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Master the Fundamental Theorem of Calculus | Key Concepts

A-Level Maths: Master Core Concepts & Problem Solving

Surds

Evaluating and simplifying radicals

auto-gen

Square and square roots

Square Root of a Perfect Square

Cubic and cube roots

Cube root of a perfect cube

Converting radicals to mixed radicals

Converting radicals to entire radicals

Adding and subtracting radicals

Adding radicals

Subtracting radicals

Multiplying and dividing radicals

Dividing radicals

Multiplying radicals

Rationalize the denominator 

Laws of Indices

Indices: Product rule <i>(a^x)(a^y) = a^(x+y)</i>

Exponents: Product rule (a^x)(a^y) = a^(x+y)

Indices: Division rule (a^x / a^y) = a^(x-y)

Exponents: Division rule: a^x / a^y = a^(x-y)

Indices: Power rule <i>(a^x)^y = a^(x * y)</i>

Exponents: Power rule: (a^x)^y = a^(xy)

Exponents: Power rule

Indices: Negative indices

Exponents: Negative exponents

Zero index: <i>a^0 = 1</i>

Exponents: Zero exponent: a^0 = 1

Rational indices

Exponents: Rational exponents

Combining laws of indices

Combining the exponent rules

Standard form

Scientific notation

Solving for indices

Solving for exponents

Linear Functions

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Gradient equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Gradient intercept form: y = mx + b

Slope intercept form: y = mx + b

General form: Ax + By + C = 0

Gradient-point form: \(y - y_1 = m (x - x_1)\)

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and gradient

Word problems of graphing linear functions

Rate of change

Parallel and perpendicular lines in linear functions

Applications of linear relations

Factorisation

Factorise Difference of Cubes

Factoring difference of cubes

Factorise Sum of Cubes

Factoring sum of cubes

Factorise by taking out the greatest common factor

Factor by taking out the greatest common factor

Factor by taking out GCF

Factorise by grouping

Factor by grouping

Factorise difference of squares:&nbsp;x^2 - y^2

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factorise trinomials

Factoring trinomials

Factoring Trinomials

Quadratic Functions

Introduction to quadratic functions

Transformations of quadratic functions

Quadratic function in general form: <i>y = ax^2 + bx + c</i>

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Quadratic function in vertex form: <i>y = a(x-p)^2 + q</i>

Completing the square

Completing the Square

Quadratic Equations

Solving quadratic equations by factorising

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

The discriminant: Nature of roots of quadratic equations

Applications of quadratic equations

Inequalities

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

Solving quadratic inequalities

Solving absolute value inequalities

Solving Absolute Value Inequalities Involving "less than"

Solving Absolute Value Inequalities Involving "greater than"

Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"

Simultaneous Equations

Simultaneous linear equations

System of linear equations

Simultaneous linear-quadratic equations

System of linear-quadratic equations

Simultaneous quadratic-quadratic equations

Operations with Algebraic Fractions

Partial fraction decomposition

Solving algebraic fraction equations

Solving rational equations

Simplifying algebraic fractions and restrictions

Simplifying rational expressions and restrictions

Adding and subtracting algebraic fractions

Adding and subtracting rational expressions

Adding and Subtracting Rational Expressions

Multiplying algebraic fractions

Multiplying rational expressions

Multiplying Rational Expressions

Multiplying Rational Expressions in Factored Form

Dividing algebraic fractions

Dividing rational expressions

Dividing Rational Expressions

Dividing Rational Expressions in Factored Form

Applications of algebraic fraction equations

Applications of rational equations

Simplifying complex fractions

Absolute Value Functions

Absolute value functions

Expressing an Absolute Value Linear Function as a Piecewise Function

Expressing an Absolute Value Quadratic Function as a Piecewise Function

Solving absolute value equations

Algebraic Division

Polynomial long division

Polynomial synthetic division 

Remainder theorem

Finding the Remainder Using Synthetic Division and the Remainder Theorem

Factor theorem

Using the Factor Theorem to Test For Factors of a Polynomial

Functions

Domain and range of a function

Composite functions

<p>Get an A with our complete A level Maths help. Whether it&apos;s for <a href="https://qualifications.pearson.com/en/qualifications/edexcel-a-levels/mathematics-2008.html">Edexcel</a> (Pearson), <a href="https://www.aqa.org.uk/subjects/mathematics/as-and-a-level">AQA</a>, <a href="https://www.ocr.org.uk/qualifications/by-subject/mathematics/">OCR</a> or <a href="http://www.wjec.co.uk/qualifications/mathematics/mathematics-gce-a-as/">WJEC</a>, our A level maths tutors  got you all covered!</p>

<p>Aligning with all exam boards&apos; A level maths specifications, our comprehensive help includes all materials you need for the exam, such as Derivatives, Integration, Factorisation, Solving simultaneous equations, Trigonometry, Vectors, Circle theorems, and more. Learn the concepts with our online video tutorials that show you step-by-step solutions to even the hardest a level maths questions. Then, reinforce your understanding with tons of a level maths practice.</p> 
<p>All our lessons are taught by experienced A-level Maths tutors. Let&apos;s finish your homework in no time, and ACE that exam.</p>

A-Level Maths

math

<h2>Is A-Level Maths Hard?</h2>
<p>Advanced level maths or "A-Level maths" as its more commonly known, builds upon the topics you would have studied in school, venturing into more complex aspects of mathematics. With that in mind, it&apos;s easy to see why so many students fear that A-Level maths may be a bit too difficult for them.</p>
<p>If you&apos;ve passed GCSE maths, you will be capable of studying A-Level maths. You may find it challenging at times, but with a firm grasp of the basics, you&apos;ll find that the problems faced in A-level maths aren&apos;t as complicated as they initially seemed.</p>
<p>To assist you in your attempts to conquer A-level maths, <a href="[[region_textbook.html|{"textbook_key":"uk-as-level-maths"}]]">StudyPug</a> has a rich collection of video lessons that cover every aspect of the curriculum. Ourvideo tutorials will explore everything you&apos;d expect to find in your exams, and we&apos;ll deliver helpful step-by-step examples in plain english, ensuring you learn and retain the necessary information.</p>
<p>Whether you need help with Cosine and Sine rule, integration by parts, circle theorems,  or any other topic, our collection of useful revision strategies and proven tips will get you learning in no time at all.</p>
<p>Furthermore, our tutors have worked to ensure that we provide simple solutions to even the trickiest of A-level maths problems. Our online content reflects the same material you&apos;d expect to find in modern textbooks, and our tutorials are suitable guides for all maths exams regardless of the examining body (AQA, Edexcel, MEI, WJEC or OCR).</p>
<p>Don&apos;t worry if you&apos;re not entirely confident in your maths ability just yet. Our "A-Level maths for dummies" approach to teaching, assumes no prior knowledge and starts from the basics, taking A-Level maths step-by-step. However, if you are relatively confident in your maths abilities, you can skip past the content or fast-forwarded the information you don&apos;t need.</p>

<h2>What is A-Level Maths Like?</h2>
<p>A-Level maths is split into two different programs of study. There is the AS Level maths course, which is a one year course, and the A-Level Maths course, which is a 2 year course that incorporates the AS program as it&apos;s first year of content.</p>
<p>Students on both courses will be given a choice as to what modules they study. The modules available are listed below:</p>
<ul type="disc">
<li>C1-4 - Core Maths</li>
<li>D1 & D2 - Decisions</li>
<li>M1 & M2 - Mechanics</li>
<li>S1 & S2 - Statistics</li>
</ul>
<p>All students on either the AS or A-level maths program will be required to study the C1-4 modules. Students on the AS program will then pick one additional module to study (statistics, mechanics, or decisions). A-Level students will be required to pick two.</p>
<p>Popular topics covered on the A-level program are as follows:</p>
<ul type="disc">
<li>Binomial expansion</li>
<li>Chain rule</li>
<li>Surds</li>
<li>Geometric series</li>
<li>Laws of indices</li>
<li>Factor theorem</li>
<li>Factorization</li>
<li>Parametric equations</li>
</ul>

<h2>How Can I Get an A in A-Level Maths?</h2>
<p>To give yourself the best possible chance of securing an A grade in your A-level exams, you&apos;ll need to revise on a regular basis, so make time and get yourself into a weekly study routine. Within the routine, you should be reviewing your class notes (which you should be taking!), completing homework, and testing yourself using online worksheets and A-Level maths past papers.</p>
<p>Using past papers will build your confidence and time management ahead of exams. They will also allow you to single out any areas of weakness that needs addressing. Make note of the areas you need to improve upon and visit our site to review our content on the topic. Watch the videos,  take notes, resit past papers, and chart your progress. Keep this routine up and you&apos;ll soon see improvements in your performance as you strengthen your knowledge across all aspects of A-level maths.</p>
<p>If you&apos;re not in a position to subscribe to our services, we do offer a collection of free A-Level Maths lessons across the following subject areas:</p>
<ul type="disc">
<li>Converting Radicals to Entire Radicals</li>
<li>Distance Formula</li>
<li>Graphing Linear Functions Using x- and y-Intercepts</li>
<li>Factor by Taking Out the Greatest Common Factor</li>
<li>Characteristics of Quadratic Functions</li>
<li>And Many More</li>
</ul>
<p>Finally, during the marking process, examiners will look to see if you&apos;ve arrived at the solution through an understanding of the problem or if you&apos;ve just employed simple memorization skills. With this in mind, attempt to show your working out where possible. Taking every step to demonstrate your understanding, could result in additional marks, turning a B grade into an A grade.</p>

<h2>What Calculators are Allowed in A-Level Maths?</h2>
<p>Each examining board follows the <u>JCQ</u> (Joint Council for Qualifications) regulations when it comes to rulings on acceptable calculators in exams.</p>
<p>The regulations below are taken directly from the <a href="https://www.jcq.org.uk/exams-office/ice---instructions-for-conducting-examinations">JCQ website</a>:</p>
<p><u>Calculators must be:</u></p>
<ul type="disc">
<li>Of a size suitable for use on the desk</li>
<li>Either battery or solar powered</li>
<li>Free of lids, cases and covers which have printed instructions or formulas</li>
</ul>
<p><u>Calculators must not:</u></p>
<ul type="disc">
<li>
Be designed or adapted to offer any of these facilities:
<ul type="circle">
<li>translators</li>
<li>Symbolic algebra manipulation</li>
<li>Symbolic differentiation or integration</li>
<li>Communication with other machines or the internet</li>
</ul>
</li>
<li>Be borrowed from another candidate during an examination for any reason</li>
<li>
Have retrievable information stored in them - this includes:
<ul type="circle">
	<li>Databanks</li>
	<li>Dictionaries</li>
	<li>Mathematical formulas</li>
	<li>Text</li>
</ul>
</li>
</ul>

<h2>Is it Worth Getting my A-Level Maths Paper Remarked?</h2>
<p>Understandably, receiving a grade lower than you were expecting can be frustrating. This is made even more frustrating when you need a specific grade for college enrollment or for employment purposes. If this happens to you, you should look into how many marks away from the next grade you were. If its fewer than 5 marks, you may find that it's worth the cost of a remark.</p>
<p>If you want a remark, you&apos;ll need to consult with your teacher and see if they also believe you could get a few extra marks. Depending on your examining body and type of remark required, you&apos;ll have to pay anywhere between &pound;10 and &pound;57.</p>
<figure class="table">
<table>
<thead>
<tr><th></th><th>AQA</th><th>Edexcel</th><th>OCR</th><th>WJEC</th><th>Deadline</th></tr>
</thead>
<tbody>
<tr><th class="text-nowrap text-right">Access to Marked Paper</th><td>&pound;13.95</td><td>Free</td><td>&pound;11.40</td><td>&pound;11.40</td><td>Aug 24</td></tr>
<tr><th class="text-nowrap text-right">Priority Remark</th><td>&pound;50.30</td><td>&pound;49.70</td><td>&pound;56.30</td><td>&pound;46</td><td>Aug 24</td></tr>
<tr><th class="text-nowrap text-right">Full Re-mark</th><td>&pound;42.25</td><td>&pound;41.70</td><td>&pound;45.60</td><td>&pound;36</td><td>Sept 21</td></tr>
<tr><th class="text-nowrap text-right">Clerical Re-check</th><td>&pound;16.10</td><td>&pound;11.20</td><td>&pound;16.40</td><td>&pound;10</td><td>Sept 21</td></tr>
</tbody>
</table>
<figcaption>Pricing as of 2017</figcaption>
</figure>
<p>As you can see, it&apos;s not a cheap process so please keep in mind that remarking should only be done if you&apos;re close to achieving a higher grade. A remark won&apos;t change a D grade to an A, but it may just change that failing grade into a pass.</p>

https://dmn92m25mtw4z.cloudfront.net/img/textbook/uk/textbook-a-level-maths.png

Fundamental Theorem of Calculus: Bridging Differentiation and Integration

What You'll Learn

Recognize that differentiation and integration are inverse operations that cancel each other out

Apply Part 1 to evaluate derivatives of integrals with variable upper limits using the chain rule

Use Part 2 to evaluate definite integrals by finding antiderivatives and computing endpoint differences

Calculate definite integrals of polynomial, trigonometric, logarithmic, and exponential functions

Identify when integrals cannot be evaluated due to discontinuities in the interval

What You'll Practice

Finding derivatives of integrals with variable upper limits like x and x³

Evaluating definite integrals using antiderivatives at upper and lower limits

Working with trigonometric integrals involving sine and cosine functions

Integrating exponential and logarithmic functions with e and ln

Checking for discontinuities before evaluating definite integrals

Why This Matters

This Unit Includes

9 Video lessons

Practice exercises

Learning resources

Skills

Definite Integrals

Antiderivatives

Chain Rule

Integration Techniques

Trigonometric Functions

Exponential Functions

Logarithmic Functions

Calculus