Find the linear approximation of f(x)=\ln x at x=1 and use it to estimate ln 1.12.
L(x)= . ?
ln 1.12 \approx ?
To find the linear approximation of a function, we use the formula:
L(x) = f(a) + f'(a)*(x - a)
where a is the point of approximation, f(a) is the value of the function at that point, and f'(a) is the derivative of the function at that point.
In this case, we want to find the linear approximation of f(x) = ln(x) at x = 1. Let's start by finding f(1) and f'(1).
f(x) = ln(x)
f(1) = ln(1) = 0
To find f'(x), we take the derivative of f(x).
f'(x) = 1/x
f'(1) = 1/1 = 1
Now that we have f(1) = 0 and f'(1) = 1, we can substitute these values into the linear approximation formula:
L(x) = f(a) + f'(a)*(x - a)
L(x) = 0 + 1*(x - 1)
L(x) = x - 1
So, the linear approximation of f(x) = ln(x) at x = 1 is L(x) = x - 1.
To estimate ln(1.12) using this linear approximation, we can substitute x = 1.12 into L(x):
ln 1.12 ≈ L(1.12)
ln 1.12 ≈ 1.12 - 1
ln 1.12 ≈ 0.12
Therefore, ln 1.12 is approximately 0.12.