Mathematics for TMUA & ESAT

Cheatsheet Content

MM1. Algebra and Functions MM1.1 Laws of Indices For $a > 0$ and any real numbers $m, n$: Definition: $a^m = a \times a \times \dots \times a$ ($m$ times) Rule 1 (Multiplication): $a^m \times a^n = a^{m+n}$ Rule 2 (Roots): $a^{1/n} = \sqrt[n]{a}$ Rule 3 (Negative Exponents): $a^{-m} = \frac{1}{a^m}$ Rule 4 (Zero Exponent): $a^0 = 1$ Rule 5 (Division): $\frac{a^m}{a^n} = a^{m-n}$ Rule 6 (Power of a Power): $(a^m)^n = a^{mn}$ Rule 7 (Fractional Exponents): $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ Important Note: Always clarify if $a$ is positive. The rules generally apply for $a > 0$. If $a$ is negative, fractional exponents can lead to complex numbers. MM1.2 Use and Manipulation of Surds Definition: A surd is an irrational root, usually a square root, that cannot be simplified to a rational number (e.g., $\sqrt{2}$). Convention: $\sqrt{a}$ always denotes the positive square root. For both positive and negative roots, write $\pm \sqrt{a}$. Simplifying Roots: $\sqrt{AB} = \sqrt{A} \times \sqrt{B}$ (e.g., $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$) Multiplying out expressions: Treat surds like variables. Example: $(2 + 3\sqrt{5})^2 = 2^2 + 2(2)(3\sqrt{5}) + (3\sqrt{5})^2 = 4 + 12\sqrt{5} + 45 = 49 + 12\sqrt{5}$. Rationalising the Denominator: Eliminate surds from the denominator. For $\frac{1}{\sqrt{a}}$, multiply by $\frac{\sqrt{a}}{\sqrt{a}}$ to get $\frac{\sqrt{a}}{a}$. For $\frac{1}{x + \sqrt{y}}$, multiply by the conjugate $\frac{x - \sqrt{y}}{x - \sqrt{y}}$. This uses the difference of squares: $(A+B)(A-B) = A^2-B^2$. TMUA/ESAT Strategy: Rationalising denominators is a common requirement. Ensure you are proficient at it. MM1.3 Quadratic Functions and Their Graphs A quadratic function is of the form $ax^2 + bx + c$ where $a \neq 0$. Solving Quadratic Equations: Factorisation: If possible, factorise the quadratic into $(px+q)(rx+s)=0$. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Completing the Square: $ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}$ Discriminant ($\Delta = b^2 - 4ac$): $\Delta > 0$: Two distinct real roots (graph cuts x-axis at two points). $\Delta = 0$: One repeated real root (graph touches x-axis at one point). $\Delta Graph Properties: Parabola shape. Opens upwards if $a > 0$, downwards if $a Vertex (min/max point): Found by completing the square or setting $\frac{dy}{dx}=0$. For $y = a\left(x - h\right)^2 + k$, vertex is $(h, k)$. For $ax^2+bx+c$, x-coordinate of vertex is $-\frac{b}{2a}$. Line of Symmetry: $x = -\frac{b}{2a}$. x-intercepts (roots): Points where $y=0$. y-intercept: Point where $x=0$, which is $(0, c)$. TMUA/ESAT Tip: Understand the interplay between algebraic forms (factorised, completed square) and graphical features (roots, vertex, symmetry). MM1.4 Simultaneous Equations Solving Linear Simultaneous Equations: Substitution: Solve one equation for one variable and substitute into the other. Elimination: Multiply equations to make coefficients of one variable equal, then add or subtract. Graphically: The solution(s) are the intersection points of the graphs. Number of Solutions for Linear Equations (Graphically): One Solution: Lines intersect at a single point (different gradients). No Solutions: Lines are parallel and distinct (same gradient, different y-intercepts). Infinitely Many Solutions: Lines are identical (same gradient, same y-intercept). Solving Linear and Quadratic Simultaneous Equations: Substitute the linear equation into the quadratic equation to get a single quadratic equation in one variable. Solve the resulting quadratic equation (using factorisation or formula). Substitute the found value(s) back into the linear equation to find the corresponding value(s) of the other variable. Number of Solutions for Linear and Quadratic Equations (Graphically): Two distinct solutions: Line crosses the quadratic at two distinct points ($\Delta > 0$ for the resulting quadratic). One repeated solution: Line is tangent to the quadratic ($\Delta = 0$ for the resulting quadratic). No real solutions: Line does not intersect the quadratic ($\Delta TMUA/ESAT Strategy: Always interpret solutions both algebraically and geometrically. For linear-quadratic systems, the discriminant is key to determining the number of real solutions. MM1.5 Solution of Linear and Quadratic Inequalities Rules for Inequalities: Adding/subtracting a number from both sides does not change the inequality sign. Multiplying/dividing by a positive number does not change the inequality sign. Multiplying/dividing by a negative number REVERSES the inequality sign. Squaring both sides can introduce extraneous solutions or change the inequality direction if negative numbers are involved. Be careful! Solving Linear Inequalities: Rearrange like equations, reversing the sign if multiplying/dividing by a negative number. Solving Quadratic Inequalities: Rearrange to have 0 on one side (e.g., $ax^2 + bx + c > 0$). Find the roots of the corresponding quadratic equation ($ax^2 + bx + c = 0$). Sketch the graph of the quadratic. Identify the regions where the inequality holds true based on the graph. Inequalities with Modulus (Absolute Value): Geometric Interpretation: $|x-a|$ represents the distance between $x$ and $a$ on the number line. Algebraic Approach: $|x| $|x| > k \implies x k$ For $|f(x)| Graphical Approach: Sketch $y = |f(x)|$ and $y = |g(x)|$ and find where one graph is above/below the other. TMUA/ESAT Trick: When dealing with modulus inequalities, squaring both sides is often the safest algebraic method. For other inequalities, sketching the graph is a robust approach. MM1.6 Algebraic Manipulation of Polynomials Expanding Brackets and Collecting Like Terms: Standard algebraic expansion and simplification. Factorisation: Common factors. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$. Sum/difference of cubes: $a^3 + b^3 = (a+b)(a^2-ab+b^2)$, $a^3 - b^3 = (a-b)(a^2+ab+b^2)$. Quadratic factorisation: $(px+q)(rx+s)$. Algebraic Division (Long Division): Allows division of polynomials by linear or quadratic expressions. Result: $\frac{f(x)}{g(x)} = q(x) + \frac{r(x)}{g(x)}$, where $f(x) = q(x)g(x) + r(x)$. Factor Theorem: For a polynomial $f(x)$, $(x-a)$ is a factor if and only if $f(a) = 0$. Remainder Theorem: When a polynomial $f(x)$ is divided by $(x-a)$, the remainder is $f(a)$. For division by $(px-q)$, the remainder is $f(q/p)$. In general, if $f(x)$ is divided by $g(x)$, $f(x) = q(x)g(x) + r(x)$, where the degree of $r(x)$ is less than the degree of $g(x)$. TMUA/ESAT Strategy: Factor and Remainder Theorems are crucial for factorising higher-degree polynomials and solving related problems. Practice long division for polynomials. MM1.7 Functions: One-to-one and Many-to-one Definition of a Function: A rule that maps each input value from its domain to exactly one output value. A clear domain must be specified. Each input must have a unique output. Types of Functions: One-to-one function: Each distinct input maps to a distinct output (no two inputs give the same output). Many-to-one function: Multiple distinct inputs map to the same output. Graphical Tests: Vertical Line Test (for a function): If any vertical line intersects the graph more than once, it is NOT a function. Horizontal Line Test (for one-to-one): If any horizontal line intersects the graph more than once, it is NOT a one-to-one function. Common Functions and Properties: $f(x) = \sqrt{x}$: Always means the positive square root. Domain $x \ge 0$. One-to-one. $f(x) = |x|$ (Modulus function): Takes the positive value of $x$. Many-to-one (e.g., $|2|=2, |-2|=2$). TMUA/ESAT Tip: Be precise with function definitions, especially domain restrictions. The graphical tests are quick ways to check function types. MM2. Sequences and Series MM2.1 Sequences and Series (General) Sequence: An ordered list of numbers (e.g., $1, 2, 3, \dots$). Series: The sum of the terms in a sequence (e.g., $1+2+3+\dots$). Progression: A general term for sequences or series. Recurrence Relation: Defines each term in relation to previous terms (e.g., $x_{n+1} = f(x_n)$). TMUA/ESAT Strategy: When dealing with recurrence relations, write out enough terms to spot a pattern. Be careful with the number of terms in sums (e.g., $\sum_{n=0}^{100} x_n$ has 101 terms). MM2.2 Arithmetic Series (Arithmetic Progressions - APs) Definition: A sequence where the difference between consecutive terms is constant (common difference $d$). First term: $a$ Common difference: $d$ $n$-th term: $u_n = a + (n-1)d$ Sum of $n$ terms: $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + u_n)$ Property: $u_{n+1} - u_n = d$ Linear combinations: If $u_n$ and $v_n$ are APs, then $\alpha u_n + \beta v_n$ is also an AP. TMUA/ESAT Tip: Understand how to derive these formulas. Practice questions involving linear combinations of APs. MM2.3 Geometric Series (Geometric Progressions - GPs) Definition: A sequence where the ratio between consecutive terms is constant (common ratio $r$). First term: $a$ Common ratio: $r$ $n$-th term: $u_n = ar^{n-1}$ Sum of $n$ terms: $S_n = \frac{a(1-r^n)}{1-r}$ (for $r \neq 1$) Sum to infinity: $S_{\infty} = \frac{a}{1-r}$ (valid when $|r| Property: $\frac{u_{n+1}}{u_n} = r$ Sigma Notation: $\sum_{k=1}^{n} ar^{k-1} = \sum_{k=0}^{n-1} ar^k$ Derived GPs: If $u_n$ is a GP, then $u_n^k$ is a GP with first term $a^k$ and common ratio $r^k$. Partial sums: $\sum_{k=m}^{n} ar^{k-1} = S_n - S_{m-1}$ TMUA/ESAT Strategy: Be comfortable with sigma notation. Pay close attention to the starting and ending terms of sums. Deriving partial sums (e.g., $ar^m + \dots + ar^n$) can be done by treating it as a new GP or as a difference of two sums. MM2.4 Binomial Expansion Binomial Theorem: For a positive integer $n$, $(a+x)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} x^k$. Combinations: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ (read as "n choose k"), represents the number of ways to choose $k$ items from $n$ distinct items without regard to order. Properties of $\binom{n}{k}$: $\binom{n}{k} = \binom{n}{n-k}$ (symmetry) $\binom{n}{0} = 1$, $\binom{n}{n} = 1$, $\binom{n}{1} = n$ Finding a specific term in $(A+B)^n$: The term containing $B^k$ (or $x^k$) is $\binom{n}{k} A^{n-k} B^k$. Example: Term with $x^7$ in $(3+2x)^8$: $\binom{8}{7} (3)^{8-7} (2x)^7 = \binom{8}{7} (3)^1 (2^7 x^7) = 8 \times 3 \times 128 x^7 = 3072 x^7$. Careful with coefficients and signs: $(2x)^7 = 2^7 x^7$, $(-3x)^5 = (-3)^5 x^5$. TMUA/ESAT Strategy: Understand the meaning of $\binom{n}{k}$ and how it relates to choosing objects. Be meticulous with powers and signs when finding specific terms. MM3. Coordinate Geometry in the (x, y) Plane MM3.1 Equation of a Straight Line Forms of a Straight Line Equation: Gradient-intercept form: $y = mx + c$ $m$: gradient (slope), rate of change of $y$ with respect to $x$. $c$: y-intercept (where the line crosses the y-axis). Point-gradient form: $y - y_1 = m(x - x_1)$ (line passing through $(x_1, y_1)$ with gradient $m$) General form: $ax + by + c = 0$ (note: $c$ here is not the y-intercept unless $b=1$ and $a=0$) Gradient ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1}$ for points $(x_1, y_1)$ and $(x_2, y_2)$. $m = \tan \theta$, where $\theta$ is the angle the line makes with the positive x-axis. Parallel Lines: $m_1 = m_2$. Perpendicular Lines: $m_1 m_2 = -1$ (for non-vertical/horizontal lines). Vertical lines ($x=k$) are perpendicular to horizontal lines ($y=c$). Their gradients are undefined and $0$ respectively. Special Cases: Horizontal line: $y = k$ (gradient $0$). Vertical line: $x = k$ (undefined gradient). x-axis: $y=0$. y-axis: $x=0$. TMUA/ESAT Strategy: Be adept at converting between different forms of line equations. Understand the geometric meaning of gradients and intercepts. Avoid sign errors in coordinate calculations. MM3.2 Coordinate Geometry of the Circle Equation of a Circle: Centre $(0,0)$, radius $r$: $x^2 + y^2 = r^2$. Centre $(a,b)$, radius $r$: $(x-a)^2 + (y-b)^2 = r^2$. General form: $x^2 + y^2 + cx + dy + e = 0$. Can be converted to centre-radius form by completing the square for $x$ and $y$ terms: $x^2 + cx + y^2 + dy + e = 0$ $\left(x + \frac{c}{2}\right)^2 - \frac{c^2}{4} + \left(y + \frac{d}{2}\right)^2 - \frac{d^2}{4} + e = 0$ $\left(x + \frac{c}{2}\right)^2 + \left(y + \frac{d}{2}\right)^2 = \frac{c^2}{4} + \frac{d^2}{4} - e$ Centre is $(-\frac{c}{2}, -\frac{d}{2})$ and radius $r = \sqrt{\frac{c^2}{4} + \frac{d^2}{4} - e}$. Conditions for a Circle: For $x^2 + y^2 + cx + dy + e = 0$ to represent a circle, the radius squared must be positive: $\frac{c^2}{4} + \frac{d^2}{4} - e > 0$. Tangent to a Circle: A line is tangent to a circle if it intersects at exactly one point ($ \Delta = 0$ when solving simultaneously). Distance from a point to a line: Used in problems involving tangents or closest distances. (Often not explicitly given but can be derived or worked out using geometry). TMUA/ESAT Strategy: Be proficient in completing the square to find circle centres and radii. Understand how the discriminant applies to line-circle intersections. Geometric intuition (e.g., shortest distance from point to line is perpendicular) is often helpful. MM3.3 Circle Properties (Circle Theorems) The perpendicular from the centre to a chord bisects the chord. The tangent at any point on a circle is perpendicular to the radius at that point. The angle subtended by an arc at the centre is twice the angle subtended by the arc at any point on the circumference. The angle in a semicircle is a right angle. Angles in the same segment are equal. The opposite angles in a cyclic quadrilateral add to $180^\circ$. The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment. TMUA/ESAT Strategy: Know all circle theorems and their converses. Practice angle chasing and adding auxiliary lines (tangents, diameters, chords) to solve problems. MM4. Trigonometry MM4.1 Sine and Cosine Rules, Area of Triangle Area of a Triangle: Area $= \frac{1}{2}ab \sin C$ (where $a, b$ are side lengths and $C$ is the included angle). Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ (where $R$ is the circumradius). Used when given: Two angles and one side. Two sides and a non-included angle (ambiguous case). Ambiguous Case: When given two sides and a non-included angle, there can be two possible triangles if $a b \sin A$. Cosine Rule: $a^2 = b^2 + c^2 - 2bc \cos A$ (and cyclic permutations for $b^2$ and $c^2$). Can be rearranged to find angles: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$. Used when given: Three sides. Two sides and the included angle. Note: If $A=90^\circ$, $\cos A = 0$, so $a^2 = b^2 + c^2$ (Pythagoras' Theorem). TMUA/ESAT Tip: Understand when to use each rule. Be aware of the ambiguous case for the Sine Rule and how to handle it. Clearly label diagrams. MM4.2 Radian Measure Definition: One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Conversions: $\pi$ radians $= 180^\circ$. $1$ radian $\approx 57.298^\circ$. Degrees to radians: $\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$. Radians to degrees: $\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}$. Standard Conversions: $30^\circ = \pi/6$ rad $45^\circ = \pi/4$ rad $60^\circ = \pi/3$ rad $90^\circ = \pi/2$ rad $180^\circ = \pi$ rad $360^\circ = 2\pi$ rad Arc Length ($L$): $L = r\theta$ (where $\theta$ is in radians). Area of Sector ($A_{sector}$): $A_{sector} = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians). TMUA/ESAT Strategy: Always ensure angles are in radians when using arc length and sector area formulas. Be familiar with standard angle conversions. MM4.3 Values of Sine, Cosine, and Tangent for Standard Angles $0^\circ$ (0 rad): $\sin 0 = 0$, $\cos 0 = 1$, $\tan 0 = 0$. $30^\circ$ ($\pi/6$ rad): $\sin 30^\circ = 1/2$, $\cos 30^\circ = \sqrt{3}/2$, $\tan 30^\circ = 1/\sqrt{3}$. $45^\circ$ ($\pi/4$ rad): $\sin 45^\circ = 1/\sqrt{2}$, $\cos 45^\circ = 1/\sqrt{2}$, $\tan 45^\circ = 1$. $60^\circ$ ($\pi/3$ rad): $\sin 60^\circ = \sqrt{3}/2$, $\cos 60^\circ = 1/2$, $\tan 60^\circ = \sqrt{3}$. $90^\circ$ ($\pi/2$ rad): $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\tan 90^\circ$ is undefined. TMUA/ESAT Trick: Learn these values or be able to quickly derive them using equilateral and isosceles right-angled triangles. MM4.4 Sine, Cosine, and Tangent Functions: Graphs, Symmetries, Periodicity Sine Function ($y = \sin x$): Period $2\pi$ ($360^\circ$). Range $[-1, 1]$. Odd function: $\sin(-x) = -\sin x$. Symmetry about $x = \frac{\pi}{2} + n\pi$. Cosine Function ($y = \cos x$): Period $2\pi$ ($360^\circ$). Range $[-1, 1]$. Even function: $\cos(-x) = \cos x$. Symmetry about $x = n\pi$. Tangent Function ($y = \tan x$): Period $\pi$ ($180^\circ$). Range $(-\infty, \infty)$. Odd function: $\tan(-x) = -\tan x$. Vertical asymptotes at $x = \frac{\pi}{2} + n\pi$. CAST Diagrams / Projections: Visualise the signs of $\sin$, $\cos$, $\tan$ in different quadrants. $\sin \theta$: y-coordinate projection. $\cos \theta$: x-coordinate projection. $\tan \theta$: slope of the radius. TMUA/ESAT Strategy: Use graphing tools (like Desmos) to explore transformations and relationships between trigonometric graphs. Understand how values repeat due to periodicity. MM4.5 Trigonometric Identities Quotient Identity: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ (derived from right-angled triangle definitions or unit circle). Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$ (derived from Pythagoras' theorem on a right-angled triangle or unit circle). TMUA/ESAT Tip: These fundamental identities are used extensively to simplify expressions and solve equations. Understand their origins. MM4.6 Solution of Simple Trigonometric Equations General Approach: Isolate the trigonometric function (e.g., $\sin x = k$). Find the principal value (basic solution) using $\arcsin$, $\arccos$, or $\arctan$. Use knowledge of graph symmetries and periodicity (or CAST diagrams) to find all solutions within the given interval. Remember to find all solutions for $A x + B$ before isolating $x$. Example for $\sin \theta = k$: Principal value $\alpha = \arcsin k$. General solutions: $\theta = n\pi + (-1)^n \alpha$ or $\theta = \alpha + 2n\pi$, $\theta = (\pi - \alpha) + 2n\pi$. Example for $\cos \theta = k$: Principal value $\alpha = \arccos k$. General solutions: $\theta = \pm \alpha + 2n\pi$. Example for $\tan \theta = k$: Principal value $\alpha = \arctan k$. General solutions: $\theta = \alpha + n\pi$. Equations involving $\sin^2 x$, $\cos^2 x$, etc.: Often require using identities (e.g., $\sin^2 x = 1 - \cos^2 x$) to convert to a single trigonometric function, then solve as a quadratic. TMUA/ESAT Strategy: Always find all possible solutions within the required range. Be careful not to lose solutions by rearranging too early. Sketching graphs helps visualise solutions. MM5. Exponentials and Logarithms MM5.1 $y = a^x$ and its Graph Definition: $y = a^x$ is an exponential function. Graph Properties: Passes through $(0,1)$ (since $a^0=1$). Horizontal asymptote at $y=0$. If $a > 1$: strictly increasing function. If $0 If $a = 1$: $y = 1^x = 1$ (horizontal line). Domain: All real numbers ($x \in \mathbb{R}$). Range: All positive real numbers ($y > 0$). TMUA/ESAT Tip: Understand how the base $a$ affects the shape of the graph. Note that $a$ is usually positive; negative bases create complications with non-integer exponents. MM5.2 Laws of Logarithms Definition: $a^b = c \iff b = \log_a c$ (logarithms are the inverse of exponentials). Conditions: Base $a > 0, a \neq 1$. Argument $c > 0$. Laws: $\log_a x + \log_a y = \log_a(xy)$ $\log_a x - \log_a y = \log_a\left(\frac{x}{y}\right)$ $k \log_a x = \log_a(x^k)$ Special Cases: $\log_a a = 1$ $\log_a 1 = 0$ (since $a^0=1$) $\log_a\left(\frac{1}{x}\right) = -\log_a x$ Change of Base Formula (not explicitly tested but useful): $\log_a b = \frac{\log_c b}{\log_c a}$. A useful consequence is $\log_a b = \frac{1}{\log_b a}$. Graph of $y = \log_a x$: Passes through $(1,0)$. Vertical asymptote at $x=0$. If $a > 1$: strictly increasing. If $0 Domain: $x > 0$. Range: All real numbers ($y \in \mathbb{R}$). TMUA/ESAT Strategy: Master the logarithm laws for simplification and solving equations. Remember the domain restrictions for logarithms ($x>0$). MM5.3 Solution of Equations with Exponentials and Logarithms Solving $a^x = b$: Take logarithms of both sides to any convenient base (e.g., base $a$ or base $10$ or $e$). $\log(a^x) = \log b \implies x \log a = \log b \implies x = \frac{\log b}{\log a}$. Choose a base that simplifies calculations (e.g., if $b$ is a power of $a$). Solving Equations Reducible to $a^x = b$ form: Example: $3^{2x} = 4 \implies (3^x)^2 = 4$. Let $y = 3^x$, then $y^2 = 4$. Solve for $y$, then solve for $x$. Example: $25^x - 3 \times 5^x + 2 = 0 \implies (5^x)^2 - 3(5^x) + 2 = 0$. Let $y = 5^x$, then $y^2 - 3y + 2 = 0$. Solving Logarithmic Equations: Use logarithm laws to combine terms (e.g., $\log_a x + \log_a y = \log_a(xy)$). Convert to exponential form: $\log_a x = y \iff x = a^y$. Always check solutions against the domain of the original logarithmic expressions ($x>0$). TMUA/ESAT Strategy: "Solve exactly" means leaving answers in terms of logarithms, not decimal approximations. Look for substitutions to turn exponential equations into quadratics. MM6. Differentiation MM6.1 The Derivative as Gradient and Rate of Change Derivative ($f'(x)$ or $\frac{dy}{dx}$): Represents the gradient of the tangent to the curve $y=f(x)$ at a given point $x$. Rate of Change: The derivative measures how fast one quantity changes with respect to another (e.g., speed is the rate of change of distance with respect to time). Second-order derivative ($f''(x)$ or $\frac{d^2y}{dx^2}$): The derivative of the first derivative. Measures the rate of change of the gradient. Notation: $\frac{dy}{dx}$: First derivative of $y$ with respect to $x$. $\frac{d^2y}{dx^2}$: Second derivative of $y$ with respect to $x$. $f'(x)$: First derivative of $f(x)$. $f''(x)$: Second derivative of $f(x)$. $\dot{s}$: First derivative of $s$ with respect to time $t$. $\ddot{s}$: Second derivative of $s$ with respect to time $t$. TMUA/ESAT Strategy: Focus on interpreting derivatives in context (e.g., velocity from position, acceleration from velocity). Differentiation from first principles is excluded. MM6.2 Differentiation of $x^n$ Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ (for any rational $n$). Constant Multiple Rule: $\frac{d}{dx}(kx^n) = knx^{n-1}$. Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$. Simplification: Often expressions need to be simplified into sums of powers of $x$ before differentiating (e.g., $\frac{(3x+2)^2}{x^2} = \frac{9x^2+12x+4}{x^2} = 9 + 12x^{-1} + 4x^{-2}$). TMUA/ESAT Tip: Be comfortable differentiating terms with negative and fractional exponents. Algebraic simplification is often the first step in differentiation problems. MM6.3 Applications of Differentiation Gradients, Tangents, Normals: Gradient of tangent at $(x_1, y_1)$ is $m_t = f'(x_1)$. Equation of tangent: $y - y_1 = m_t(x - x_1)$. Gradient of normal at $(x_1, y_1)$ is $m_n = -\frac{1}{m_t}$ (if $m_t \neq 0$). Equation of normal: $y - y_1 = m_n(x - x_1)$. Stationary Points (Turning Points): Occur where $f'(x) = 0$. Classifying Stationary Points: Second Derivative Test: If $f''(x) > 0$: Local Minimum. If $f''(x) If $f''(x) = 0$: Test is inconclusive (could be minimum, maximum, or point of inflexion). First Derivative Test (Sign Change): Examine the sign of $f'(x)$ just before and just after the stationary point. $+$ to $0$ to $-$ : Local Maximum. $-$ to $0$ to $+$ : Local Minimum. $+$ to $0$ to $+$ OR $-$ to $0$ to $-$ : Point of Inflexion. Increasing/Decreasing Functions: If $f'(x) > 0$ on an interval, the function is strictly increasing on that interval. If $f'(x) Points of Inflexion: Occur where the concavity changes. A qualitative understanding is expected. Not all points where $f''(x)=0$ are inflexion points. TMUA/ESAT Strategy: Be able to use both first and second derivative tests to classify stationary points. Understand the relationship between $f'(x)$ and the function's behaviour (increasing/decreasing). MM7. Integration MM7.1 Definite Integration and Area Indefinite Integral: The reverse of differentiation. If $\frac{d}{dx}F(x) = f(x)$, then $\int f(x) dx = F(x) + C$. $C$ is the constant of integration. Definite Integral: $\int_a^b f(x) dx = F(b) - F(a)$. Represents the net signed area between the curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$. Areas above the x-axis are positive. Areas below the x-axis are negative. Area between Curve and x-axis: To find the total area (always positive), split the integral at x-intercepts and take the absolute value of each segment: Total Area $= \int_a^b |f(x)| dx$. TMUA/ESAT Tip: Clearly distinguish between a definite integral (net signed area) and the total area. Sketching the graph helps identify regions above/below the x-axis. MM7.2 Integrals of $x^n$ Power Rule for Integration: $\int kx^n dx = \frac{kx^{n+1}}{n+1} + C$ (for $n \neq -1$). Special Case ($n = -1$): $\int \frac{1}{x} dx = \ln|x| + C$ (this is typically outside the TMUA/ESAT scope, so $n \neq -1$ is usually specified). Simplification: As with differentiation, algebraic expressions often need to be expanded or rewritten as sums of powers of $x$ before integration. Linearity: $\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx$. TMUA/ESAT Strategy: Ensure you add the constant of integration $C$ for indefinite integrals. Simplify expressions into power form before integrating. MM7.3 Fundamental Theorem of Calculus Part 1: If $F'(x) = f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$. This links differentiation and integration. Part 2: $\frac{d}{dx} \int_a^x f(t) dt = f(x)$. This shows that differentiation "undoes" integration. Properties of Definite Integrals: $\int_a^b f(x) dx = -\int_b^a f(x) dx$ (swapping limits changes sign). $\int_a^c f(x) dx + \int_c^b f(x) dx = \int_a^b f(x) dx$ (combining contiguous ranges). This holds even if $c$ is not between $a$ and $b$. TMUA/ESAT Tip: Understand the fundamental link between integration and differentiation. Use the properties of definite integrals to simplify calculations. MM7.5 Approximation of Area (Trapezium Rule) Trapezium Rule: Approximates the area under a curve by dividing it into trapezoids. For $n$ strips of equal width $h$, the area is approximately: $A \approx \frac{h}{2} [y_0 + y_n + 2(y_1 + y_2 + \dots + y_{n-1})]$ where $h = \frac{b-a}{n}$ and $y_k = f(x_k)$. Overestimate/Underestimate: If the curve is concave down (curved downwards), the trapezium rule gives an underestimate. If the curve is concave up (curved upwards), the trapezium rule gives an overestimate. If the curve changes concavity, it's not immediately clear (may need to split). TMUA/ESAT Strategy: Memorise the trapezium rule formula. Be able to determine if the approximation is an overestimate or underestimate by looking at the curve's concavity. MM7.6 Solving Differential Equations ($\frac{dy}{dx} = f(x)$) Definition: A differential equation relates a function to its derivatives. Solving $\frac{dy}{dx} = f(x)$ means finding the function $y(x)$. General Solution: Integrate $f(x)$ with respect to $x$ to find $y$. Remember to include the constant of integration $C$. $y = \int f(x) dx$. Particular Solution: If "boundary conditions" or "initial conditions" (a specific point $(x_1, y_1)$ the curve passes through) are given, substitute them into the general solution to find the value of $C$. Using Definite Integrals to find Particular Solution: $\int_{y_1}^{y} dy = \int_{x_1}^{x} f(t) dt \implies y - y_1 = \int_{x_1}^{x} f(t) dt$. This directly gives the particular solution without explicitly finding $C$. TMUA/ESAT Tip: Be clear about whether a general solution (with $C$) or a particular solution (with $C$ determined) is required. Both methods (finding $C$ first or using definite integrals) are valid. MM8. Graphs of Functions MM8.1 Recognise and Sketch Common Function Graphs Lines: $y = mx+c$. Straight line. Quadratics: $y = ax^2+bx+c$. Parabola. Cubics: $y = ax^3+bx^2+cx+d$. S-shape or monotonic. Quartics: $y = ax^4+\dots$. W or M shape. Exponential: $y=a^x$. Asymptote at $y=0$. Logarithmic: $y=\log_a x$. Asymptote at $x=0$. Square Root: $y=\sqrt{x}$. Starts at $(0,0)$, increases. Modulus: $y=|f(x)|$. Reflects parts of $f(x)$ below x-axis to above. Trigonometric: $\sin x, \cos x, \tan x$. Periodic waves. TMUA/ESAT Strategy: Practice sketching these basic shapes. Understand how the leading coefficient ($a$) affects the overall shape (e.g., $ax^n$ for large $x$). MM8.2 Effect of Simple Transformations Let $y=f(x)$ be the original graph. Vertical Stretch/Compression: $y = af(x)$ If $|a| > 1$: stretch vertically by factor $a$. If $0 If $a Vertical Translation: $y = f(x) + a$ Shift up by $a$ units if $a > 0$. Shift down by $a$ units if $a Horizontal Translation: $y = f(x+a)$ Shift left by $a$ units if $a > 0$. Shift right by $a$ units if $a Horizontal Stretch/Compression: $y = f(ax)$ If $|a| > 1$: compress horizontally by factor $a$. If $0 If $a Modulus Transformations: $y = |f(x)|$: Reflect any part of $f(x)$ below the x-axis to above the x-axis. $y = f(|x|)$: Reflect the part of $f(x)$ for $x>0$ into the region $x TMUA/ESAT Trick: Always be careful with horizontal transformations ($f(x+a)$ and $f(ax)$) as they are often counter-intuitive. Use Desmos or similar tools to verify your understanding. MM8.3 Understanding $y=mx+c$ (Revisited) The transformations of $y=x$ to $y=mx+c$ can be seen as: $y=x \to y=mx$ (vertical stretch/compression by $m$, or reflection if $m $y=mx \to y=mx+c$ (vertical translation by $c$). TMUA/ESAT Tip: Visualise these transformations to deepen your understanding of linear equations. MM8.4 Understanding $y=a(x+b)^2+c$ The transformations of $y=x^2$ to $y=a(x+b)^2+c$ can be seen as: $y=x^2 \to y=(x+b)^2$ (horizontal translation by $-b$). $y=(x+b)^2 \to y=a(x+b)^2$ (vertical stretch/compression by $a$, or reflection if $a $y=a(x+b)^2 \to y=a(x+b)^2+c$ (vertical translation by $c$). The vertex of the parabola $y=a(x+b)^2+c$ is at $(-b, c)$. TMUA/ESAT Strategy: Break down complex transformations into simpler steps. This form directly reveals the vertex and overall shape. MM8.5 (See MM6.3) Applications of differentiation to determine graph shape, stationary points, and increasing/decreasing intervals. MM8.6 Intersections with Coordinate Axes and Real Roots x-intercepts (roots): Set $y=0$ and solve for $x$. The number of real roots corresponds to the number of times the graph crosses the x-axis. y-intercept: Set $x=0$ and solve for $y$. Number of Real Roots: For a polynomial of degree $n$, there can be at most $n$ real roots. The discriminant helps determine the number of real roots for quadratics. TMUA/ESAT Tip: Identifying intercepts is a key step in sketching graphs and understanding function behavior. MM8.7 Geometric Interpretation of Simultaneous Equations (Revisited) The solutions to simultaneous equations correspond to the intersection points of their graphs. For two functions $y=f(x)$ and $y=g(x)$, solving $f(x)=g(x)$ gives the x-coordinates of their intersection points. TMUA/ESAT Strategy: Always connect algebraic solutions of equations to the geometric interpretation of intersecting graphs.