The circles are my drawings on top of the lines. We do not need any rigor here just an intuitive feel that as the lines get closer and closer the point of intersection becomes fuzzier. For instance if all three drawings were on graph paper it would be easy to read off the point of intersection for the first one with better accuracy than the third one.

Yes, it represents a certain amount of error in the calculation. Like your eyes, algorithms that try to find the points of intersection of the third example will experience instability and have more round off error and consequently give poorer results. So lines that cross at nearly right angles are more accurate in a sense than lines that cross at low angles. That is the point.

When we curve fit using ordinary polynomials x, x^2, x^3, x^4, x^5,...as the basis we can see by graphing how much they are like example 3. Look at that mess around the origin. All of them on top of each other. That is why it is not recommended to curve fit a function using powers higher than say 10. The accumulated error makes them very difficult to get accurate results.

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Hi;

When we use an orthogonal basis we get this plot. Notice the high angles of intersection.

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Taylor series are not orthogonal but they are osculating so they have some benefits.

Now in the plot of e^x and the Fourier series you will not see the orthogonal basis in the plot itself. They are present in the formation of the series.

That I do not know for sure. The Fourier fit is also least squares or minimax, I am not sure.

Least squares minimize the square of the error between the fit equation and the data. First you start with an overdetermined system.

It happens automatically. It is inherent in the fit.

Most of the time I use Mathematica. Today algebra is sort of like square roots of numbers. If you had to evaluate √ (234.176253) you would turn to your calculator. If you needed to multiply 102536 * 776241 you would turn to your calculator. At one time people did them both with pencil and paper. Same thing now with mechanical symbolic math. When you are learning do it by hand, when you know it use a CAS!

That is always a danger and some of that will definitely happen. Everyone has slowed down with mental calculation.

What is the upside?

1) You are freed up from the drudgery that takes much time.

2) You can think about the problem without getting sidetracked with the details of solving it.

3) By getting passed things you could not get passed before you now can delve into deeper mathematics because you can use your package to help read journal articles. Incidentally, they do not admit it in the article but most journal articles are now written with the help of a CAS.

4) And this is a big one you never make a careless mistake that blows a solution away. For instance you might have wondered about that faulty solution for many months.