To find the linear approximation of the function f(x) = 1/x at a point, we typically use the formula:
L(x) = f(a) + f'(a)(x – a)
where:
- L(x) is the linear approximation
- f(a) is the value of the function at point a
- f'(a) is the derivative of the function at point a
- x is the variable
- a is the point at which we are approximating the function
However, it’s essential to note that the function f(x) = 1/x is undefined at x = 0. Therefore, we cannot evaluate f(0) or f'(0). As such, we need to choose a point near x = 0 but not at x = 0. Let’s consider using a = 1 for our approximation.
1. Calculate f(1):
f(1) = 1/1 = 1
2. Now, compute the derivative f'(x):
f'(x) = -1/x^2
3. Evaluate the derivative at a = 1:
f'(1) = -1/1^2 = -1
4. Now we can plug these values into our linear approximation formula:
L(x) = f(1) + f'(1)(x – 1)
L(x) = 1 – 1(x – 1)
L(x) = 1 – (x – 1) = 2 – x
Thus, the linear approximation of the function f(x) = 1/x near the point x = 1, which is the closest point to x = 0, is:
L(x) = 2 – x
In summary, when dealing with functions that are undefined at the point of interest, it’s crucial to select a nearby point for approximation to ensure accurate results.