MATH1011 - Lecture 10

Skeleton Slides

Let $f (x)$ and $g (x)$ be continuously differentiable scalar functions
If $f (x)$ has a local extremum at some point $c$ subject to the constraint $g (x) = k$ where $\nabla g (c) \neq = 0$ , then there exists a number $λ \in R$ such that $\nabla f (c) = λ \nabla g (c)$
- $λ$ is know as a Lagrange multiplier
- Watch lecture recording for worked example
When solving minimums/maximums, simplify functions down to make the question easier
- e.g. $x^{2} + y^{2} + z^{2}$ has the same minimum as $(x^{2} + y^{2} + z^{2})$
- Using $(x^{2} + y^{2} + z^{2})$ will make the question much easier

Let $f (x)$ be a real-valued function of one variable with continuous derivatives $f^{(n)} (x)$ on $I$ for some integer $n \geq 1$
- Let $a$ be an interior point of $I$
The $n^{t h}$ degree Taylor polynomial of $f$ about $a$ is defined by $T_{n, a} (x) = f (a) + \frac{f ^{'} ( a )}{1 !} (x - a) + \frac{f ^{''} ( a )}{2 !} (x - a)^{2} + ... + \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}$
$T_{1, a} (x)$ is the equation of tangent line to the curve $y = f (x)$ when $x = a$
The Taylor polynomial is an approximation of the function $f (x)$
- $f (x) \approx T_{n, a} (x)$
  - This can be written as: $f (x) = T_{n, a} (x) + R_{n, a} (x)$
    - where $R_{n, a} (x)$ is the error term or remainder
- As $n$ increases, the domain in which $T_{n, a} (x)$ is a good approximation increases in size

Taylor’s formula - Lagrange form:

Assume that $f$ has continuous derivatives up to order $n + 1$ on some interval $I$ and $a$ is an interior point of $I$
Then for any $x \in I$ there exists $z$ between $a$ and $x$ such that: $R_{n, a} (x) = \frac{f ^{(n + 1)} ( z )}{( n + 1 )!} (x - a)^{n + 1}$
i.e. for a given Taylor polynomial calculated using point $a$ , we can choose another point $x$ (variable, not a constant) to plug into $T_{n, a} (x)$ , which will produce an estimate for $f (x)$ . This estimate will have an error (difference from $f (x)$ ) given exactly by $R_{n, a} (x)$ , a function of $x$ . The variable $z$ is always unknown, it is some arbitrary value between $c$ and $x$ .

Only for certain functions such as trigonometric functions, can we choose a value for $z$ where we maximise (overestimate) the error. e.g. since $sin (z)$ will always have a maximum value of $1$ for all $z$ , whereas $e^{z}$ could stretch into infinity.

Annotated Slides

Quartz 5