+c=0$.

Discriminant

The discriminant of a quadratic equation $a{x}^2+bx+c=0$ is $b^2-4ac$.

Meaning

$\Delta >0$ has two distinct real roots. $\Delta =0$ has one repeated real root. $\Delta <0$ has no real roots.

Line of symmetry

The line of symmetry is at $x=\frac{-b}{2a}=\frac{x_1+x_2}{2}$.

Polynomials

Factor theorem

$(ax-b)$ is a factor of $f(x)$ if and only if $f(\frac{b}{a})=0$.

Graph sketching

Consider:

• The basic shape based on the degree and the lead coefficient.
• The y-intercept.
• The x-intercepts (roots).
• How the curve meets each root (e.g. repeated roots, point of inflection).

Graph transformations

Translation vertically by $\left(\begin{array}{c}0\\ a\end{array}\right)$

$f(x)+a$

Translation horizontally by $\left(\begin{array}{c}a\\ 0\end{array}\right)$

$f(x-a)$

Vertical stretch, scale factor a, relative to the $x$-axis

$af(x)$

Horizontal stretch, scale factor $\frac{1}{a}$, relative to the $y$-axis

$f(ax)$

Reflection in the $x$-axis

$-f(x)$

Reflection in the $y$-axis

$f(-x)$

Coordinate geometry

Straight line equations

$y=mx+c$ $y-y_1=m(x-x_1)$ $ax+by+c=0$

Midpoint of two points

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Distance between two points

$\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}$

Perpendicular lines

Perpendicular lines have gradients that multiply to -1.

Circle equation

$(x-a{)}^2+(y-b{)}^2=r^2$ for a circle centred at $(a,b)$ with radius $r$.

Circle theorems

• The angle in a semicircle is a right angle.
• The perpendicular from the centre of a circle to a chord bisects the chord.
• The radius of a circle is perpendicular to a tangent at the circumference.

Intersections of circles

There are 5 cases (where $r_1>r_2$):

• $d>r_1+r_2$: The two circles do not intersect, with the smaller circle outside the larger circle.
• $d=r_1+r_2$: The two circles intersect in one place, where they are tangent, with the smaller circle outside the larger circle (externally tangent).
• $r_1-r_2<d<r_1+r_2$: The two circles intersect in two places.
• $d=r_1-r_2$: The two circles intersect in one place, where they are tangent, with the smaller circle inside the larger circle (internally tangent).
• $d<r_1-r_2$: The two circles do not intersect, with the smaller circle inside the larger circle.

Logarithms

Rules of logarithms

• Converting: $a^b=c⟺{\log }_a(c)=b$
• Multiplication${\log }_b(xy)={\log }_b(x)+{\log }_b(y)$
• Division: ${\log }_b(\frac{x}{y})={\log }_b(x)-{\log }_b(y)$
• Exponentiation: ${\log }_b(x^y)=y{\log }_b(x)$

Specific bases

Base $e$

Logarithms in base $e$ are written as $\ln x$.

Base 10

Logarithms in base 10 are written as $\log x$.

Graph sketching

A logarithmic function of the form $y={\log }_b(x)$ has:

• An $x$-intercept of $(1,0)$
• The $y$-axis as an asymptote.

Exponentials

Graph sketching

An exponential of the form $y=a^x$ has:

• A $y$-intercept of $(0,1)$
• All $y>0$
• The $x$-axis as an asymptote.

Growth and decay

For an exponential of the form $y=a^x$:

• If $a>1$, then $y$ increases as $x$ increases - exponential growth.
• If $0<a<1$, then $y$ decreases as $x$ increases - exponential decay.

Modelling

Exponential functions are used to model situations where the rate of growth or decay is proportional to the current amount. This is because the derivative of $e^{kx}$ is $k{e}^{kx}$.

Equation

A model with equation $y=A{e}^{kt}$ has:

• Initial value ($t=0$) of $y=A$
• A rate of change of $ky$.

Straightening graphs

Exponential

For a graph $y=A{b}^x$, taking $\ln$ on both sides gives:

$$\ln y=x\ln b+\ln A$$

Thus when plotting $\ln y$ against $x$, the gradient is $\ln b$ and $y$-intercept is $\ln A$.

Polynomial

For a graph $y=A{x}^b$, taking $\ln$ on both sides gives:

$$\ln y=b\ln x+\ln A$$

Thus when plotting $\ln y$ against $\ln x$, the gradient is $b$ and the $y$-intercept is $\ln A$.

Binomial expansion

The expansion of $(a+b{)}^n$ can be found by using the expansion:

$$(a+b{)}^n=\left(\begin{array}{c}n\\ 0\end{array}\right)a^n{b}^0+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}{b}^1+\left(\begin{array}{c}n\\ 2\end{array}\right)a^{n-2}{b}^2+\cdots +\left(\begin{array}{c}n\\ n\end{array}\right)a^0{b}^n$$

Binomial coefficient

The binomial coefficient, written ${}^n{C}_r$ or $\left(\begin{array}{c}n\\ r\end{array}\right)$ is given by: $\frac{n!}{r!(n-r)!}$

Trigonometry

Trig identities

$$\begin{array}{r}{\sin }^2\theta +{\cos }^2\theta =1\\ \\ \tan \theta =\frac{\sin \theta }{\cos \theta }\end{array}$$

Sine rule

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$

When using the sine rule to find angles, sometimes there may be two possible answers, $A$ and $180°-A$.

Cosine rule

$$a^2=b^2+c^2-2bc\cos A$$

Area of a triangle

$$\frac{1}{2}ab\sin C$$

Vectors

A vector has both magnitude and direction.

Magnitude

The magnitude of a vector is denoted $|\mathbf{a}|$ and can be found using the Pythagorean theorem.

Direction

The direction of a vector is measured anticlockwise from the positive $x$ axis and can be found using ${\tan }^{-1}$.

Unit vector

A unit vector has a magnitude of 1.

Operations

Vectors can be, added, subtracted, and multiplied by a scalar.

Parallel vectors

If vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel then for some scalar $k$, $\mathbf{a}=k\mathbf{b}$.

Position vectors

Vectors can represent positions of points or the displacement between two points. The position vector of a point is a vector from the origin to that point.

Displacement vector

For two points $A$ and $B$ with position vectors $\mathbf{a}$ and $\mathbf{b}$, the displacement vector from $A$ to $B$, $\overrightarrow{AB}$, is given by $\mathbf{b}-\mathbf{a}$.

Distance

The distance between two points $A$ and $B$ with position vectors $\mathbf{a}$ and $\mathbf{b}$ is the magnitude of the displacement vector: $AB=|\overrightarrow{AB}|=|\mathbf{b}-\mathbf{a}$.

Geometric properties

Midpoint

The midpoint of the line segment of two points $A$ and $B$ with position vectors $\mathbf{a}$ and $\mathbf{b}$ is $\frac{1}{2}(\mathbf{a}+\mathbf{b})$.

Parallelograms

Parallelograms have vectors for opposite sides with equal magnitude.

Rhombuses

Rhombuses have vectors for all four sides with equal magnitude.

Differentiation

First principles

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

Rules

Derivative of $y=x^n$

$\frac{\text{d}y}{\text{d}x}=n{x}^{n-1}$

Derivative of $y=kf(x)$, where $k$ is a constant:

$\frac{\text{d}y}{\text{d}x}=k{f}^{\prime }(x)$

Derivative of $y=f(x)+g(x)$

$\frac{\text{d}y}{\text{d}x}=f^{\prime }(x)+g^{\prime }(x)$

Increasing and decreasing functions

The sign of the derivative at a point shows whether the function is increasing or decreasing:

• If $\frac{\text{d}y}{\text{d}x}>0$, the function is increasing.
• If $\frac{\text{d}y}{\text{d}x}<0$, the function is decreasing.

Higher derivatives

The second derivative is denoted $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}$ or $f^{\prime \prime }(x)$. It measures the gradient of the gradient, or the rate of change of the gradient with respect to $x$.

Tangents and normals

For a point on $y=f(x)$:

• The gradient of the tangent is $f^{\prime }(x)$,
• The gradient of the normal is $-\frac{1}{f^{\prime }(x)}$

Stationary points

At the local maxima and minima, $\frac{dy}{dx}=0$. This can be used for optimisation questions.

Nature

Given a stationary point:

• If $\frac{d^{2}x}{d{y}^2}<0$ at that point, then it is a local maximum.
• If $\frac{d^{2}x}{d{y}^2}>0$ at that point, then it is a local minimum.
• If $\frac{d^{2}x}{d{y}^2}=0$ at that point, then no conclusion can be reached.

Integration

Fundamental theorem of calculus

$$\int f(x)dx=F(x)+c⟺f(x)=\frac{d}{dx}F(x)$$

Integration is the opposite of differentiation.

Integral of polynomials

For all rational $n\ne 1$:

$$\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C$$

Definite integrals

The definite integral ${\int }_a^{b}f(x)dx$ is found by evaluating the integrated expression at the upper limit $b$ and subtracting the integrated expression evaluated at the lower limit $a$. This gives the signed area.

If the integrand changes sign between the two limits, then the interval needs to be split to find the total area.