Calculus AR

What is a continuous function?

A function $f(x)$ is continuous at a point $x=c$ iff it meets the following three conditions:

1. $f(c)$ exists (c lies in the domain of f).
2. $\lim x\to cf(x)$ exists (f has a limit as x tends towards c).
3. $\lim x\to cf(x)=f(c)$ (the limit equals the function value).

We define a continuous function as one that is continuous at every point in its domain.

State the mean value theorem and its corollaries?

$\frac{f(b)-f(a)}{b-a}=f^{\prime }(c)$

1. Only constant functions have zero derivates. If $f^{\prime }(c)$ is zero for all $x$ in the interval $[a,b]$ then $f(b)$ is equal to $f(a)$.
2. Matching curvatures if $f^{\prime }(x)=g^{\prime }(x)$ for all $x\in [a,b]$ then there exists a constant $C$ for all $x\in [a,b]$. $f-g$ is a constant function on the interval.

Rolle’s Theorem if there is a horizontal line touching the graph twice then there is a point in this interval whose derivative is zero

State the general idea and formal version of Newtons method?

Use tangent lines of the graph $f(x)$ to approximate a solution for $f(x)=0$ (the roots of the function).

The idea is to start with an initial educated guess $x_0$ we construct the tangent line to the curve at $x_0$.

If we look at the linearization of a line at point $x_n$,

$y=f(x_n)+f^{\prime }(x_n)(x-x_n)$ .

And we’re interested in the point $x$ where $y=0$ then rearranging gets us

$x=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$

This is finding the x-intercept of the tangent line to a point $(x_n,f(x_n))$. This becomes the new x value and the process is iterated.