This module aims to introduce the student to fundamental methods in applied mathematics, to give the student experience in using the computer to solve mathematical problems, and to reinforce A level calculus skills.
Jeremy Levesley, University of Leicester
Definition of power series with explanation of summation; radius of convergence; finding radius of convergence. An understanding of series is necessary before tackling power series.
Defining convergence and testing for convergence including the alternating series test, ratio test, and comparison test
Explaining absolute convergence with a look at conditional convergence, rearrangements, and product rule
Official definition and examples, including demonstration of using L'hopital's rule
Differentiation is the process by which we extract the gradient of a line from the formula of a line.
Sequences are understood by first defining convergence: a sequence ends up at a certain value when it gets to infinity. Limits and closeness.
Referring to sequences as a whole; null sequences; can ignore a finite number of sequence terms without affecting anything; domination; bounded sequences; convergent sequences; subsequences
Covers the two types of diverging sequences: sequences tending to infinity, and oscillating sequences
An introduction to series. It is necessary to understand sequences before tackling series. Sigma notation, summation, and geometric series are explained as well.
Bounds are properties of finite and infinite sets. Upper and lower bounds of these sets are explained, along with maximum elements. Least upper bound and greatest lower bound.
Prof Jeremy Levesley talks about the solution of second order differential equations with constant coefficients.
Prof Jeremy Levesley talks about integration over a rectangle and integration over regions where the range of integration is given. He also explains how to use polar coordinates for regions with circular symmetry.
Prof Jeremy Levesley talks about the solution of differential equations of linear and homogenous types.
Prof Jeremy Levesley talks about the solution of first order differential equations using direct integration and separable equations.
Prof Jeremy Levesley explains how to compute the Taylor series for a function and how to estimate the error approximating a function with its Taylor series.
Prof Jeremy Levesley explains how to find and categorise the local maxima and minima using the second derivative test. He also explains how to find the global maxima and minima by checking for the interior local maxima and minima and then points on the interiors of edges and corners.
Prof Jeremy Levesley explains how to compute derivatives in given directions and the maximum slope at a point.
Prof Jeremy Levesley talks about constructing tangent lines to curves in space and tangent planes to surfaces.
Prof Jeremy Levesley talks about partial differentiation and the chain rule for partial differentiation.
Prof Jeremy Levesley explains how to calculate the differentiation of vectors which depend on a parameter, the tangent vector to a space curve, the speed and velocity of a particle, and the length of a space curve defined parametrically.
Prof Jeremy Levesley explains how to calculate the vector and Cartesian equations of lines and planes, and the scalar and vector products.
Prof Jeremy Levesley explains how to compute the angle between two vectors using the scalar product, the area of a parallelogram included between two vectors, the scalar product, the vector product, and the length of a vector.
Prof Jeremy Levesley explains how to add vectors, multiply vectors by a scalar, compute the scalar (dot) and vector (cross) products of vectors and calculate the length of a vector.
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