How do you find the linearization of $f(x) = e^(5 x)$ at the point x=0?

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Answer 1 · 2016-12-29T23:14:08

$e^(5x) ~= 1+5x$

The linearization of $f(x)$ is its approximation using the tangent line:

$y(x) = f(x_0) + f'(x_0)(x-x_0)$

in our case:

$f(x) = e^(5x)$
$f'(x) = 5e^(5x)$

So for $x_0=0$

$f(0) = 1$
$f'(0) = 5$

and the approximation is:

$e^(5x) ~= 1+5x$

How do you find the linearization of f(x) = e^(5 x)f(x) = e^(5 x) at the point x=0?