24

One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations. For example, if you are asked to find the cosine of an angle and the only computing device you have is a four-function calculator, then you can get a good approximation of the cosine ...


19

An excellent introductory example would be exponential function $\exp(x) = e^x$. By definition, this is the function that is its own derivative, i.e. $\exp'(x) = \exp(x)$. That's all nice and swell from a mathematical stand point, and it makes it easy to prove interesting properties of the function. But how do you actually compute it? The definition above is ...


12

I feel it a bit strange that no one mentioned it, but a famous example is how Newton quickly estimated $\pi$ by the Taylor series. Here is a quick sketch: First note that the area of a quarter circle is given by the integral: $$ \int_0^1 \sqrt{1-x^2} dx = \frac{\pi}{4} \tag{1}$$ Now look at the left side.. $x<1$ .... which means... BINOMIAL EXPANSION TIME!...


6

I suggest looking at the deduction for the equation of a simple pendulum. While a differential equation is involved, it is very simple. And it is made simple by the fact that one can safely replace, for small enough $\theta$, the sine $\sin\theta$ with $\theta$.


5

Your question could apply generally to why should anyone learn the math "behind" anything, if they can easily compute the answer on a computer. I don't think they ALWAYS should. There should be a reason. For example, I had an old professor who said that when calculators first became common, some professors insisted that students should still learn ...


5

If they're not impressed by small errors, point out catastrophic cancellation. As a computer programmer, these issues are the kind of thing that we pay great attention to. We even modify the quadratic formula to avoid these problems. Where you really get in trouble is when you use equations that are built around exact assumptions, like "energy is ...


5

Following the link you have in your post I found an answer mentioning combinatorics. Formal power series (generating functions) are used often in probability and combinatorics. After a bit of search on the internet I found an interesting (for me) example. The coefficients of the expansion of $f(x)=\dfrac{x}{1-x-x^2}$ are the numbers in the Fibonacci series....


4

they can just run a simple simulation Simulations outside the classroom are often anything but simple. Brute force simulations are often a lot of work to set up, take a long time to calculate, and take even more computational resources to make sufficiently accurate. It is therefore necessary to have more advance tools in one's toolbox. Here is a very nice ...


4

I really like the idea of discovery fictions because they capture an essential component of how mathematics is best understood and communicated. Almost all ideas in mathematics follow simply and inexorably from previous ideas, and understanding any mathematical discipline consists almost entirely of figuring out how you could have developed these ideas on ...


3

I think that it makes sense to introduce continuity in the same lesson that you introduce limits. Here is a sketch of a lesson plan: Give them this link https://www.desmos.com/calculator/rlu2zgcjyf Group work: Is $f(2)$ defined or undefined? As $t$ approaches $2$, what are the values of $f(t)$ approaching? Make a table of values for $t = 1.9, 1.99, 1.999$ ...


3

Here is one more example: A student if often taught to memorize the l'hopital rule : Hawken you see a $\frac{0}{0}$, you plug in the numbers for the limit then algorithmically like a computer you must take derivative of numerator and denominator and then you have the true value of the limit. OK, but why does that work? A rigorous answer is difficult but ...


2

This may be too advanced for some students in a first-semester linear algebra course, but those who have had some Physics may be impressed by the diagonalization of a moment of inertia tensor. Take some irregularly-shaped 3-dimensional object, and form a $3 \times 3$ matrix describing all of the components of the moments of inertia around various axes; the ...


2

I assume your students have seen Linear Algebra. Remember in Linear Algebra how you sometimes have to solve $Ax = b$ for a matrix $A$ with more columns than rows? You usually get free variables, right? So after reducing to echelon form you have something like: $$\begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \...


2

In statistics, business and science, I keep hearing about how precision = consistency = low variability = closeness of the data points = reproducibility & repeatability. In numerical analysis, however, I think of precision not as a measure of data dispersion, but as a measure of the resolution of our recording or the instrument. In this sense, ...


2

Significance is precision which is reproducability. Accuracy is a degree of correctness. Consider two 1 foot rulers lying on a table. Both are marked in nominal inches: One ruler is made of Invar-42 low expansion nickel-iron alloy but is obviously only about 10 inches in length despite being numbered to 12. The other looks to be 12 inches in length but is ...


2

Another Newton example: Physics is full of differential equations, and one may ask how Newton dealt with them when most of the tips and tricks we know in the modern days were not yet discovered. Well, with his mighty series methods of course. In most introductory Physics classes, simple harmonic motion is introduced as: $$ \frac{d^2 x}{dt^2 } = -k x$$ For ...


1

Your question prompted me to reread a comment that John Tukey wrote in the March 1979 edition of the Journal of the American Statistical Association (pp. 121-122). He referred to what he calls "the whole data analyst-statistician." Such a person is able to "take quite different views and adopt quite different styles as the needs change." ...


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