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<p>
<span class="sidetocthetitle">MAT1001 Differential Calculus: Lecture Notes</span>
</p>
<p>
Contents
</p>
</div>
<div class="sidetoccontents">
<p>
<a href="index.html" class="linkhome" >
Home</a>
</p>
<p>
<a href="Why-Do-We-Need-Calculus.html#autosec-5" class="tocchapter" >
<span class="sectionnumber">1</span> Why Do We Need Calculus</a>
</p>
<p>
<a href="Why-Do-We-Need-Calculus.html#autosec-6" class="tocsection" >
<span class="sectionnumber">1.1</span> Course Overview</a>
</p>
<p>
<a href="Why-Do-We-Need-Calculus.html#autosec-7" class="tocsection" >
<span class="sectionnumber">1.2</span> Why Calculus</a>
</p>
<p>
<a href="Functions.html#autosec-17" class="tocchapter" >
<span class="sectionnumber">2</span> Functions</a>
</p>
<p>
<a href="Functions.html#autosec-18" class="tocsection" >
<span class="sectionnumber">2.1</span> What is a function</a>
</p>
<p>
<a href="Functions.html#autosec-38" class="tocsection" >
<span class="sectionnumber">2.2</span> Trigonometric functions</a>
</p>
<p>
<a href="Functions.html#autosec-54" class="tocsection" >
<span class="sectionnumber">2.3</span> Logarithms and Exponentials</a>
</p>
<p>
<a href="Functions.html#autosec-65" class="tocsection" >
<span class="sectionnumber">2.4</span> Hyperbolic functions</a>
</p>
<p>
<a href="Functions.html#autosec-70" class="tocsection" >
<span class="sectionnumber">2.5</span> Limits and asymptotics</a>
</p>
<p>
<a href="Functions.html#autosec-84" class="tocsection" >
<span class="sectionnumber">2.6</span> Continuity and differentiability</a>
</p>
<p>
<a href="Differentiation.html#autosec-101" class="tocchapter" >
<span class="sectionnumber">3</span> Differentiation</a>
</p>
<p>
<a href="Differentiation.html#autosec-102" class="tocsection" >
<span class="sectionnumber">3.1</span> Differentiation from first principles</a>
</p>
<p>
<a href="Differentiation.html#autosec-111" class="tocsection" >
<span class="sectionnumber">3.2</span> Differentiation techniques</a>
</p>
<p>
<a href="Differentiation.html#autosec-131" class="tocsection" >
<span class="sectionnumber">3.3</span> Multiple derivatives</a>
</p>
<p>
<a href="Differentiation.html#autosec-133" class="tocsection" >
<span class="sectionnumber">3.4</span> Applications of differentiation</a>
</p>
<p>
<a href="Integration.html#autosec-157" class="tocchapter" >
<span class="sectionnumber">4</span> Integration</a>
</p>
<p>
<a href="Integration.html#autosec-158" class="tocsection" >
<span class="sectionnumber">4.1</span> Integration and antiderivatives</a>
</p>
<p>
<a href="Integration.html#autosec-168" class="tocsection" >
<span class="sectionnumber">4.2</span> Techniques for integration</a>
</p>
<p>
<a href="Integration.html#autosec-177" class="tocsection" >
<span class="sectionnumber">4.3</span> Definite integrals</a>
</p>
<p>
<a href="Integration.html#autosec-204" class="tocsection" >
<span class="sectionnumber">4.4</span> Applications of integrals</a>
</p>
<p>
<a href="Differential-Equations.html#autosec-216" class="tocchapter" >
<span class="sectionnumber">5</span> Differential Equations</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-218" class="tocchapter" >
<span class="sectionnumber">6</span> Numerical Methods</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-219" class="tocsection" >
<span class="sectionnumber">6.1</span> Solving equations numerically</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-251" class="tocsection" >
<span class="sectionnumber">6.2</span> Approximating functions</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-277" class="tocsection" >
<span class="sectionnumber">6.3</span> Numerical integration</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-303" class="tocsection" >
<span class="sectionnumber">6.4</span> Numerical differentiation</a>
</p>
<p>
<a href="Numerical-Methods.html#autosec-324" class="tocsection" >
<span class="sectionnumber">6.5</span> Numerical approaches to differential equations</a>
</p>
<p>
<a href="Calculus-in-Computer-Science.html#autosec-326" class="tocchapter" >
<span class="sectionnumber">7</span> Calculus in Computer Science</a>
</p>
<p>
<a href="Calculus-in-Computer-Science.html#autosec-327" class="tocsection" >
<span class="sectionnumber">7.1</span> LLMs and AI</a>
</p>
<p>
<a href="Calculus-in-Computer-Science.html#autosec-334" class="tocsection" >
<span class="sectionnumber">7.2</span> Robotics</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-336" class="tocchapter" >
<span class="sectionnumber">8</span> Advanced Topics</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-337" class="tocsection" >
<span class="sectionnumber">8.1</span> L’Hôpital’s rule for evaluating limits</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-341" class="tocsection" >
<span class="sectionnumber">8.2</span> Trigonometric Derivatives in Degrees</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-342" class="tocsection" >
<span class="sectionnumber">8.3</span> Functions of two variables</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-343" class="tocsection" >
<span class="sectionnumber">8.4</span> Multiple integrals</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-344" class="tocsection" >
<span class="sectionnumber">8.5</span> Optimisation Problems</a>
</p>
<p>
<a href="Advanced-Topics.html#autosec-347" class="tocsection" >
<span class="sectionnumber">8.6</span> Polynomial approximation</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-350" class="tocchapter" >
<span class="sectionnumber">9</span> Background Mathematics</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-351" class="tocsection" >
<span class="sectionnumber">9.1</span> Background and References</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-352" class="tocsection" >
<span class="sectionnumber">9.2</span> Polynomials and roots</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-354" class="tocsection" >
<span class="sectionnumber">9.3</span> Trigonometry Primer</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-358" class="tocsection" >
<span class="sectionnumber">9.4</span> Radians and Degrees</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-365" class="tocsection" >
<span class="sectionnumber">9.5</span> Rearranging Equations</a>
</p>
<p>
<a href="Background-Mathematics.html#autosec-370" class="tocsection" >
<span class="sectionnumber">9.6</span> More on Limits</a>
</p>
<p>
<a href="Further-Reading.html#autosec-373" class="tocchapter" >
<span class="sectionnumber">10</span> Further Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-374" class="tocsection" >
<span class="sectionnumber">10.1</span> Why Calculus Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-375" class="tocsection" >
<span class="sectionnumber">10.2</span> Functions Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-376" class="tocsection" >
<span class="sectionnumber">10.3</span> Differentiation Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-377" class="tocsection" >
<span class="sectionnumber">10.4</span> Integration Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-378" class="tocsection" >
<span class="sectionnumber">10.5</span> Differential Equations Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-379" class="tocsection" >
<span class="sectionnumber">10.6</span> Numerical Methods Extra Reading</a>
</p>
<p>
<a href="Further-Reading.html#autosec-380" class="tocsection" >
<span class="sectionnumber">10.7</span> Calculus in Computer Science</a>
</p>
<p>
<a href="Further-Reading.html#autosec-381" class="tocsection" >
<span class="sectionnumber">10.8</span> Advanced Topics Extra Reading</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-383" class="tocchapter" >
<span class="sectionnumber">11</span> Tutorial Sheets</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-384" class="tocsection" >
<span class="sectionnumber">11.1</span> Week 1</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-400" class="tocsection" >
<span class="sectionnumber">11.2</span> Week 2</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-424" class="tocsection" >
<span class="sectionnumber">11.3</span> Week 3</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-442" class="tocsection" >
<span class="sectionnumber">11.4</span> Week 4</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-450" class="tocsection" >
<span class="sectionnumber">11.5</span> Week 5</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-473" class="tocsection" >
<span class="sectionnumber">11.6</span> Week 6</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-493" class="tocsection" >
<span class="sectionnumber">11.7</span> Week 7</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-518" class="tocsection" >
<span class="sectionnumber">11.8</span> Week 8</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-528" class="tocsection" >
<span class="sectionnumber">11.9</span> Week 9</a>
</p>
<p>
<a href="Tutorial-Sheets.html#autosec-540" class="tocsection" >
<span class="sectionnumber">11.10</span> Week 10</a>
</p>
<p>
<a href="Extra-Proofs-Derivations.html#autosec-551" class="tocchapter" >
<span class="sectionnumber">12</span> Extra Proofs and Derivations</a>
</p>
<p>
<a href="Standard-Derivatives-Integrals.html#autosec-554" class="tocchapter" >
<span class="sectionnumber">13</span> Standard Derivatives and Integrals</a>
</p>
<p>
<a href="Standard-Derivatives-Integrals.html#autosec-555" class="tocsection" >
<span class="sectionnumber">13.1</span> Derivatives</a>
</p>
<p>
<a href="Standard-Derivatives-Integrals.html#autosec-558" class="tocsection" >
<span class="sectionnumber">13.2</span> Integrals</a>
</p>
<p>
<a href="Bibliography.html#autosec-562" class="tocchapter" >
Bibliography</a>
</p>
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<h1>MAT1001 Differential Calculus: Lecture Notes</h1>
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<h3 id="autosec-17">Chapter <span class="sectionnumber">2 </span>Functions</h3>
<a id="mat1001-autopage-17"></a>
<a id="mat1001-autofile-2"></a>
<a id="sec:functions"></a>
<div class="epigraph" role="note">
<div class="qitem">
<p>
The art of doing mathematics is finding that special case that contains all the germs of generality.
</p>
<div class="epigraphsource">
<p>
<i>David Hilbert</i>
</p>
</div>
</div>
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<h4 id="autosec-18"><span class="sectionnumber">2.1 </span>What is a function</h4>
<a id="mat1001-autopage-18"></a>
<p>
The concept of a function is essential to understand not just calculus but also computer programming. We can think of a function as being a black box that takes in information, potentially changes it in some way, and then outputs information.<br />
</p>
<figure id="autoid-5" class="figure ">
<div class="center">
<p>
<span
id="lateximage-mat1001-2"
class="lateximagesource"
><!--
1
2 A
B
3 C
4 D
Domain Codomain
--><img
src="mat1001-images\image-2.svg"
alt="(A schematic of how a function behaves.)"
style=""
class="lateximage"
></span>
</p>
<div class="figurecaption">
Figure 2.1: A schematic of how a function behaves, it maps objects \(1,2,3,\dots \) in the Set \(X\) to objects \(A,B,C,D,\dots \) in the set \(Y\).
</div>
<a id="fig: function schematic"></a>
</div>
</figure>
<p>
Being more precise, a function is a map between two spaces, the domain (\(X\)), and the codomain (\(Y\)). e.g.
</p>
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