Differentiating the variable held fixed

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

1. 01

   Let $f(x, y) = x\,e^{2y} - 3x^{2}y + y^{4}$. Find $f_x(x, y)$.

   ·Functions of two variables and partial derivatives·Introductory

2. 02

   Let $f(x, y) = \left(x + y^{2}\right)e^{xy}$. Find $f_y(2, 1)$.

   ·Functions of two variables and partial derivatives·Standard

3. 03

   A firm’s output is $Q(K, L) = 12K^{1/4}L^{3/4}$ units, where $K$ is units of capital and $L$ is units of labour. It now uses $K = 16$ and $L = 81$. Which statement correctly gives and interprets the marginal product of labour, $\dfrac{\partial Q}{\partial L}$, there?

   ·Functions of two variables and partial derivatives·Standard

4. 04

   Let $f(x, y) = \arctan\!\left(\dfrac{y}{x}\right)$ for $x > 0$. Find $f_y(x, y)$.

   ·Functions of two variables and partial derivatives·Standard

5. 05

   A company makes two models of e-bike, which compete for the same buyers. To sell $x$ of the first model and $y$ of the second in a month, it must charge $p_{1} = 2000 - 4x - y$ dollars for the first and $p_{2} = 1600 - x - 3y$ dollars for the second. Its monthly revenue is $R(x, y) = x\,p_{1} + y\,p_{2}$ dollars. Find the marginal revenue from the first model, $\dfrac{\partial R}{\partial x}$.

   ·Functions of two variables and partial derivatives·Standard

6. 06

   The surface $z = f(x, y) = 9 - x^{2} + xy - y^{2}$ is sliced by the vertical plane $y = 1$. The slice is the curve $z = f(x, 1)$, drawn below with $x$ across and $z$ up. Which statement about the slice at the marked point, where $x = 2$, is correct?

   ·Functions of two variables and partial derivatives·Standard