Question

Assume points P and Q are close. Estimate g(Q). Round your answer to 3 decimal places. P = (5, 8), Q = (4.97, 7.99), g(P) = 7, gx(P) = −0.3, gy(P) = −0.4.

Answer 1

To estimate g(Q), we can use the partial derivatives gx(P) and gy(P) to find the rate of change of g with respect to x and y at point P. We can then use the coordinates of point Q to estimate the change in g at that point. By multiplying the rate of change of g with respect to x and y by the change in coordinates, we can estimate g(Q) to be approximately 7.013.

Step-by-step explanation:

To estimate g(Q), we can use the partial derivatives gx(P) and gy(P) to find the rate of change of g with respect to x and y at point P. We can then use the coordinates of point Q to estimate the change in g at that point.

First, let's find the rate of change of g with respect to x and y at point P:

delta_g_x = gx(P) = -0.3

delta_g_y = gy(P) = -0.4

Next, let's find the change in coordinates from P to Q:

delta_x = Q_x - P_x = 4.97 - 5 = -0.03

delta_y = Q_y - P_y = 7.99 - 8 = -0.01

Finally, we can estimate g(Q) by multiplying the rate of change of g with respect to x and y by the change in coordinates:

g(Q) = g(P) + delta_g_x * delta_x + delta_g_y * delta_y

g(Q) = 7 + (-0.3) * (-0.03) + (-0.4) * (-0.01) = 7 + 0.009 + 0.004 = 7.013

Therefore, the estimated value of g(Q) is approximately 7.013.

Answer 2

To estimate g(Q), use the formula g(Q) = g(P) + gx(P) * (Qx - Px) + gy(P) * (Qy - Py). The value of g(Q) is approximately 7.009.

Step-by-step explanation:

To estimate g(Q), we can use the formula g(Q) = g(P) + gx(P) * (Qx - Px) + gy(P) * (Qy - Py), where P and Q are points and g(P), gx(P), and gy(P) are given values.

Plugging in the values from the question, we have g(Q) = 7 + (-0.3) * (4.97 - 5) + (-0.4) * (7.99 - 8).

Simplifying the equation, we get g(Q) = 7 + (-0.3) * (-0.03) + (-0.4) * (-0.01).

Calculating further, we find that g(Q) is approximately 7.009.