Let
𝑓
(
𝑥
,
𝑦
)
=
20
−
𝑥
2
−
7
𝑦
2
. Use the linear Taylor approximation of
𝑓
(
𝑥
,
𝑦
)
about
(
2
,
1
)
to estimate the value of
𝑓
(
𝑥
,
𝑦
)
𝑎
𝑡
(
1.5
,
2
)
.
What is the estimated value of
𝑓
(
𝑥
,
𝑦
)
at
(
1.5
,
2
)
?

Note 1: There is an error margin of 0.5 in this question. That is, if the answer is 7.42, then any answer in the interval [6.92, 7.92] will be accepted as correct.

Note 2: Just write the number in the space provided below. For example, if the estimated value is 6.6 when x=5, then just write: 6.6

Question

Let 
𝑓
(
𝑥
,
𝑦
)
=
20
−
𝑥
2
−
7
𝑦
2
. Use the linear Taylor approximation of 
𝑓
(
𝑥
,
𝑦
)
about 
(
2
,
1
)
to estimate the value of  
𝑓
(
𝑥
,
𝑦
)
𝑎
𝑡
(
1.5
,
2
)
.
What is the estimated value of 
𝑓
(
𝑥
,
𝑦
)
at 
(
1.5
,
2
)
?

Note 1: There is an error margin of 0.5 in this question. That is, if the answer is 7.42, then any answer in the interval [6.92, 7.92] will be accepted as correct.

Note 2: Just write the number in the space provided below. For example, if the estimated value is 6.6 when x=5, then just write: 6.6

BlackTom AI · Accepted Answer

We start by identifying the function and the point around which we linearize.
f(x,y) = 20 - x^2 - 7y^2, and we will use the linear Taylor approximation about (a,b) = (2,1).

Next, evaluate the function at the center: f(2,1) = 20 - (2)^2 - 7*(1)^2 = 20 - 4 - 7 = 9.

Compute the partial derivatives at (2,1):
f_x = ∂f/∂x = -2x, so f_x(2,1) = -4.
f_y = ∂f/∂y = -14y, so f_y(2,1) = -14.

Determine the changes in x and y from the center to the target point (x,y) = (1.5, 2):
dx = x - a = 1.5 - 2 = -0.5,
dy = y - b = 2 - 1 = 1.

Apply the linear approximation:
f(x,y) ≈ f(a,b) + f_x(a,b)*dx + f_y(a,b)*dy
= 9 + (-4)*(-0.5) + (-14)*(1).

Calculate each term: (-4)*(-0.5) = 2, and (-14)*(1) = -14.
Thus, f(1.5,2) ≈ 9 + 2 - 14 = -3.

Interpretation: The linear approximation uses the value at (2,1) and adjusts by the slope in the x and y directions scaled by how far we move from (2,1). The computed estimate at (1.5,2) is -3, reflecting how the function decreases with y (since f_y is negative) and increases modestly with a small move left in x (since f_x is negative, moving x left increases f slightly).

Note: The actual function value at (1.5,2) is f(1.5,2) = 20 - (1.5)^2 - 7*(2)^2 = 20 - 2.25 - 28 = -10.25, which deviates from the linear estimate, illustrating the approximation’s limited accuracy when moving farther from the expansion point.

ECON6003/ECON6703 Mathematical Methods of Econ Analysis Assignment 2

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