Exercise 25

Let $g$ be a function that is differentiable at $(-2,5)$ and suppose that its tangent plane at this point is given by $z = -7 + 4(x+2) - 3(y-5)$.

1. Determine the values of $g(-2,5)$, $g_x(-2,5)$, and $g_y(-2,5)$. Write one sentence to explain your thinking.

2. Estimate the value of $g(-1.8, 4.7)$. Clearly show your work and thinking.

3. Given changes of $dx = -0.34$ and $dy = 0.21$, estimate the corresponding change in $g$ that is given by its differential, $dg$.

4. Suppose that another function $h$ is also differentiable at $(-2,5)$, but that its tangent plane at $(-2,5)$ is given by $3x + 2y - 4z = 9.$ Determine the values of $h(-2,5)$, $h_x(-2,5)$, and $h_y(-2,5)$, and then estimate the value of $h(-1.8, 4.7)$. Clearly show your work and thinking.