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AS Further Maths - January Exam Revision Resources

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Investigating the vector equation of a line

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Looking for maths A level teaching resources and notes? Colmanweb features loads of tried and tested classroom resources.

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One of the solutions to the equation is , correct to three significant figures.

Write down the upper and lower bound of 2.55.

Hence, use the sign change rule to confirm that this is a solution
(to three significant figures) to the equation .

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Numerical Solutions of Equations | CIE A Level Maths: Pure 3 Questions & Answers 2020 (Medium)

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CIE A Level Maths: Pure 3 exam revision with questions, model answers & video solutions for Numerical Solutions of Equations. Made by expert teachers.

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Cambridge International A Level Questions By Category (formerly CIE) | A Level Maths Revision

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The following questions have been chosen to aid with past paper revision. The questions are carefully chosen from a variety of exam boards to give a breadth

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Play the animation to see a demonstration of the conics, hyperbola, parabola, ellipse and circle.

Stop the animation and explore additional special cases by changing the angle of the plane (drag the orange point) and the location of the plane (drag the brown point).

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Conic section as the intersection of a plane and a cone.

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GeoGebra Apps for A-Level Further Pure Mathematics

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GeoGebra Apps for A-Level Further Pure Mathematics

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The worksheets have been grouped by chapter order from the Pearson Edexcel Textbook
Mark schemes at the end of each file. ​

We all make mistakes!
​If you find one, please let me know at devinaj90@gmail.com or hit the contact button on the header bar

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A-level practice papers, questions, starters and more

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for use with complex numbers FMSP activity always, sometimes, never

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for use with complex numbers FMSP activity always, sometimes, never

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Use this content finder to find SolutionBank, GeoGebra interactives and Casio calculator support for Pure Mathematics Year 1/AS. The chapter links below take you the interactives and calculator support for each chapter and to full worked solutions for each exercise.
Pure 1 combined SolutionBank
1 Algebraic expressions

* Page 31: Explore how the value of the discriminant changes with k using GeoGebra.

3 Equations and inequalities

4 Graphs and transformations

Review exercise 1

* Page 90: Explore the gradient formula using GeoGebra.

* Page 174: Explore the cosine rule using GeoGebra.

* Page 231: Explore vector addition using GeoGebra.

* Page 257: Explore the gradient of the chord using GeoGebra.

Review exercise 3

Practice exam paper

Technical support

If you are having issues accessing the links in this content folder, check this list of common problems.

Alternatively you can get in touch by phone or email. Our digital support team are available from 8 am to 5 pm Monday to Friday (excluding bank holidays).

Email digital.support@pearson.com, tel. 0845 313 8888*

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Where we do need to call you, if possible, please be at the computer or workstation that is experiencing technical difficulty.

* Calls to 0845 numbers cost 3p per minute, plus your phone company's access charge. International call charges vary.

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Pure Mathematics Year 1 AS Extra online content

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Use this content finder to find SolutionBank, GeoGebra interactives and Casio calculator support for Pure Mathematics Year 1/AS. The chapter links below take you the interactives and calculator support for each chapter and to full worked solutions for each exercise. Pure 1 combined SolutionBank * Page 31: Explore how the value of the discriminant changes with k using GeoGebra.

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Core Pure Mathematics Book 1 AS Extra online content

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Unlock more content

This is only a selection of our papers. Registered Cambridge International Schools can access the full catalogue of teaching and learning materials including papers from 2018 through our School Support Hub.

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Cambridge International AS & A Level Mathematics (9709)

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Cambridge International AS & A Level Mathematics (9709) syllabus information and updates. Find out more on our website.

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CAIE October/November 2023 Session Starts

If any paper is still missing, please report using the Contact Us! tab.
Common Search Terms:
Papers A Levels Mathematics (9709), Papers A Levels Mathematics (9709) Past Papers, Papers A Levels Mathematics (9709) Question Papers, Papers A Levels Mathematics (9709) Marking Schemes, Papers A Levels Mathematics (9709) Grade Thresholds

Resource Guide for File Naming System.

Click the image to view.

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Past Papers | Papers | A Levels | Mathematics (9709) | GCE Guide

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CAIE Past Papers for Cambridge O Level, Cambridge Int'l AS and A Level and Cambridge IGCSE subjects

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Group theory, abstraction, and the 196,883-dimensional monster

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application of eigenvectors - velocity of particles

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application of eigenvectors - velocity of particles

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TFM2 Eigenvector velocity of particle demo

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TFM2 Eigenvector velocity of particle demo

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This file is reproduced with the kind permission of Steven Sutcliffe, who created it following an AMSP Teaching Further Mathematics 2 study day. This is from Steven's notes. This is a system of two identical masses of mass m and 3 identical springs of stiffness k. The two masses are given an initial displacement from their equilibrium position, x0 for mass 1 and y0 for mass 2. The red and blue curves show the displacement of the two masses after release. The slider for t is just keeping track of time. Use the play button to start the animation. The slider for H simply changes the length of the middle spring (i.e. how far apart the masses are). This doesn't change any of the analysis and it's not really important, it's just for a visual aid more than anything else. Keep it at around 12.5 if you want to "see" the system working. Put it at zero if you want the displacement curves on top of each other to analyse. If you want to analyse the horizontal system rather than the vertical one, just set the g slider to zero. Check the box if you want to see the relative displacement between the two masses i.e. the blue curve minus the red curve. You'll notice that this brown curvealways looks nice and simple - just a single sinusoidal wave! The frequency of this wave is known as a beat frequency - if you want to know why, build one of these systems, stick a speaker (playing a single note) on each mass and listen to what happens!

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Complex Numbers - Calculations and Equations - Casio Calculators

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Choose Complex settings, calculate and solve equations.

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Polar curves with dotted lines for negative r

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Three files containing teaching ideas for A level mechanics. Friction experiments using a simple bench pulley, friction plates and blocks. Moments contains simple experiments using a ruler, bulldog clips and some small weights. What can you do with some curtain rail contains a variety expriments us…

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Mechanics in Action is an introduction to using practical approaches to teaching mechanics. It includes a background to modeling and 53 practical investigations with photocopyable worksheets and teachers' notes.

A knowledge of mechanics is necessary in the study of mathematics, science and engineering. This book aims to help teachers make the subject interesting and accessible.

* It is for teachers who wish to explore the use of practical investigations in mechanics. It provides exciting ideas for experienced teachers and plenty of support for people who lack confidence in the subject.

* There are photocopiable worksheet masters with ideas for investigations.

* Each investigation has teachers' notes. Some of these are written up in detail with lesson plans.

* There is a section giving essential mathematical background.

* A section about common misconceptions includes a questionnaire to help identify problem areas.

* Includes practical investigations on forces, pulleys, moments, friction, kinematics, energy, circular motion, springs, oscillations, simple harmonic motion and impact.

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Mechanics in Action is an introduction to using practical approaches to teaching mechanics. It includes a background to modeling and 53 practical investigations with photocopyable worksheets and teachers’ notes. A knowledge of mechanics is necessary in the study of mathematics, science and engineeri…

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Advance Information from the Exam Boards

Formula Sheets - all examination boards, OCR formula tests

Corbett Maths - all examination boards - such good resources

Dr Frost Maths, Edexcel, AQA and OCR skills aligned to advance information

Sparx Maths - all examination boards, Edexcel IGCSE included also

Diagnsotic Questions (Edexcel and AQA)

Mathsbox - Skills Checks Edexcel, OCR

1st Class Maths - Foundation and Higher papers based on AQA advance information

AHS Maths - Edexel - Hegarty, Corbett Maths and Just Maths Revision Guide links

Just Maths - Take 5 (Edexcel)

Maths White Board (Edexcel)

Richard Morely - videos, each with a notes jotter

Ben Gordon - Mixed Six (Edexcel)

Maths Teacher Hub question papers

GCSE Maths Tutor - Edexcel practice booklets, videos and revision check-lists AQA, Edexcel, and OCR

Potential crossover topics for GCSE Edexcel Maths 2022 - Bharti Karadia

Brooke Hunter - student-friendly AQA level 2 Further Maths - Dr Frost Skills

Custom Maths Revision Starters

A-Level Resources

From Pearson/Edexcel, comes a very valuable resource, under Summer 2022, Support, A Level Mathematics Topic Tests, is a new set of topic tests (no login required) for Pure Mathematics (10 tests), Mechanics (5 tests) and Statistics (5 tests) are provided. These are very comprehensive, not only do we have the questions and mark schemes but further notes. Looking at an example, the first test of the Pure Mathematics tests is on Proof, 5 varied questions cover the knowledge, skills and understanding relevant to Pearson's qualification. The tests are designed to be used with the advance information for the subject as well as general marking guidance for the qualification.

Whilst not written for the advance information, the following resources may be useful if you are searching by topic for A level Maths and Further Maths:

Advance Information for summer 2022 exams is now available.
Formula Sheets
From OCR, higher and foundation formula sheet tests.

The Exams Office publishes regular updates, available to all. This is a really useful resource and includes all their updates including from JCQ, DfE and all the examination boards. As well as AQA, OCR and Pearson you will find for example City and Guilds and International Examinations.

Whilst we have this advance information, it should perhaps be regarded with some caution; we cannot assume too much about what will not be on the examination. It is still important to teach the complete specification, many students will be going on to study A level, but the many revision resources appearing based on the advance information for Mathematics will provide a useful focus for revision. I will keep this post updated as new resources are created.

To start, from TeachitMaths an AQA Maths partnership resource: GCSE mathematics-small things make a big difference; many really useful and important reminders for students.

On Dr Frost Maths, we have the skills involved in Edexcel's Foundation GCSE papers 2022, as per the official guidance, thank you to of New College Durham. You can find Edexcel papers including the higher papers and IGCSE on the Edexcel courses page on Dr Frost Maths; scroll to the last courses on the page.

OCR courses including 2022 GCSE Papers 1-6, which students can work through to practise by topic from @rachaelhandley2 of @ThomasEstley school can be found .

From Sparx Maths, 2022 GCSE and IGCSE Advance Information - Sparx Topics and Key Questions. This is a brilliant resource, a whole collection of carefully selected questions. Great to see questions for IGCSE.

Craig Barton has provided links for quizzes on Diagnostics Questions mapped to Edexcel's and AQA's Higher and Foundation advance information.

On Mathsbox, we have some really useful skills checks - one or two-page documents with answer sheets aligned to the Edexcel GCSE advance information. OCR Paper 1 is available, AQA to follow.

From 1st Class Maths, papers using the AQA advance information AQA Paper 1F, , , , QR codes link to video solutions on the 1st Class Maths YouTube channel.

Returning to Corbett Maths, has (and for those who have them - Hegarty Maths and the revision guide from Just Maths) to the Edexcel advanced information for GCSE Foundation and Higher.

Look for the Corbett video number here. Remember that with Corbett Maths, not only do we have the videos but worked examples by topic too.

On Maths White Board, Matt Woodfine has created worksheets based on the advanced information from AQA, Edexcel and OCR; each time you create a worksheet, you get new questions, answers are included.

See also on Maths White Board:

From MathsTeacherHub questions based on the Edexcel advanced information for Higher paper 1 and Foundation Paper 1.

🚨 Paper 3 practice booklets are live and ready to download NOW 🚨

Exclusively for the 2022 Advanced Information topics!

🔺 Higher and Foundation
🔺 Full Video Tutorial for Each Booklet
🔺 Completely FREE!
🔺 Available Now 🙌🏼

Please share 🙏🏼

Download: https://t.co/xGaUvnOSQt pic.twitter.com/wxyNEIDPOh
- The GCSE Maths Tutor (@gcsemathstutor) February 24, 2022

From Bharti Karadia, this clear document (note the tabs at the bottom) helps to identify potential crossover topics for Edexcel GCSE 2022.

From Brooke Hunter, for AQA, Level 2 Further Maths, a student-friendly version of the advanced information linked to DrFrostMaths key skill numbers throughout.

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Advance Information from the Exam Boards Formula Sheets - all examination boards, OCR formula tests Corbett Maths - all examination boards - such good resources Dr Frost Maths, Edexcel, AQA and OCR skills aligned to advance information Sparx Maths - all examination boards, Edexcel IGCSE included also Diagnsotic Questions (Edexcel

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STEM SMART (Subject Mastery and Attainment Raising Tuition) is a widening participation initiative from the University of Cambridge in association with Isaac Physics, to provide free, complementary teaching and support to UK (non-fee paying) students who: have either experienced educational disadvantage or belong to a group that is statistically less likely to progress to

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Dropbox – University Applications – Simplify your life

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Posts about A-level Diagnostic questions written by berwickmaths

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Vicky Neale on the fundamental theorem of arithmetic

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Article by Vicky Neale

In this article we shall look at some elementary results in Number Theory, partly because they are interesting in themselves, partly because they are useful in other contexts (for example in olympiad problems), and partly because they will give you a flavour of what Number Theory is about.

You will require a little knowledge in advance, but not much: you need to be familiar with congruence notation, and you need to know that if $a$ and $n$ are coprime (have highest common factor $1$) then $a$ has a multiplicative inverse mod $n$ (that is, there is an integer $b$ such that $a\times b\equiv 1 \text{ mod } n$). You can find an explanation of all of this in the article called Modular Arithmetic . You might also find it useful to know that an integer is a whole number ($\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ are all integers), and by natural number , I mean a positive integer ($1, 2, 3, \ldots$). Sometimes people include $0$ as a natural number, but I'm not going to in this article.

We say that $x$ divides $y$, written $x|y$, if there is an integer $c$ such that $y=c x$. So for example, $3|6$ (as $6=2\times 3$), but $3\nmid7$. We need a little property of primes to help us later. The property is that if $p$ is a prime and $p|a b$, then $p|a$ or $p|b$. Is this obvious? Well, no, not really, because it's not true unless $p$ is a prime. For example, $4$ divides $6\times 10=60$, but $4$ does not divide $6$ and $4$ does not divide $10$. Let's prove it (for primes, of course!). We know that $p|a b$. If $p|a$ or $p|b$, then obviously the result is true, so let's suppose that it doesn't. Now we're going to use Bezout's Theorem, which says that $m$ and $n$ are coprime if and only if there exist integers $h$ and $k$ such that $h m+k n=1$. Since $p\nmid a$, $a$ and $p$ are coprime (remember, $p$ is a prime). So there exist integers $h$ and $k$ such that $a h+p k=1$. Similarly, there are integers $H$ and $K$ such that $b H+p K=1$. Let's multiply these equations together. Then $1=(a h+p k)(b H+p K)=a b h H+p k b H+p K a h+p^2 k K= (a b)(h H)+p(k b H+K a h+p k K)$. But $p|a b$ and certainly $p|p$, so $p$ divides the right-hand side, so $p|1$. But this is absurd. So $p|a$ or $p|b$.

We can use this result and induction to prove the following very important theorem:

The Fundamental Theorem of Arithmetic:

Every natural number $n> 1$ can be expressed in an essentially unique way as the product of prime numbers.

By "essentially unique", I mean "counting different orderings of the primes as the same": $12=2^2\times3=3\times2^2$, but I'm counting these products as essentially the same. You should also note the very important fact that $1$ is not a prime number - otherwise this theorem would clearly be false! I'm not going to prove this result here, but you might like to have a go yourself, or you can look it up in any introductory book on number theory.

The first theorem we're going to prove is called Fermat's Little Theorem , sometimes, confusingly, known as FLT (confusing because FLT is also used to refer to Fermat's Last Theorem, which is something quite different!). Here's what the theorem says:

Theorem: Let $p$ be a prime and $a$ a natural number not divisible by $p$. Then \begin{equation} a^{p-1}\equiv 1 \text{ mod } p. \end{equation} This is in essence the same as the following statement:

Let $p$ be a prime and $a$ a natural number. Then \begin{equation} a^p\equiv a \text{ mod } p. \end{equation} Why is this really the same? If $a\equiv 0 \text{ mod } p$, then it's pretty obvious that $a^p\equiv a \text{ mod } p$. If $a\not\equiv 0 \text{ mod } p$, then it has an inverse, so we can multiply both sides of (2) by the inverse and get back (1). Also, I hope it's clear that we can multiply both sides of (1) by $a$ to get (2).

Let's prove the theorem (we'll prove (1)). Consider the set $\{1,2,3,\ldots,p-1\}$. This contains all of the distinct non-zero elements mod $p$ (that is, they are all different, and none of them is zero mod $p$). Since they are all non-zero, if we multiply them all together, we'll get something non-zero mod $p$ (repeatedly using the result from earlier that if $p|a b$ then $p|a$ or $p|b$). Now consider the set $\{a,2a,3a,\ldots,(p-1)a\}$, where we reduce each element mod $p$ (so it lies between 0 and $p-1$). Now none of these numbers is 0 mod $p$, and they are all distinct (check this, if you don't believe me!). So it's the set of all of the non-zero elements mod $p$. So certainly multiplying all the numbers in the second set gives the same answer as multiplying all the numbers in the first set (mod $p$). In symbols, $1\times 2\times \ldots\times (p-1)\equiv a\times(2a)\times\ldots\times(p-1)a\equiv a^{p-1}\times 1\times 2\ldots\times (p-1) \text{ mod } p$. Since $1\times 2\times \ldots\times (p-1)\not\equiv 0 \text{ mod } p$, we can divide by it, leaving $a^{p-1}\equiv 1 \text{ mod } p$.

The next theorem is a generalisation of Fermat's Little Theorem, but first we need to define a new function.

The Euler phi function , also known as the Euler totient function , is defined as the function $\phi:\mathbf{N}\rightarrow\mathbf{N}$ (that is, taking values in the natural numbers and giving values in the natural numbers) where $\phi(n)$ is the number of natural numbers less than or equal to $n$ that are coprime to $n$. So $\phi(p)=p-1$ for all primes $p$ (because everything less than $p$ is coprime to $p$), for example. Another example: $\phi(15)=8$, since $1, 2, 4, 7, 8, 11, 13$ and $14$ are exactly the natural numbers less than and coprime to $15$.

This function has a special property. If $m$ and $n$ are coprime , then $\phi(m n)=\phi(m)\phi(n)$. I'm going to leave the proof of this to you as an exercise, because with a couple of hints it's not too hard. We've already worked out what $\phi(p)$ is if $p$ is prime. Now work out $\phi(p^2)$ and then work out $\phi(p^k)$ where $k$ is a natural number. Now work out $\phi(p q)$, where $p$ and $q$ are different primes, and work up to $\phi(p^m q^n)$, again where $p$ and $q$ are distinct primes. Spotted the pattern? Now use the Fundamental Theorem of Arithmetic (see above) to prove that $\phi(m n)=\phi(m)\phi(n)$ for coprime $m$, $n$.

Now let's look at a generalisation of Fermat's Little Theorem, sometimes called the Fermat-Euler Theorem .

Theorem: Let $n> 1$ be a natural number and $a$ an integer coprime to $n$. Then $a^{\phi(n)}\equiv 1 \text{ mod } n$.

Why is this a generalisation of Fermat's Little Theorem? Well, consider when $n$ is prime. Then $\phi(n)=n-1$, and we get back exactly the statement of Fermat's Little Theorem.

I'm going to leave the proof as an exercise, as it's very similar to the proof of Fermat's Little Theorem.

We now come to the last theorem in this article, called Wilson's Theorem .

Theorem: Let $p$ be a prime number. Then $(p-1)!\equiv -1 \text{ mod } p$ (where ! denotes factorial, and $5! = 1\times 2 \times 3 \times 4 \times 5 $).

At first sight, it might seem totally unclear how one could go about proving this, but there is a beautiful and simple proof that I shall outline now. Remember that each of the numbers $1, 2, 3, \ldots, p-1$ has an inverse $\text{ mod } p$? So we can pair off numbers, so that a number in the pair is the inverse of the other number in the pair. But a number could be the inverse of itself. When does this happen? If you think about it, you'll see that $x$ is its own inverse only if $x^2\equiv 1 \text{ mod } p$. That is, $x$ is its own inverse only if $(x-1)(x+1)\equiv 0 \text{ mod } p$, and since $p$ is prime, we see that we must have $x-1\equiv 0 \text{ mod } p$ or $x+1\equiv 0 \text{ mod } p$, so the only numbers between $1$ and $p-1$ that are their own inverses are $1$ and $p-1$. Now let's think back to $(p-1)!$. Each pair $(a,a^{-1})$ contributes $1$ to this product, so the only things we have left to worry about are $1$ and $p-1$. So $(p-1)!\equiv 1\times(p-1)\equiv -1 \text{ mod } p$, as we wanted.

I hope this has given you a flavour of what Number Theory is about; there are numerous books available that continue to develop the theory, and large numbers of Olympiad problems you might like to tackle with your new knowledge!

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Article by Vicky Neale In this article we shall look at some elementary results in Number Theory, partly because they are interesting in themselves, partly because they are useful in other contexts (for example in olympiad problems), and partly because they will give you a flavour of what Number Theory is about.

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A friend of Plus, Vicky Neale, has written a book about why people study maths at university, and to give a taste of what it's like to study a maths degree. We asked her to tell us about her book and why she wrote it...

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Search Results for "a level maths" - MathematicsAndCoding

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My class are doing the Mechanics and Statistics applied options for the AQA Further Maths A-Level. because of this my warmup sheet is focussed on these two areas - sorry I didn't have time to do one for each applied grouping. Next Year I shall make sure I have a sheet for each of the [...]

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Return to A Level Further Mathematics Menu

AMSP Videos for Further Maths

From amsp this brilliant collection of short videos produced by the legacy Further Mathematics Support Program supports the Further Maths Specification. I have used many of these successfully in class and recommended them to students to support their studies. Look at any of the examination boards to see the coverage for the course.
Exemplar Materials

Pearson Sample Assessment Materials - Exemplar Answers with examiner comments (freely available, no login required, scroll down the page).

Pearson - Teacher Support - Further Mathematics AQA Teaching Guidance - Further Maths Topic Tests

A set of really useful Topic Tests are available for both Maths and Further Maths A Levels from AQA. I like the index provided by the mapping documents, one for Maths and one for Further Maths. For each test there is a clear statement of what is assessed in the test which comprises two sections. The questions in section A test basics of the topic and those in section B require a bit more thinking. Mark schemes are provided for all tests.
(How to get access to AQA's free portal - All About Maths).

Dr Frost - Further Mathematics resources for Edexcel.

Mr Southern Maths:

MathedUp Further Maths Takeaway

On Mohammed Ladak's 'MathedUp' see his A Level Further Maths Takeaway, a wonderful source of exam questions by topic with mark schemes. AQA questions have been used here.
CrashMATHS

crashMATHS practice papers

From crashMATHS, note the A Level Practice Papers and mark schemes.
For Further Mathematics, AS and A Level Practice papers are available for Edexcel Further Mathematics (see Bronze Sets A and B).

On Exam Solutions you will find tutorials and solutions to past papers for AQA, Edexcel and OCR. If we look at the Edexcel page, we can see worked solutions to the June 18 A Level papers. Further Mathematics worked solutions will be made available when the papers are published by the examination board.

Owen (Owen134866 on TES) has a library of Further Mathematics teaching resources, these are really clearly structured with step by step examples.

A Level Core Pure Mathematics Year 1
A-level Core Pure Mathematics Year 2/A2
Further Mechanics 1

Joe Berwick has written a complete set of notes for Further Statistics 1, these are for the Edexcel and AQA specifications and include many very clear worked examples.

Sebastian Bicen (@BicenMaths) has created this really useful hyperlinked PowerPoint and pdf which provides Further Mathematics exam questions by topic. The exam papers included are as follows:
AS and A2 SAMs
AS and A2 Specimen
AS 2018
A2 Mock Paper

Tayyub Majeed has a very useful exam paper pack with questions on Pure Maths, Statistics and Mechanics, his exam pack is available on Google Drive, here.

For Further Maths, note the pack and Further Mechanics. Questions are taken from old and new specifications, mostly Edexcel.

On Dr Tom Bennison's blog, you will find many excellent teaching resources for Further Mathematics (and also Mathematics).

AMSP Resources
You can register your school free with AMSP for many benefits to support the teaching and learning of AS/A level Further Mathematics. Already mentioned above. from amsp is this brilliant collection of short videos produced by the legacy Further Mathematics Support Program

Use of Technology
MEI GeoGebra Tasks for AS/A Level Further Maths
From The GeoGebra Institute of MEI, Use of GeoGebra in Further Maths: examples. See also Using GeoGebra for A Level Further Mathematics, a GeoGebra Book providings ideas and suggestions for using GeoGebra to enhance teaching and learning A level Further Mathematics.

Jack Brown has created thousands of videos covering the complete A Level specification; the easiest way to navigate the videos is through his website, TLMaths.com.

Jack also has Further Maths teaching videos and exam paper walkthroughs.
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TL Maths - Further Maths

From the legacy specification but still some really useful and clear PowerPoints for Further Mathematics from Dan Walker.

Jonny Griffiths investigative activities for the pure A Level Mathematics classroom are well known (details are included in the A Level resources pages, see RISPS. He has now published Further Risps, forty rich tasks for the pure Further Mathematics classroom.

Two editions are available - a or pdf version. The pdf version is now available free from www.further-risps.co.uk. On you can see two trial activities, Further Risps 1 and 2.

The pdf not only provides the forty problems but also full teachers notes for each. The notes for each task begin with the topic or topics covered, identify the type of task, for example, introductory and state any preliminary knowledge required. This is a valuable resource for teaching Further Mathematics.

Investigating the first Risp on Matrices at the session this certainly would help students with fluency in finding the determinant of a matrix. Rich tasks like this can provide students with a greater understanding than just a traditional exercise and will hopefully stick for longer!

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Return to A Level Further Mathematics Menu AMSP Videos for Further Maths From amsp this brilliant collection of short videos produced by the legacy Further Mathematics Support Program supports the Further Maths Specification. I have used many of these successfully in class and recommended them to students to support their studies.

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Show Mobile Notice Show All Notes Hide All Notes

Here are the notes for my Calculus I course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Calculus I or needing a refresher in some of the early topics in calculus.

I've tried to make these notes as self-contained as possible and so all the information needed to read through them is either from an Algebra or Trig class or contained in other sections of the notes.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasn't covered in class.

2. Because I want these notes to provide some more examples for you to read through, I don't always work the same problems in class as those given in the notes. Likewise, even if I do work some of the problems in here I may work fewer problems in class than are presented here.

3. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible when writing these up, but the reality is that I can't anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I've not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.

4. This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.

Here is a listing (and brief description) of the material that is in this set of notes.

Review - In this chapter we give a brief review of selected topics from Algebra and Trig that are vital to surviving a Calculus course. Included are Functions, Trig Functions, Solving Trig Equations and Equations, Exponential/Logarithm Functions and Solving Exponential/Logarithm Equations.

Functions - In this section we will cover function notation/evaluation, determining the domain and range of a function and function composition.
Inverse Functions - In this section we will define an inverse function and the notation used for inverse functions. We will also discuss the process for finding an inverse function.
Trig Functions - In this section we will give a quick review of trig functions. We will cover the basic notation, relationship between the trig functions, the right triangle definition of the trig functions. We will also cover evaluation of trig functions as well as the unit circle (one of the most important ideas from a trig class!) and how it can be used to evaluate trig functions.
Solving Trig Equations - In this section we will discuss how to solve trig equations. The answers to the equations in this section will all be one of the "standard" angles that most students have memorized after a trig class. However, the process used here can be used for any answer regardless of it being one of the standard angles or not.
Solving Trig Equations with Calculators, Part I - In this section we will discuss solving trig equations when the answer will (generally) require the use of a calculator ( i.e. they aren't one of the standard angles). Note however, the process used here is identical to that for when the answer is one of the standard angles. The only difference is that the answers in here can be a little messy due to the need of a calculator. Included is a brief discussion of inverse trig functions.
Solving Trig Equations with Calculators, Part II - In this section we will continue our discussion of solving trig equations when a calculator is needed to get the answer. The equations in this section tend to be a little trickier than the "normal" trig equation and are not always covered in a trig class.
Exponential Functions - In this section we will discuss exponential functions. We will cover the basic definition of an exponential function, the natural exponential function, i.e. \({\bf e}^{x}\), as well as the properties and graphs of exponential functions.
Logarithm Functions - In this section we will discuss logarithm functions, evaluation of logarithms and their properties. We will discuss many of the basic manipulations of logarithms that commonly occur in Calculus (and higher) classes. Included is a discussion of the natural (\(\ln(x)\)) and common logarithm (\(\log(x)\)) as well as the change of base formula.
Exponential and Logarithm Equations - In this section we will discuss various methods for solving equations that involve exponential functions or logarithm functions.
Common Graphs - In this section we will do a very quick review of many of the most common functions and their graphs that typically show up in a Calculus class.

Limits - In this chapter we introduce the concept of limits. We will discuss the interpretation/meaning of a limit, how to evaluate limits, the definition and evaluation of one-sided limits, evaluation of infinite limits, evaluation of limits at infinity, continuity and the Intermediate Value Theorem. We will also give a brief introduction to a precise definition of the limit and how to use it to evaluate limits.

Tangent Lines and Rates of Change - In this section we will introduce two problems that we will see time and again in this course : Rate of Change of a function and Tangent Lines to functions. Both of these problems will be used to introduce the concept of limits, although we won't formally give the definition or notation until the next section.
The Limit - In this section we will introduce the notation of the limit. We will also take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us. We will be estimating the value of limits in this section to help us understand what they tell us. We will actually start computing limits in a couple of sections.
One-Sided Limits - In this section we will introduce the concept of one-sided limits. We will discuss the differences between one-sided limits and limits as well as how they are related to each other.
Limit Properties - In this section we will discuss the properties of limits that we'll need to use in computing limits (as opposed to estimating them as we've done to this point). We will also compute a couple of basic limits in this section.
Computing Limits - In this section we will looks at several types of limits that require some work before we can use the limit properties to compute them. We will also look at computing limits of piecewise functions and use of the Squeeze Theorem to compute some limits.
Infinite Limits - In this section we will look at limits that have a value of infinity or negative infinity. We'll also take a brief look at vertical asymptotes.
Limits At Infinity, Part I - In this section we will start looking at limits at infinity, i.e. limits in which the variable gets very large in either the positive or negative sense. We will concentrate on polynomials and rational expressions in this section. We'll also take a brief look at horizontal asymptotes.
Limits At Infinity, Part II - In this section we will continue covering limits at infinity. We'll be looking at exponentials, logarithms and inverse tangents in this section.
Continuity - In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval.
The Definition of the Limit - In this section we will give a precise definition of several of the limits covered in this section. We will work several basic examples illustrating how to use this precise definition to compute a limit. We'll also give a precise definition of continuity.

Derivatives - In this chapter we introduce Derivatives. We cover the standard derivatives formulas including the product rule, quotient rule and chain rule as well as derivatives of polynomials, roots, trig functions, inverse trig functions, hyperbolic functions, exponential functions and logarithm functions. We also cover implicit differentiation, related rates, higher order derivatives and logarithmic differentiation.

The Definition of the Derivative - In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function.
Interpretation of the Derivative - In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function.
Differentiation Formulas - In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers.
Product and Quotient Rule - In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate.
Derivatives of Trig Functions - In this section we will discuss differentiating trig functions. Derivatives of all six trig functions are given and we show the derivation of the derivative of \(\sin(x)\) and \(\tan(x)\).
Derivatives of Exponential and Logarithm Functions - In this section we derive the formulas for the derivatives of the exponential and logarithm functions.
Derivatives of Inverse Trig Functions - In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent.
Derivatives of Hyperbolic Functions - In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine.
Chain Rule - In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule!
Implicit Differentiation - In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e.g. y = f(x) and yet we will still need to know what f'(x) is. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates (the next section).
Related Rates - In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one (or more) quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work.
Higher Order Derivatives - In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives.
Logarithmic Differentiation - In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients (sometimes easier than using product and quotient rule). More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i.e. there are variables in both the base and exponent of the function.

Applications of Derivatives - In this chapter we will cover many of the major applications of derivatives. Applications included are determining absolute and relative minimum and maximum function values (both with and without constraints), sketching the graph of a function without using a computational aid, determining the Linear Approximation of a function, L'Hospital's Rule (allowing us to compute some limits we could not prior to this), Newton's Method (allowing us to approximate solutions to equations) as well as a few basic Business applications.

Rates of Change - In this section we review the main application/interpretation of derivatives from the previous chapter (i.e. rates of change) that we will be using in many of the applications in this chapter.
Critical Points - In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions.
Minimum and Maximum Values - In this section we define absolute (or global) minimum and maximum values of a function and relative (or local) minimum and maximum values of a function. It is important to understand the difference between the two types of minimum/maximum (collectively called extrema) values for many of the applications in this chapter and so we use a variety of examples to help with this. We also give the Extreme Value Theorem and Fermat's Theorem, both of which are very important in the many of the applications we'll see in this chapter.
Finding Absolute Extrema - In this section we discuss how to find the absolute (or global) minimum and maximum values of a function. In other words, we will be finding the largest and smallest values that a function will have.
The Shape of a Graph, Part I - In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative (or local) minimum and maximum values of a function and where a function will be increasing and decreasing. We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum.
The Shape of a Graph, Part II - In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points (i.e. where concavity changes) that a function may have. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points (but not all) as relative minimums or relative maximums.
The Mean Value Theorem - In this section we will give Rolle's Theorem and the Mean Value Theorem. With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter.
Optimization Problems - In this section we will be determining the absolute minimum and/or maximum of a function that depends on two variables given some constraint, or relationship, that the two variables must always satisfy. We will discuss several methods for determining the absolute minimum or maximum of the function. Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc.
More Optimization Problems - In this section we will continue working optimization problems. The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the 'simple' geometric objects we looked at in the previous section.
L'Hospital's Rule and Indeterminate Forms - In this section we will revisit indeterminate forms and limits and take a look at L'Hospital's Rule. L'Hospital's Rule will allow us to evaluate some limits we were not able to previously.
Linear Approximations - In this section we discuss using the derivative to compute a linear approximation to a function. We can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this. We give two ways this can be useful in the examples.
Differentials - In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not be able to discuss it until then.
Newton's Method - In this section we will discuss Newton's Method. Newton's Method is an application of derivatives will allow us to approximate solutions to an equation. There are many equations that cannot be solved directly and with this method we can get approximations to the solutions to many of those equations.
Business Applications - In this section we will give a cursory discussion of some basic applications of derivatives to the business field. We will revisit finding the maximum and/or minimum function value and we will define the marginal cost function, the average cost, the revenue function, the marginal revenue function and the marginal profit function. Note that this section is only intended to introduce these concepts and not teach you everything about them.

Integrals - In this chapter we will give an introduction to definite and indefinite integrals. We will discuss the definition and properties of each type of integral as well as how to compute them including the Substitution Rule. We will give the Fundamental Theorem of Calculus showing the relationship between derivatives and integrals. We will also discuss the Area Problem, an important interpretation of the definite integral.

Indefinite Integrals - In this section we will start off the chapter with the definition and properties of indefinite integrals. We will not be computing many indefinite integrals in this section. This section is devoted to simply defining what an indefinite integral is and to give many of the properties of the indefinite integral. Actually computing indefinite integrals will start in the next section.
Computing Indefinite Integrals - In this section we will compute some indefinite integrals. The integrals in this section will tend to be those that do not require a lot of manipulation of the function we are integrating in order to actually compute the integral. As we will see starting in the next section many integrals do require some manipulation of the function before we can actually do the integral. We will also take a quick look at an application of indefinite integrals.
Substitution Rule for Indefinite Integrals - In this section we will start using one of the more common and useful integration techniques - The Substitution Rule. With the substitution rule we will be able integrate a wider variety of functions. The integrals in this section will all require some manipulation of the function prior to integrating unlike most of the integrals from the previous section where all we really needed were the basic integration formulas.
More Substitution Rule - In this section we will continue to look at the substitution rule. The problems in this section will tend to be a little more involved than those in the previous section.
Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals. We will be approximating the amount of area that lies between a function and the \(x\)-axis. As we will see in the next section this problem will lead us to the definition of the definite integral and will be one of the main interpretations of the definite integral that we'll be looking at in this material.
Definition of the Definite Integral - In this section we will formally define the definite integral, give many of its properties and discuss a couple of interpretations of the definite integral. We will also look at the first part of the Fundamental Theorem of Calculus which shows the very close relationship between derivatives and integrals.
Computing Definite Integrals - In this section we will take a look at the second part of the Fundamental Theorem of Calculus. This will show us how we compute definite integrals without using (the often very unpleasant) definition. The examples in this section can all be done with a basic knowledge of indefinite integrals and will not require the use of the substitution rule. Included in the examples in this section are computing definite integrals of piecewise and absolute value functions.
Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals. The only real requirements to being able to do the examples in this section are being able to do the substitution rule for indefinite integrals and understanding how to compute definite integrals in general.

Applications of Integrals - In this chapter we will take a look at some applications of integrals. We will look at Average Function Value, Area Between Curves, Volume (both solids of revolution and other solids) and Work.

Average Function Value - In this section we will look at using definite integrals to determine the average value of a function on an interval. We will also give the Mean Value Theorem for Integrals.
Area Between Curves - In this section we'll take a look at one of the main applications of definite integrals in this chapter. We will determine the area of the region bounded by two curves.
Volumes of Solids of Revolution / Method of Rings - In this section, the first of two sections devoted to finding the volume of a solid of revolution, we will look at the method of rings/disks to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation.
Volumes of Solids of Revolution / Method of Cylinders - In this section, the second of two sections devoted to finding the volume of a solid of revolution, we will look at the method of cylinders/shells to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation.
More Volume Problems - In the previous two sections we looked at solids that could be found by treating them as a solid of revolution. Not all solids can be thought of as solids of revolution and, in fact, not all solids of revolution can be easily dealt with using the methods from the previous two sections. So, in this section we'll take a look at finding the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.
Work - In this section we will look at is determining the amount of work required to move an object subject to a force over a given distance.

Extras - In this chapter proofs of many of the facts/properties/theorems given through out the material are given. Also included are a brief review of summation notation, a discussion on the different 'types' of infinity and a discussion about a subtlety involved with the constant of integration from indefinite integrals.

Proof of Various Limit Properties - In this section we prove several of the limit properties and facts that were given in various sections of the Limits chapter.
Proof of Various Derivative Facts/Formulas/Properties - In this section we prove several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.

Proof of Trig Limits - In this section we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions using the definition of the derivative.
Proofs of Derivative Applications Facts/Formulas - In this section we prove many of the facts that we saw in the Applications of Derivatives chapter.
Proof of Various Integral Facts/Formulas/Properties - In this section we prove some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter.
Area and Volume Formulas - In this section we derive the formulas for finding area between two curves and finding the volume of a solid of revolution.
Types of Infinity - In this section we have a discussion on the types of infinity and how these affect certain limits. Note that there is a lot of theory going on 'behind the scenes' so to speak that we are not going to cover in this section. This section is intended only to give you a feel for what is going on here. To get a fuller understanding of some of the ideas in this section you will need to take some upper level mathematics courses.
Summation Notation - In this section we give a quick review of summation notation. Summation notation is heavily used when defining the definite integral and when we first talk about determining the area between a curve and the \(x\)-axis.
Constant of Integration - In this section we have a discussion on a couple of subtleties involving constants of integration that many students don't think about when doing indefinite integrals. Not understanding these subtleties can lead to confusion on occasion when students get different answers to the same integral. We include two examples of this kind of situation.

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Here is a set of notes used by Paul Dawkins to teach his Calculus I course at Lamar University. Included are detailed discussions of Limits (Properties, Computing, One-sided, Limits at Infinity, Continuity), Derivatives (Basic Formulas, Product/Quotient/Chain Rules L'Hospitals Rule, Increasing/Decreasing/Concave Up/Concave Down, Related Rates, Optimization) and basic Integrals (Basic Formulas, Indefinite/Definite integrals, Substitutions, Area Under Curve, Area Between Curves, Volumes of Revolution, Work).

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What one fool can do, another can.

(Ancient Simian Proverb.)

Table of Contents

About this book

Calculus Made Easy is a book on calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject.

I read "Calculus Made Easy" by Silvanus P. Thompson and it's still to this day my inspiration for explaining complex technical topics to lay people. It's a fantastic book, and even if you know math you must read it if you want to understand how to teach complexity to others. (source)

Thompson creates a warm, inviting environment where students will learn and grasp the true essence of calculus without any added fluff or overt technicality. (source)

Most college calculus texts weigh a ton; this one does not - it just gets to the point. This is how I learned calculus: my uncle gave me a copy. (source)

Buy paper version

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Visual Complex Analysis* is legendary. Just as Calculus Made Easy, it uses a geometrical approach instead of the usual symbolic one.

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About this edition & thanks
This text is based on the PDF version from Project Gutenberg converted to html by hand.
Thanks to Paula Appling, Don Bindner, Chris Curnow, Andrew D. Hwang and Project Gutenberg Online Distributed Proofreading Team for preparing the original PDF.

The theme is borrowed from Dive Into HTML5 by Mark Pilgrim released under the CC-BY-3.0 license.

Full legal notices. Github page.

Please send corrections, suggestions, and comments to nadvornik.jiri@gmail.com

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What one fool can do, another can. (Ancient Simian Proverb.) About this book Calculus Made Easy is a book on calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject. I read "Calculus Made Easy" by Silvanus P.

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In Discussing distance we mentioned speed and velocity. Average speed is the rate your distance changes in a set period of time, for example \(\quantity{50}{km}\) in \(1\) hour. If you walk to school you will speed up, slow down, perhaps even stop. Your speed will not stay the same, but we can talk about your average speed for the trip.

Velocity is similar but, like displacement, also depends on direction. If you average \(\quantity{3}{km/h}\) on your walk to school, and walk home in the same amount of time, your velocity for the return part of your journey would be \(\quantity{-3}{km/h}\).

Both displacement and velocity involve two pieces of information. How far (or fast) you travel, and the direction. Quantities like this are known as vectors.

Try to sketch the speed-time graph and the velocity-time graph from these distance and displacement graphs of the same journey. Think about how you will work out the speeds and velocities. Will you use a formula? Or can you use features from the graphs?

Given this velocity-time graph below, can you sketch the displacement-time graph?

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Speed vs velocity | Introducing Calculus | Underground Mathematics

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In Discussing distance we mentioned speed and velocity. Average speed is the rate your distance changes in a set period of time, for example \(\quantity{50}{km}\) in \(1\) hour. If you walk to school you will speed up, slow down, perhaps even stop.

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New graph to see the point P at the bottom of the page move, while the displacement-time graph for its motion is shown at the top of the page, along with a description of the movement. You can pause the motion using the button in the bottom left corner of the graph. What does the gradient of the graph represent? How can you tell which direction P is moving in by looking at the graph?

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New graph to see the point P at the bottom of the page move, while the displacement-time graph for its motion is shown at the top of the page, along with a description of the movement. You can pause the motion using the button in the bottom left corner of the graph.

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Average speed | Introducing Calculus | Underground Mathematics

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This resource has two interesting situations that require students to think carefully about how average speed is calculated, and in turn, think abo...

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Discussing distance | Introducing Calculus | Underground Mathematics

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This resource asks students to decide what two unlabelled graphs represent. It goes on to ask students to think carefully about the difference betw...

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finding approximate solutions to tricky polynomials

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finding approximate solutions to tricky polynomials

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can be used as an introduction to numerical methods and iterative procedures, or as a review exercise

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extension/challenge

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Imagine that the rate of a chemical reaction only depends on the concentration of a single chemical $A$, with concentration denoted by $[A]$. The reaction is called $m$th order with respect to $[A]$ if it satisfies a rate equation of the following type
$$
-\frac{d[A]}{dt} =k[A]^m
$$
This problem involves trying to find various solutions to rate equations from a purely mathematical perspective. As a rate equation is non-linear in $[A]$ if $m\neq 1$ it is difficult to say how many different sorts of solution it might possess.

To begin with, can $[A] =\lambda+\mu t$ ever be a solution to a rate equation? What would be the order of the reaction? What possible values could $\lambda$ and $\mu$ take?

Can $[A]=(\lambda+\mu t)^n$ ever solve a rate equation when $n$ is a positive whole number other than $1$? A negative whole number? Zero?

What are the possibilities if $n$ is not a whole number?

Which other solutions can you find to a rate equation?

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Imagine that the rate of a chemical reaction only depends on the concentration of a single chemical $A$, with concentration denoted by $[A]$. The reaction is called $m$th order with respect to $[A]$ if it satisfies a rate equation of the following type $$ -\frac{d[A]}{dt} =k[A]^m $$ This problem involves trying to find various solutions to rate equations from a purely mathematical perspective.

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investigating solutions to a set of DEs

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can use when teaching integration and differentiation

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In Curve match we asked you to match the curves on the diagram below with the following functions.

\[ y=\sqrt{x} \quad y=x^2 \quad y=x \quad y=\dfrac{x^2+x}{2} \quad \text{and } \quad y=2\sqrt{x}-x\]

Now we'd like you to look at the different regions enclosed by the curves.

Which do you think is the smallest region? How can you check?

How big are the other regions?

Try to explain why certain regions are the size they are.

Think about what techniques you can use and how you can reduce the number of calculations you need to do.

You might find it helpful to start by matching the curves, or you could check this in Curve match.

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Meaningful areas | Calculus of Powers | Underground Mathematics

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In Curve match we asked you to match the curves on the diagram below with the following functions. \[ y=\sqrt{x} \quad y=x^2 \quad y=x \quad y=\dfrac{x^2+x}{2} \quad \text{and } \quad y=2\sqrt{x}-x\] Now we'd like you to look at the different regions enclosed by the curves. Which do you think is the smallest region?

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Investigating the derivative of the sine curve in relation to radians

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Demonstration of differentiating sin x and cos x

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Can you locate these values on this interactive logarithmic scale?

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Demonstrates the derivative of the natural logarithm function

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Investigating the gradient of exponential graphs

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practising chain, product and quotient rule

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practising chain, product and quotient rule

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Shows the relationship between the graphs of y = a sin x + b cos x and y = R sin(x + alpha)

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Geometrical demonstration of the double angle formulae

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Geometrical demonstration of the compound angle formula

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Which is bigger, \(\frac{2}{5}\) or \(\frac{1}{3}\)?

How do you know?

Is there more than one way of convincing yourself that one is bigger than the other?

What if you had the fractions \(\frac{2x}{5}\) and \(\frac{x}{3}\)? Would your answer be the same?

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Which is bigger? | Thinking about Algebra | Underground Mathematics

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Which is bigger, \(\frac{2}{5}\) or \(\frac{1}{3}\)? How do you know? Is there more than one way of convincing yourself that one is bigger than the other? What if you had the fractions \(\frac{2x}{5}\) and \(\frac{x}{3}\)? Would your answer be the same?

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more challenging

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Graphs & transformations 2: trigonometric graphs Activity 1 Option 1 (AS Pure)

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This sorting activity is based on a task from the Integral Maths resources for Year 1 AS Pure Graphs and transformations. For more details about the resources see: https://integralmaths.org/

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Graphs & transformations 2: trigonometric graphs Activity 1 Option 2 (AS Pure)

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This sorting activity is based on a task from the Integral Maths resources for Year 1 AS Graphs and Transformations. For more details about the resources see: https://integralmaths.org/

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Shows how the equation of a circle relates to Pythagoras' theorem.

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Coordinate Geometry 1: Points & straight lines Activity 6 Option 2 - Label sketch rather than card sort (AS Pure)

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This sorting activity is based on a task from the Integral Maths resources for Year 1 AS Coordinate Geometry. For more details about the resources see: https://integralmaths.org/

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Coordinate Geometry 1: Points & straight lines Activity 6 Option 1 (AS Pure)

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This sorting activity is based on a task from the Integral Maths resources for Year 1 AS Coordinate Geometry. For more details about the resources see: https://integralmaths.org/

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\(\sqrt{x}\) or \(x^{\frac{1}{2}}\)? Whichever we prefer, we need to be comfortable working with indices and roots in either form.

Below is a series of equations, for which you should think about the following questions.

For what values of \(x\) and \(y\) do these statements make sense?

What are the possible values for the expressions?

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Index issues | Thinking about Algebra | Underground Mathematics

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(\sqrt{x}\) or \(x^{\frac{1}{2}}\)? Whichever we prefer, we need to be comfortable working with indices and roots in either form. Below is a series of equations, for which you should think about the following questions. For what values of \(x\) and \(y\) do these statements make sense? What are the possible values for the expressions?

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Differential Equations - Complete Review Course | Online Crash Course

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Differential Equations: Lecture 1.1-1.2 Definitions and Terminology and Initial Value Problems

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Differential Equations 1: Oxford Mathematics 2nd Year Student Lecture

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Calculus 1 Lecture 5.1:  Finding Area Between Two Curves

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Calculus 2 Lecture 7.6:  Improper Integrals

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Math 2B. Calculus. Lecture 03. Definite Integral.

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Calculus 1 Lecture 2.2:  Techniques of Differentiation (Finding Derivatives of Functions Easily)

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Lec 3 | MIT 18.01 Single Variable Calculus, Fall 2007

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Lec 2 | MIT 18.01 Single Variable Calculus, Fall 2007

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Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007