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Master the Average Value Formula in Calculus | Function Analysis

A Level Maths: Master Core Concepts & Skills

Sine Rule and Cosine Rule

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Sine rule

Law of sines

Ambiguous case: SSA triangles

Cosine rule

Applications of the sine rule and cosine rule

Applications of the sine law and cosine law

Limits

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Finding limits from graphs

Continuity

Discontinuities of Rational Functions (denominator=0)

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Squeeze theorem

Quadratic Function with Integer Vertex

Differentiation

Slope and equation of tangent line

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Derivative of inverse trigonometric functions

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Implicit differentiation

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Higher order derivatives

Higher Order Derivatives

First and second derivatives

First and Second Derivatives

Second derivatives with implicit differentiation

Implicit Differentiation

Higher Order Derivatives with Repeating Patterns

Integration

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Fundamental Theorem of Calculus

Definite integral

Definite Integral

Integration Applications

Areas between curves

Volumes of solids of revolution - Disc method

Volumes of solids with known cross-sections

Volumes of solid with known cross-sections

Volumes of solids of revolution - Shell method

Volumes of solid of revolution - Shell method

Arc length

Average value of a function

Quadratic Equations and Inequalities

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solving quadratic inequalities

Reciprocal functions

Graphing reciprocals of linear functions

Graphing reciprocals of quadratic functions

Parametric Equations and Polar Coordinates

Defining curves with parametric equations

Polar coordinates

Set Theory

Set notation

Understanding How to Use Set Notation

Set builder notation

Intersection and union of 2 sets

Intersection and Union of 2 Sets

Intersection and union of 3 sets

Intersection and Union of 3 Sets

Bearings

Introduction to bearings

Bearings and direction word problems

Bearings and Direction Word Problems

Angle of elevation and depression

Linear Equations

Introduction to linear equations

Introduction to nonlinear equations

Special case of linear equations: Horizontal lines

Special case of linear equations: Vertical lines

Parallel line equation

Perpendicular line equation

Combination of both parallel and perpendicular line equations

Applications of linear equations

Linear Inequalities

Compound inequalities

Analyze the Alternate Form of Compound Inequalities: AND

Compound linear inequalities

Evaluate Compound Inequalities: And

Express linear inequalities graphically and algebraically

Graph the Inequality

Linear Inequalities – Word Problem 

Linear inequalities

Solving one-step linear inequalities

One-step Linear Inequalities

One-step linear inequalities

One-step Linear Inequalities Decimal

One-step Linear Inequalities Decimal with Fraction

One-step Linear Inequalities Fraction

Solve One Step Linear Inequalities Find Solution– Word Problem

One-step Linear Inequalities — Word Problem

Solving multi-step linear inequalities

Multi-step Linear Inequalities

Two-step Linear Inequalities

Multi-step linear inequalities

Two-step linear inequalities

Solve Multi-step Linear Inequalities – Word Problem 

Solve Multi Step Linear Inequalities Solve Equation – Word Problem

Solve Multi Step Linear Inequalities Equation – Word Problem

Inequalities in Two Variables

What is linear programming?

Finding Optimal Values Given Constraints In Slope-Intercept Form

We&apos;ve got you covered with our complete help for Year 12 maths, whether it&apos;s for Lower Sixth form, <a href="https://www.gov.uk/government/publications/gce-as-and-a-level-mathematics">AS-Level Maths</a>, Core 1 and Core 2 Maths, or <a href="https://www.gov.uk/government/publications/functional-skills-criteria-for-mathematics">Functional skills level 3</a>.

Aligned with your class or textbook, you will get year 12 maths help on topics like Trigonometric identities, Derivatives, Integrals, Solving simultaneous equations, Factorising quadratic equations, Sequences and series, and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest algebra problems. Then, strengthen your understanding with tons of Algebra 2 practice. 

All our lessons are taught by experienced Year 12 Maths teachers. Let&apos;s finish your homework in no time, and ACE that exam.

Year 12 Maths

math

<h2>How Can I Pass A Year 12 Maths Exam?</h2>
To help you obtain a passing grade in your year 12 maths exam, you should be doing maths revision on a regular basis. Try and take the time to get yourself into a weekly study routine that involves reviewing your class notes (which if you&apos;re not taking, you should be!), working on homework assignments, and testing your knowledge via online maths aids like worksheets and past year 12 maths exam papers.
Use these past papers to build your confidence and improve your time management ahead of your actual exams. Assess how long it takes for you to answer questions and see how long you should realistic spend on each section. If a section proves to be too difficult, skip it and come back to it later. It's more important to get through the paper, answering as many questions as possible, so don&apos;t waste time on one problem.
Using these past papers, you can also single out any topics that are giving you trouble. By Highlighting these weaker areas, you will able able to build much more effective revision strategies.
Once you know where you need to improve, use StudyPug for some additional year 12 maths help. You can find the lesson you need help with via the easy to use search function and from there, we&apos;ll help you tackle the problems by introducing step-by-step examples that work to build your understanding.
Our year 12 maths tutors will cover all the complicated A level maths questions and we&apos;ve made sure that our content reflects the current Year 12 maths curriculum and the A level maths syllabus. You&apos;ll also have access to your very own study planner that you can use to track your progress and keep you moving towards your goals.
Once you&apos;re comfortable using our videos to study maths, try sitting additional past papers and attempt more worksheets to chart your progress. You&apos;ll soon see improvements in your performance and you&apos;ll be in a much better position to pass your upcoming exams.
Finally, when you&apos;re actually in your exam, please keep in mind that the external markers will be looking for proof that you understood the problem. They want to see that you knew how to solve the problems beyond memorizing formulae and other tools. Where possible, show your working out! This will show them your thought process and your mathematical reasoning abilities. In doing so, you may earn additional marks and receive a bump in your final grade.

<h2>Should I Try to Take an A-level Exam Early?</h2>
If you feel confident enough to take the test ahead of time, you can, but please be aware that it's not necessarily something that will help you in your university applications. The UCAS points you earn will still count towards your overall UCAS score, but keep in mind that universities are looking for more than just UCAS points.
The Russell Group, an association of United Kingdom-based universities, founded in 1994, had this to say in regards to taking A-levels early and how it may impact your university applications.
<blockquote>
Some universities or their individual subject departments may want to see that you have taken a number of Advanced level qualifications all at the same time; for example, they may want to see three A-levels taken in Year 13.
This can be because they want to know that you can comfortably manage a workload of this size&hellip; Admissions policies may therefore differ in relation to A-levels taken early, and whether these are included in offers made or not.
For example, some courses that typically make a conditional offer of AAB may take account of an A-level A grade achieved at the end of Year 12 and, as a result, make a conditional offer of AB for A-levels taken in Year 13. Others may still make a conditional offer of AAB on subjects taken at the end of Year 13 and will not include the A-level already taken in their conditional offer.	
</blockquote>

<h2>What Units Will I Study in A-Level Maths?</h2>
Choosing to study <a href="[[region_textbook.html|{"textbook_key":"uk-a-level-maths"}]]">A level maths</a> will not only benefit your university applications, but a qualification in A-Level maths, will help you in your future career too. It demonstrates to potential employers that you have the ability to apply analytical thinking, finding logical solutions to solve a variety of problems.
We&apos;re not suggesting that employers are specifically looking for a candidate that can solve maths equations or knows how to calculate angels (unless it&apos;s an architectural role), but they&apos;ll certainly see the value in the various transferable skills that are associated with solving those problems.
Once you&apos;ve decided to study A/AS-level maths, you&apos;ll need to narrow down that actual units you&apos;d like to cover throughout the program. Within A-Level mathematics, there are two different programs of study.
Firstly, there&apos;s AS Level maths, this is a one year course that will cover 2 core maths units plus one additional unit from either mechanics, decisions, or statistics. If you decide to study a full 2 year A-level course, you&apos;ll cover all 4 core maths units plus two additional units.
With this in mind, you should select the optional units that will benefit you best in your future university applications and career. For example, if you&apos;re looking to enter into engineering, you may want to consider studying mechanics.
To give you an idea of what to expect, here is a breakdown of topics you&apos;ll likely cover in the three optional units (not a complete list).

Statistics
<ul type="disc">
<li>Statistical models</li>
<li>Summarising data</li>
<li>Probability</li>
<li>Correlation and Regression</li>
<li>Discrete Random Variables</li>
<li>Continuous Random Variables</li>
<li>Continuous Distributions</li>
<li>Hypothesis tests</li>
<li>Combinations of random variables</li>
</ul>

Mechanics
<ul type="disc">
<li>Vectors</li>
<li>Kinematics</li>
<li>Dynamics</li>
<li>Statics</li>
<li>Moments</li>
<li>Centres of mass</li>
<li>Work and energy</li>
<li>Collisions</li>
<li>Statics of rigid bodies</li>
</ul>

Decision
<ul type="disc">
<li>Algorithms</li>
<li>Algorithms on graphs</li>
<li>Route inspection</li>
<li>Critical path analysis</li>
<li>Linear programming</li>
<li>Matchings</li>
<li>Flows in a network</li>
</ul>

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