Oxford Calculus: Taylor's Theorem Explained with Examples and Derivation

University of Oxford mathematician Dr Tom Crawford derives Taylor's Theorem for approximating any function as a polynomial and explains how the expansion works with two detailed examples. Test yourself with some exercises on Taylor's Theorem with this FREE worksheet in Maple Learn: https://learn.maplesoft.com/doc/cbv2uybyym/trm-taylors-theorem-worksheet The video begins by introducing the idea of approximating a function by a polynomial and the condition we choose to implement that allows the coefficients to be derived. By ensuring that both the function and polynomial approximation are equal for all derivatives at a single point, Cauchy's Mean Value Theorem can be applied to give equality. Taylor's Theorem is demonstrated with two fully worked examples. First, the power series expansion for cos is derived by expanding around zero. Next, a third degree polynomial approximation is calculated for small x, when expanding the function ln(1+sin(x)). Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: https://play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en Apple: https://apps.apple.com/us/app/maple-companion/id1466659419 Other videos in the Oxford Calculus series can be found here: https://www.youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S Finding critical points for functions of several variables: https://youtu.be/Leomuu82-u8 Classifying critical points using the method of the discriminant: https://www.youtube.com/watch?v=5M_ts8Q2LEM Partial differentiation explained: https://youtu.be/RVwcBGzQcT8 Second order linear differential equations: https://youtu.be/F54yhRB9qDI Integrating factors explained: https://www.youtube.com/watch?v=ftqKuOfOX3E Solving simple PDEs: https://youtu.be/uztjxrGY6Jw Jacobians explained: https://www.youtube.com/watch?v=YqMelRryG8U Separation of variables integration technique explained: https://youtu.be/zk41c0vs9XQ Solving homogeneous first order differential equations: https://youtu.be/uqvqjbAcbL8 Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: https://www.youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths