Question

A researcher is trying to determine how class attendance and sleep impacts student test scores. From her data, she generates a function G = f(x, z) that approximates the grade a student gets on an exam when the student attends x lectures and sleeps for z hours. Some values of G = f(x, z) are given in the table below.

A. Estimate fₓ(6,8) using a central difference.

B. Interpret the meaning of fₓ(6,8) in the context of the situation.

Answer 1

fx(6,8) is calculated using the central difference method, it estimates the rate of change of a student's exam grade with respect to number of lectures attended, keeping the hours of sleep constant. This derivative represents the sensitivity of exam grades to class attendance.

Step-by-step explanation:

The student question is focused on partial derivatives, which is a concept in multivariate calculus. The function G = f(x, z) represents the grade a student achieves based on two variables: x (the number of class lectures attended) and z (the number of hours slept).

To estimate fx(6,8) using a central difference, we would need two points that are 'near' x = 6 while holding z constant. Assuming that the researcher has provided values of G for x = 5 and x = 7 at z = 8, we can calculate the central difference with the formula:

fx(6,8) ≈ [f(7, 8) - f(5, 8)] / (7 - 5)

Given the values from the table for f(7, 8) and f(5, 8), this computation will provide an estimated rate of change of the grade with respect to the number of lectures attended, while keeping the number of hours slept constant.

The interpretation of fx(6,8) in this context is that it represents the estimated change in a student's exam grade when they attend one more lecture from 6 lectures, assuming the amount of sleep (8 hours) remains the same. It indicates how sensitive the grade is to changes in class attendance.