The problem asks to approximate the change in $z$ using the total differential $dz$ as $(x, y)$ moves from point $P$ to point $Q$. Then, it asks to calculate the exact change $\Delta z$ using a calculator. The function is $z = 2x^2y^3$, and the points are $P(1, 1)$ and $Q(0.99, 1.02)$.

1. Problem Description

The problem asks to approximate the change in $z$ using the total differential $dz$ as $(x, y)$ moves from point $P$ to point $Q$. Then, it asks to calculate the exact change $\Delta z$ using a calculator. The function is $z = 2x^2y^3$, and the points are $P(1, 1)$ and $Q(0.99, 1.02)$.

2. Solution Steps

First, find the partial derivatives of $z$ with respect to $x$ and $y$.

$\frac{\partial z}{\partial x} = 4xy^3$

$\frac{\partial z}{\partial y} = 6x^2y^2$

Next, evaluate the partial derivatives at point $P(1, 1)$.

$\frac{\partial z}{\partial x}(1, 1) = 4(1)(1)^3 = 4$

$\frac{\partial z}{\partial y}(1, 1) = 6(1)^2(1)^2 = 6$

Then, find the changes in $x$ and $y$.

$\Delta x = dx = 0.99 - 1 = -0.01$

$\Delta y = dy = 1.02 - 1 = 0.02$

Now, calculate the total differential $dz$.

$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy$

$dz = 4(-0.01) + 6(0.02) = -0.04 + 0.12 = 0.08$

Next, compute the exact change $\Delta z$.

$z(1, 1) = 2(1)^2(1)^3 = 2$

$z(0.99, 1.02) = 2(0.99)^2(1.02)^3 = 2(0.9801)(1.061208) = 2(1.040016) \approx 2.080032$

$\Delta z = z(0.99, 1.02) - z(1, 1) = 2.080032 - 2 = 0.080032$

3. Final Answer

Approximate change in $z$: $dz = 0.08$

Exact change in $z$: $\Delta z = 0.080032$