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# Tangent Lines and errors in approximations. watch

1. I have been emailing my lecturer, and he is rather bad (i think) at teaching. Although I am sure he is rather clever.

Can anyone decipher his babble and tell me what he is on about as I have been trying to figure it out and have no idea what he is saying.

He has written:

"For the function x^2, can you find delta x to guarantee
that the error in the function is not more than 0.1?

The word error has many meanings.
The error in the function means the variance of the value of the function
when you vary the argument within some neighbourhood of a fixed point.
For instance, you know the value f(a) at a simple point a, but
would like to estimate f(x) for some point close to a.
Then you consider the difference f(a+delta)-f(a).

Use the first derivative to find the variance
of the linear part (approximation) of f(x).
This gives a rough (linear) approximation to f(a+delta).
Use the second derivative to estimate the error (difference)
between the actual function f(x) and its linear approximation.
The resulting variance (error in f) is not greater than
the sum of these two variances (differences):
| f(a+delta) - linear part at (a+delta) | +
| linear part at (a+delta) - f(a) |.

The case of many variables is absolutely similar,
because all logic is based on 1-variable functions.

The linear part is the linear approximation, i.e. tangent line.
For f=x^2, the tangent line at x=a is the line y=a^2+2ax.
The variance of x^2 around x=a is (a+delta)^2-a^2=2a*delta+delta^2.
If this variance is less than 0.1, then choose delta such that
delta(2a+delta)<0.1, e.g. delta=min{0.1/3|a|,1} if a not 0."

Thank for any help given.

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