Consider the function
$$ z(x,y) = xy^2 + 3y.$$
Use the property of the partial derivaties to approximate the change in the function value with respect to $z(3,1)$ when $y$ increases with $\tfrac{1}{10}$ while $x$ is held constant.
$$ z(x,y) = xy^2 + 3y.$$
Use the property of the partial derivaties to approximate the change in the function value with respect to $z(3,1)$ when $y$ increases with $\tfrac{1}{10}$ while $x$ is held constant.
Antwoord 1 correct
Correct
Antwoord 2 optie
$\Delta z \approx 0.1.$
Antwoord 2 correct
Fout
Antwoord 3 optie
$\Delta z \approx 1.$
Antwoord 3 correct
Fout
Antwoord 4 optie
Cannot be determined, because the change in the input variable $x$ is not given.
Antwoord 4 correct
Fout
Antwoord 1 optie
$\Delta z \approx 0.9.$
Antwoord 1 feedback
Correct: The change in the function value is approximately equal to
$$\Delta z \approx z'_x(x_0,y_0) \Delta x + z'_y(x_0,y_0) \Delta y.$$
It is given that
$$(x_0,y_0) = (3,1), \qquad \Delta x = 0 \qquad \text{and} \qquad \Delta y = \tfrac{1}{10}.$$
The partial derivatives in $(3,1)$ are
$$
\begin{align*}
z'_x(x,y) &= y^2 &&& z'_x(3,1) &= 1^2 = 1,\\
z'_y(x,y) &= 2xy+3 &&& z'_y(3,1) &= 2\cdot3\cdot1 + 3 = 9.
\end{align*}
$$
Hence, the change in the function value is approximately equal to
$$\Delta z \approx z'_x(3,1)\Delta x + z'_y(3,1)\Delta y = 1 \cdot 0 + 9 \cdot \tfrac{1}{10} = \tfrac{9}{10} = 0.9.$$
Go on.
$$\Delta z \approx z'_x(x_0,y_0) \Delta x + z'_y(x_0,y_0) \Delta y.$$
It is given that
$$(x_0,y_0) = (3,1), \qquad \Delta x = 0 \qquad \text{and} \qquad \Delta y = \tfrac{1}{10}.$$
The partial derivatives in $(3,1)$ are
$$
\begin{align*}
z'_x(x,y) &= y^2 &&& z'_x(3,1) &= 1^2 = 1,\\
z'_y(x,y) &= 2xy+3 &&& z'_y(3,1) &= 2\cdot3\cdot1 + 3 = 9.
\end{align*}
$$
Hence, the change in the function value is approximately equal to
$$\Delta z \approx z'_x(3,1)\Delta x + z'_y(3,1)\Delta y = 1 \cdot 0 + 9 \cdot \tfrac{1}{10} = \tfrac{9}{10} = 0.9.$$
Go on.
Antwoord 2 feedback
Wrong: What is the change in $x$? And what is the change in $y$?
See Property partial derivatives, Example 1 and Example 2.
See Property partial derivatives, Example 1 and Example 2.
Antwoord 3 feedback
Antwoord 4 feedback
Wrong: You can deduce the change in the input variable $x$ from the given facts..
See possibly Property partial derivatives, Example 1 and Example 2 and try again.
See possibly Property partial derivatives, Example 1 and Example 2 and try again.