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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>
</div>
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<main class="bodycontainer">
<section class="textbody">
<h1>MAT1001 Differential Calculus: Lecture Notes</h1>
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<h3 id="autosec-218">Chapter <span class="sectionnumber">6 </span>Numerical Methods</h3>
<a id="mat1001-autopage-218"></a>
<a id="mat1001-autofile-6"></a>
<a id="sec:numerics"></a>
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<h4 id="autosec-219"><span class="sectionnumber">6.1 </span>Solving equations numerically</h4>
<a id="mat1001-autopage-219"></a>
<p>
It frequently happens that when confronted with a problem you end up with a mathematical problem or expression that cannot be evaluated analytically. This may be a differential equation that we cannot solve, an integral that has no closed form solution, or even an
algebraic equation which does not yield an explicit solution. This is where we have to turn to computer packages or follow a numerical method. Nowadays many computer programs can implement these approaches as standard. However, it is a good idea to understand
how to implement a few of these approaches by hand as you may end up writing one of your own at some stage.<br />
</p>
<p>
Chapter 27 of [Riley et al., 2006] contains a lot of information about this topic and is a good place to go for more details than we will discuss here.<br />
</p>
<p>
Before moving on to numerical differentiation and integration we will first discuss how to solve algebraic equations numerically. This is all about finding the real roots of an equation \(f(x)=0\). This equation can either be algebraic, with \(f(x)\) being a polynomial, or
transcendental, if \(f(x)\) includes trig, log, or exponential terms. We will be using iterative schemes to solve the equations, where we make successive approximations to the true solution, which converge to the true solution. For a method to be successful we need both
that the approximations converge to the true solution, and also that we only have a finite number of solutions.<br />
</p>
<p>
It is important to note that different iteration methods will be better at solving certain kinds of problems, in practice some combination of methods is likely to be used. We will see four methods, <b>Rearrangement</b>, <b>Linear interpolation</b>, and
<b>Bisection</b>, before discussing <b>Newton</b>’s method, which is the most important method to know in this module.
</p>
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<h5 id="autosec-220">Rearrangement</h5>
<a id="mat1001-autopage-220"></a>
<p>
To explain what we mean by an iterative scheme it is useful to consider the first example method, recasting our equation \(f(x)=0\) as
</p>
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\begin{equation*}
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for some slowly varying function \(\phi (x)\). Our iteration scheme starts from a guess, \(x_{0}\) which will not solve \(f(x_{0})=0\) but should be picked so that it is close to zero<sup>1</sup>. We then have that \(x_{1}=\phi (x_{0})\) and can keep repeating this
process, i.e. \(x_{n+1}=\phi (x_{n})\). The solution we are looking for will be a value of \(x\) that is left unchanged by applying \(\phi \).<br />
</p>
<p>
The different methods are different ways of building \(\phi (x)\) so that we can start the iteration scheme.<br />
</p>
<p>
As an example of how to use the rearrangement iteration consider
</p>
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