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\contentsline {chapter}{\numberline {1}{\ignorespaces Why Do We Need Calculus\relax }}{5}{}
\contentsline {section}{\numberline {1.1}{\ignorespaces Course Overview\relax }}{6}{}
\contentsline {section}{\numberline {1.2}{\ignorespaces Why Calculus\relax }}{7}{}
\contentsline {chapter}{\numberline {2}{\ignorespaces Functions\relax }}{17}{}
\contentsline {section}{\numberline {2.1}{\ignorespaces What is a function\relax }}{18}{}
\contentsline {section}{\numberline {2.2}{\ignorespaces Trigonometric functions\relax }}{38}{}
\contentsline {paragraph}{Squares:}{50}{}
\contentsline {paragraph}{Multiple angles:}{51}{}
\contentsline {paragraph}{Further trig functions:}{52}{}
\contentsline {section}{\numberline {2.3}{\ignorespaces Logarithms and Exponentials\relax }}{54}{}
\contentsline {section}{\numberline {2.4}{\ignorespaces Hyperbolic functions\relax }}{65}{}
\contentsline {section}{\numberline {2.5}{\ignorespaces Limits and asymptotics\relax }}{70}{}
\contentsline {section}{\numberline {2.6}{\ignorespaces Continuity and differentiability\relax }}{84}{}
\contentsline {chapter}{\numberline {3}{\ignorespaces Differentiation\relax }}{101}{}
\contentsline {section}{\numberline {3.1}{\ignorespaces Differentiation from first principles\relax }}{102}{}
\contentsline {section}{\numberline {3.2}{\ignorespaces Differentiation techniques\relax }}{111}{}
\contentsline {section}{\numberline {3.3}{\ignorespaces Multiple derivatives\relax }}{131}{}
\contentsline {section}{\numberline {3.4}{\ignorespaces Applications of differentiation\relax }}{133}{}
\contentsline {chapter}{\numberline {4}{\ignorespaces Integration\relax }}{157}{}
\contentsline {section}{\numberline {4.1}{\ignorespaces Integration and antiderivatives\relax }}{158}{}
\contentsline {section}{\numberline {4.2}{\ignorespaces Techniques for integration\relax }}{168}{}
\contentsline {section}{\numberline {4.3}{\ignorespaces Definite integrals\relax }}{177}{}
\contentsline {section}{\numberline {4.4}{\ignorespaces Applications of integrals\relax }}{204}{}
\contentsline {chapter}{\numberline {5}{\ignorespaces Differential Equations\relax }}{216}{}
\contentsline {chapter}{\numberline {6}{\ignorespaces Numerical Methods\relax }}{218}{}
\contentsline {section}{\numberline {6.1}{\ignorespaces Solving equations numerically\relax }}{219}{}
\contentsline {section}{\numberline {6.2}{\ignorespaces Approximating functions\relax }}{251}{}
\contentsline {section}{\numberline {6.3}{\ignorespaces Numerical integration\relax }}{277}{}
\contentsline {section}{\numberline {6.4}{\ignorespaces Numerical differentiation\relax }}{303}{}
\contentsline {section}{\numberline {6.5}{\ignorespaces Numerical approaches to differential equations\relax }}{324}{}
\contentsline {chapter}{\numberline {7}{\ignorespaces Calculus in Computer Science\relax }}{326}{}
\contentsline {section}{\numberline {7.1}{\ignorespaces LLMs and AI\relax }}{327}{}
\contentsline {section}{\numberline {7.2}{\ignorespaces Robotics\relax }}{334}{}
\contentsline {chapter}{\numberline {8}{\ignorespaces Advanced Topics\relax }}{336}{}
\contentsline {section}{\numberline {8.1}{\ignorespaces L'Ho\begingroup \LWR@fontfortags {HTMLentity}{\LWR@origpound {}x0302}\LWR@origampersand \LWR@origpound {}x0302;\endgroup pital's rule for evaluating limits\relax }}{337}{}
\contentsline {section}{\numberline {8.2}{\ignorespaces Trigonometric Derivatives in Degrees\relax }}{341}{}
\contentsline {section}{\numberline {8.3}{\ignorespaces Functions of two variables\relax }}{342}{}
\contentsline {section}{\numberline {8.4}{\ignorespaces Multiple integrals\relax }}{343}{}
\contentsline {section}{\numberline {8.5}{\ignorespaces Optimisation Problems\relax }}{344}{}
\contentsline {section}{\numberline {8.6}{\ignorespaces Polynomial approximation\relax }}{347}{}
\contentsline {chapter}{\numberline {9}{\ignorespaces Background Mathematics\relax }}{350}{}
\contentsline {section}{\numberline {9.1}{\ignorespaces Background and References\relax }}{351}{}
\contentsline {section}{\numberline {9.2}{\ignorespaces Polynomials and roots\relax }}{352}{}
\contentsline {section}{\numberline {9.3}{\ignorespaces Trigonometry Primer\relax }}{354}{}
\contentsline {section}{\numberline {9.4}{\ignorespaces Radians and Degrees\relax }}{358}{}
\contentsline {section}{\numberline {9.5}{\ignorespaces Rearranging Equations\relax }}{365}{}
\contentsline {section}{\numberline {9.6}{\ignorespaces More on Limits\relax }}{370}{}
\contentsline {chapter}{\numberline {10}{\ignorespaces Further Reading\relax }}{373}{}
\contentsline {section}{\numberline {10.1}{\ignorespaces Why Calculus Extra Reading\relax }}{374}{}
\contentsline {section}{\numberline {10.2}{\ignorespaces Functions Extra Reading\relax }}{375}{}
\contentsline {section}{\numberline {10.3}{\ignorespaces Differentiation Extra Reading\relax }}{376}{}
\contentsline {section}{\numberline {10.4}{\ignorespaces Integration Extra Reading\relax }}{377}{}
\contentsline {section}{\numberline {10.5}{\ignorespaces Differential Equations Extra Reading\relax }}{378}{}
\contentsline {section}{\numberline {10.6}{\ignorespaces Numerical Methods Extra Reading\relax }}{379}{}
\contentsline {section}{\numberline {10.7}{\ignorespaces Calculus in Computer Science\relax }}{380}{}
\contentsline {section}{\numberline {10.8}{\ignorespaces Advanced Topics Extra Reading\relax }}{381}{}
\contentsline {chapter}{\numberline {11}{\ignorespaces Tutorial Sheets\relax }}{383}{}
\contentsline {section}{\numberline {11.1}{\ignorespaces Week 1\relax }}{384}{}
\contentsline {paragraph}{Functions}{385}{}
\contentsline {paragraph}{Challenge Problems}{396}{}
\contentsline {section}{\numberline {11.2}{\ignorespaces Week 2\relax }}{400}{}
\contentsline {paragraph}{Polynomials}{401}{}
\contentsline {paragraph}{Trig functions}{404}{}
\contentsline {paragraph}{Exponentials and Logarithms}{408}{}
\contentsline {paragraph}{Limits and Asymptotes}{411}{}
\contentsline {paragraph}{Challenge Problems}{417}{}
\contentsline {paragraph}{Hyperbolic Trig functions}{421}{}
\contentsline {section}{\numberline {11.3}{\ignorespaces Week 3\relax }}{424}{}
\contentsline {paragraph}{Differentiation}{425}{}
\contentsline {paragraph}{Challenge Problems}{438}{}
\contentsline {section}{\numberline {11.4}{\ignorespaces Week 4\relax }}{442}{}
\contentsline {paragraph}{Trig, Exp, and Log}{443}{}
\contentsline {section}{\numberline {11.5}{\ignorespaces Week 5\relax }}{450}{}
\contentsline {paragraph}{Product, Quotient, and Chain Rules}{451}{}
\contentsline {paragraph}{Antiderivatives}{459}{}
\contentsline {paragraph}{Integration and Area}{463}{}
\contentsline {paragraph}{Challenge Problems}{468}{}
\contentsline {section}{\numberline {11.6}{\ignorespaces Week 6\relax }}{473}{}
\contentsline {paragraph}{Integration and Area}{474}{}
\contentsline {paragraph}{Techniques for Integration}{479}{}
\contentsline {paragraph}{Improper Integrals}{485}{}
\contentsline {paragraph}{Challenge Problems}{488}{}
\contentsline {section}{\numberline {11.7}{\ignorespaces Week 7\relax }}{493}{}
\contentsline {paragraph}{Approximating Functions}{494}{}
\contentsline {paragraph}{Numerical Solutions of Equations}{505}{}
\contentsline {paragraph}{Improper Integrals}{511}{}
\contentsline {paragraph}{Challenge Problems}{514}{}
\contentsline {section}{\numberline {11.8}{\ignorespaces Week 8\relax }}{518}{}
\contentsline {paragraph}{Numerical Integration}{519}{}
\contentsline {paragraph}{Challenge Problems}{525}{}
\contentsline {section}{\numberline {11.9}{\ignorespaces Week 9\relax }}{528}{}
\contentsline {paragraph}{Numerical Differentiation}{529}{}
\contentsline {paragraph}{Multiple Differentiation}{534}{}
\contentsline {paragraph}{Challenge Problems}{537}{}
\contentsline {section}{\numberline {11.10}{\ignorespaces Week 10\relax }}{540}{}
\contentsline {paragraph}{Optimisation}{541}{}
\contentsline {paragraph}{Applications}{546}{}
\contentsline {chapter}{\numberline {12}{\ignorespaces Extra Proofs and Derivations\relax }}{551}{}
\contentsline {chapter}{\numberline {13}{\ignorespaces Standard Derivatives and Integrals\relax }}{554}{}
\contentsline {section}{\numberline {13.1}{\ignorespaces Derivatives\relax }}{555}{}
\contentsline {section}{\numberline {13.2}{\ignorespaces Integrals\relax }}{558}{}
\contentsline {chapter}{Bibliography}{562}{}