For each function below find the best linear approximation (linearization) at the given value.
f(x) = x^-1 at x = 2
y = 1/2
f ' (x) = -1x^-2 at x = 2 : -1/4
y - 1/2 = -1/4 (x-2)
y - 1/2 = -1/4x + 1/2
y = -1/4x + 1
Use the appropriate linearization above to find approximate values for the following, then use your calculator to find the exact values and compute the error in your approximation.
1/2.04
The error is 0.0001961 but I'm not sure how to get it. Can someone please help me?
well, the error is the approximate value minus the real value.
To find the approximate value using the linearization, substitute 1/2.04 into the linear function:
y = -1/4x + 1
Substituting x = 1/2.04:
y = -1/4 * (1/2.04) + 1
y = -1/8.16 + 1
y ≈ 0.8789
To find the exact value using a calculator:
1/2.04 ≈ 0.4901961
Now, we can calculate the error in the approximation:
Error = Exact value - Approximate value
Error = 0.4901961 - 0.8789
Error ≈ -0.3887039
The error in the approximation is approximately -0.3887039.