Answer to Question #114013 in Calculus for Lizwi

Question

What is the first approximation of the function f(x)=(1−x)^100 near x=0?

Answer 1

It is necessary to find the first approximation of the function $f(x) = (1-x)^{100}$ near x=0?

The first approximation of the function is a line with the equation $y- y_0=k(x-x_0),$ where $x_0=0$ by the condition. $y_0=f(x_0 )=(1-0)^{100}=1.$

To find k we must find the first derivative of f(x) and substitute $x_0=0$. So:

$\frac{df(x)}{dx}=-100(1-x)^{99}$

Then:

$k=\frac{df(x_0)}{dx}=-100(1-0)^{99}=-100$

Write an equation of the straight line:

$y-1=-100x$

Answer: the first approximation of the function $f(x) = (1-x)^{100}$ is the line $y=-100x+1$.