偏导数与等值线

One variable at a time

Level curves as landscapes

The slope recipe

Reading steepness economically

What does the partial derivative $\frac{\partial f}{\partial x_1}$ hold fixed, and why is that the right concept for bundles of goods?

For $f(x_1, x_2) = x_1 x_2$, what are the partial derivatives at the bundle $(3, 4)$?

What is the slope of the level curve of $f = x_1 x_2$ through the bundle $(3, 4)$, where the curve is $x_1 x_2 = 12$?

Compare the slope of $x_1 x_2 = 12$ at $(8, 1.5)$ with the slope at $(3, 4)$. What does the change say economically?

For $f(x_1, x_2) = x_1 x_2$, compute the partial derivative $\frac{\partial f}{\partial x_1}$ at the bundle $(3, 4)$, and write the number.

The level curve of $f = x_1 x_2$ through $(3, 4)$ has equation $x_1 x_2 = k$. Compute the level value $k$, and write the number.

Which of these bundles does NOT lie on the level curve x1 times x2 equal to 12?

Why is the slope of a level curve negative when both goods are desirable?

At (3, 4) the slope is minus 4 thirds, and at (8, 1.5) it is about minus 0.19. What economic reading fits?

In the slope formula minus partial f over partial x1 divided by partial f over partial x2, what does the numerator count?

At the bundle (8, 1.5) the slope formula gives minus 1.5 over 8. Compute the magnitude to three decimals and write it.

For f of x1 times x2, compute the partial with respect to x2 at the bundle (5, 6), and write the number.