Archive for the ‘Lectures’ Category

Least squares and pseudo-inverses

Posted: 5th February 2014 by seanmathmodelguy in Lectures

To appreciate the connections between solutions of the system \(Ax=b\) and least squares, we begin with two illustrative examples. Overdetermined systems: \(A\) is \(m \times n\) with \(m > n\). In this case there are more equations than unknowns, \(A^{\top}A\) is \(n\times n\) and \(AA^{\top}\) is \(m\times m\). The connection with the pseudo-inverse is that […]

Intermediate Value Theorem – Limits and Continuity

Posted: 12th February 2013 by seanmathmodelguy in Lectures

Intermediate Value Theorem To begin with, let’s start with the basic statement of the theorem. Theorem If \(f(x)\) is continuous on a closed interval \([a,b]\) and \(N\) is any number \(f(a) < N < f(b)\) then there exists a value \(c \in (a,b)\) such \(f(c) = N\). The illustration corresponding to the theorem is to […]

Continuity – Limits and Continuity

Posted: 12th February 2013 by seanmathmodelguy in Lectures

Continuity A function \(f(x)\) is said to be continuous at a point \(a\) in its domain if the following three properties hold. \(\displaystyle \lim_{x \to a} f(x)\) exists. This takes three steps to show in itself. \(f(a)\) has to exist, \(\displaystyle \lim_{x \to a} f(x) = f(a)\). Continuity connects the behaviour of a function in […]

Strategy to Calculate Limits – Limits and Continuity

Posted: 9th February 2013 by seanmathmodelguy in Lectures

Strategy to Calculate Limits   To compute \(\displaystyle \lim_{x \to a} f(x)\): A. Try to plug the value of \(a\) directly into the function. If we get a number or the limit ‘blows up’ then we are done! You should be so lucky. Typically the value is undefined, having the form \(\displaystyle \frac{0}{0}\) or \(\displaystyle […]