Question

If f(2) = 10 and f'(x)=x³/(x 6), which of the following is the best approximation for f(2.03) using local linearization?

1) 10.06
2) 10.09
3) 10.12
4) 10.15

Answer 1

The best approximation for f(2.03) using local linearization is 2) 10.09.

Step-by-step explanation:

To approximate f(2.03) using local linearization, we can utilize the formula for local linearization, which is given by

Here, is the point at which we know the function value, and is the derivative of the function. In this case, First, we find by substituting into the derivative expression, resulting in Now, we can use the local linearization formula to approximate

Therefore, the best approximation for using local linearization is 10.075. Among the given options, 2) 10.09 is the closest value.

Local linearization is a powerful method for approximating values of a function near a known point. It involves using the linear equation of the tangent line to the graph of the function at that point. The accuracy of the approximation depends on how close the point of interest is to the known point, and it's a useful technique in calculus and mathematical mo deling for estimating values in the vicinity of known data points.