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Please sir, explain: "increasing and decreasing functions" and "Approximations"

An increasing function is one that has an upward slope.
For example,
 x for all x,
 3x-4 for all x,
 x^2 + 5 for all x >= 0, and
 tan(x) for all x where tan(x) is defined.

One function can be used to approximate any other function, though most of the time it is a poor approximation.  For example, x=0 is an approximation of sin(x), but a very poor approximation.
Over all, the error is always between -1 and 1, but it fits the curve poorly.

An approximation might be x for sin(x) near x=0, for they look pretty much the same near there.
When x is taken to be a little farther from 0 than 0.8, the approximation starts to get very poor (it goes outside of 10% error).

For the function e^x, if -0.7 < x < 1.1 and the approximation function is
f(x) = 1 + x + x/2 + x/6, then the error is within 3%.

The less accuracy is desired, the wider the limits could  be.
The more accuracy that is gotten, the better on function
is said to be an approximation of the other.

Increasing and decreasing are used in calculus, for when f'(x) is positive, f(x) is increasing and where f'(x) is negative, f(x) is decreasing.  When f'(x) = 0, there is a local maximum, minimum, or an inflection point.


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