Find the linear approximation of f(x)=\ln x at x=1 and use it to estimate \ln 1.47.
To find the linear approximation of a function at a specific point, we can use the tangent line to the graph of the function at that point. This is given by the equation of a straight line:
L(x) = f(a) + f'(a) * (x - a)
Where,
- L(x) is the linear approximation of f(x),
- f(a) is the value of the function at point x = a,
- f'(a) is the derivative of the function evaluated at x = a, and
- (x - a) represents the difference between the given point x and the point of approximation a.
In this case, we want to find the linear approximation of f(x) = ln(x) at x = 1.
Step 1: Calculate f(a)
For a = 1, f(a) = ln(1) = 0, since the natural logarithm of 1 is always 0.
Step 2: Calculate f'(a)
We need to find the derivative of f(x) = ln(x). Using the derivative rule for natural logarithm, we have:
f'(x) = 1 / x
Then, evaluate f'(1) as follows:
f'(1) = 1 / 1 = 1
Step 3: Plug the values into the linear approximation formula
We can now use the values of f(a) and f'(a) in the linear approximation formula to obtain L(x):
L(x) = f(a) + f'(a) * (x - a)
= 0 + 1 * (x - 1)
= x - 1
The linear approximation of f(x) = ln(x) at x = 1 is L(x) = x - 1.
To estimate ln(1.47) using this linear approximation, substitute x = 1.47 into the linear approximation equation:
L(1.47) = (1.47) - 1
= 0.47
Therefore, the estimated value of ln(1.47) using the linear approximation is approximately 0.47.