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I am in need of some help with an old question of mine (got it wrong on homework a while ago- I was out sick the day this material was taught- never quite got the question).

Find the linearlization L(x) of f(x)=tan(x) near x=pi/4

To find the linearization of a function at some point, we need the slope of the function at that point.  Since x=pi/4 and f(x) = tan(x), it is known that y = tan(pi/4) = root(2)/2.

Since the derivative of f(x) = tan(x) is f'(x) = secē(x) and x is pi/4.
Since sec(pi/4) = sqrt(2), secē(pi/4) = 2.

This gives the equation of the line in point slope form.
It is y - root(2)/2 = 2(x - pi/4).
This reduces to y = 2x  + (root(2) - pi)/2.


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