Final answer:
To estimate the values of f(7, -1), f(8, -2), and f(8, -1) using the given information, we can use linear approximation. The estimates are 11 for f(7, -1), 6 for f(8, -2), and 10 for f(8, -1).
Step-by-step explanation:
To estimate the values of f(7, -1), f(8, -2), and f(8, -1) using the given information, we can use the concept of linear approximation. The linear approximation formula is given by:
f(x, y) ≈ f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
where (a, b) is the given point and f_x(a, b) and f_y(a, b) are the partial derivatives at that point.
a) Estimating f(7, -1):
Using the formula, we have:
f(7, -1) ≈ f(7, -2) + f_x(7, -2)(7 - 7) + f_y(7, -2)(-1 - (-2))
f(7, -1) ≈ 7 + (-1)(0) + 4(1)
f(7, -1) ≈ 7 + 0 + 4
f(7, -1) ≈ 11
Therefore, the estimate of f(7, -1) is 11.
b) Estimating f(8, -2):
Using the formula, we have:
f(8, -2) ≈ f(7, -2) + f_x(7, -2)(8 - 7) + f_y(7, -2)(-2 - (-2))
f(8, -2) ≈ 7 + (-1)(1) + 4(0)
f(8, -2) ≈ 7 - 1 + 0
f(8, -2) ≈ 6
Therefore, the estimate of f(8, -2) is 6.
c) Estimating f(8, -1):
Using the formula, we have:
f(8, -1) ≈ f(7, -2) + f_x(7, -2)(8 - 7) + f_y(7, -2)(-1 - (-2))
f(8, -1) ≈ 7 + (-1)(1) + 4(1)
f(8, -1) ≈ 7 - 1 + 4
f(8, -1) ≈ 10
Therefore, the estimate of f(8, -1) is 10.