Homework 05

Assessment: Each item is worth 1 point. The total for this homework is 20 points. There are no partial points. The best, rather than just correct, solutions are presented well and have a logical flow in the reasoning. Write the solutions for yourself, your future self, and an audience.

If you want to have your homework checked, please send an email to oer.dlsu.soe@gmail.com.

Exercise A

Let \(f(x,y)=x^2+2xy+y^2\).

  1. Compute the following Newton quotients, where both \(h\neq 0\) and \(k\neq 0\): \[\frac{f(x+h,y)-f(x,y)}{h},\ \frac{f(x,y+k)-f(x,y)}{k}\]
  2. Compute \(f^\prime_x\) and \(f^\prime_y\) using the appropriate limits.

Exercise B

Consider the function \(f(x,y)=x^3y+x^2y^2+x^2y^2+x+y^2\).

  1. Use the 3D graphical capabilities of Desmos to visualize this function. Explore the functionality of Desmos.
  2. Use the 2D graphical capabilities of Desmos to visualize the level curves of \(f\).
  3. Connect the level curves you have visualized to the slices of \(f\) parallel to the \(xy\)-plane.

Exercise C

This exercise gives you a chance to practice computing partial derivatives by using the rules of derivatives developed before.

  1. Find \(f_x^\prime\) if \(f(x,y)=e^{xy}\).
  2. Find \(f_s^\prime\) if \(f(s,t)=As^at^b\), where \(A\), \(a\), and \(b\) are constants.
  3. Find \(f_y^\prime\) if \(f(x,y)=\sqrt{1-3x^2-y^3}\).
  4. Find \(f_x^\prime\) and \(f_y^\prime\) if \(f(x,y)=e^x\ln(1+y)\)
  5. Find \(f_K^\prime\) if \(f(K,L)=(K^a+L^a)^{1/a}\), where \(a\) is a constant.
  6. Find \(f_x^\prime\) if \(f(x,y)=\left(\dfrac{x^2-y^2}{x^2+y^2}\right)^h\), where \(h\) is a constant.

Exercise D

The idea of elasticity can be extended to functions of two variables. Define the partial elasticity of \(z=f(x,y)\) with respect to \(x\) as \[\mathsf{el}_x z = \frac{x}{z}\cdot\frac{\partial z}{\partial x}.\]

  1. Define the partial elasticity of \(z\) with respect to \(y\).

  2. Find both partial elasticities if

    • \(z=x+y\)
    • \(z=x^ay^b\), where \(a\) and \(b\) are constant

  3. The demand for a product typically depends on the price of the product itself and the price of a substitute for that product. Let \(p_1\) and \(p_2\) be those prices, respectively. Let that demand function be given by \(D(p_1, p_2)=a-bp_1p_2^{-\alpha}\) where \(a\), \(b\), and \(\alpha\) are positive constants and \(\alpha<1\).

    • Find the partial elasticities of demand with respect to each of these prices.
    • What are the signs of these partial elasticities? Comment on the economic meaning of these partial elasticities.

Exercise E

Consider the function in Exercise B.

  1. Find \(f_x^\prime\) and \(f_y^\prime\). Are these also functions of two variables \(x\) and \(y\)?
  2. You have two first-order partial derivatives, similar to our discussion before first derivatives. How would you think about the idea of second derivatives in the context of partial differentiation?

Exercise F

  1. Return to Exercise C and check your answers using SymPy. As demonstration, consider the function in Exercise A.
from sympy import *
x, y = symbols('x y', real = True)
expr = x**2+2*x*y+y**2
diff(expr, x)

\(\displaystyle 2 x + 2 y\)

diff(expr, y)

\(\displaystyle 2 x + 2 y\)

  1. Verify using SymPy the partial elasticities you have computed in Exercise D.