Find the linearization L(x) of the function f(x)=e^−2x at x=0. Please help
To find the linearization L(x) of a function at a given point, we need to use the formula for a linear approximation. The formula is given by:
L(x) = f(a) + f'(a)(x - a)
where f(a) represents the value of the function at the point a, and f'(a) represents the derivative of the function at the point a.
In this case, we have f(x) = e^(-2x) and we want to find the linearization at x = 0.
Step 1: Find the value of the function at x = 0
To find f(0), we simply substitute x = 0 into the function:
f(0) = e^(-2(0)) = e^0 = 1
Step 2: Find the derivative of the function
To find f'(x), we need to differentiate f(x) = e^(-2x) with respect to x. The derivative of e^(-2x) can be found using the chain rule:
f'(x) = -2e^(-2x)
Step 3: Substitute the values into the linear approximation formula
Now we have all the necessary values to calculate L(x):
L(x) = f(0) + f'(0)(x - 0)
L(x) = 1 + (-2e^(0))(x)
L(x) = 1 - 2x
Therefore, the linearization of f(x) = e^(-2x) at x = 0 is L(x) = 1 - 2x.