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Chapter 8: Problem 2

Find the total differential, given (a) \(z=3 x^{2}+x y-2 y^{3}\) \((b) U=2 x_{1}+9 x_{1} x_{2}+x_{2}^{2}\)

Short Answer

Expert verified
The total differentials are: (a) \( dz = (6x + y) \, dx + (x - 6y^2) \, dy \), (b) \( dU = (2 + 9x_2) \, dx_1 + (9x_1 + 2x_2) \, dx_2 \).

Step by step solution

01

Understand the Problem

We need to find the total differential for two functions. The total differential of a function of two variables is a way to approximate how the function changes, taking into account changes in both variables.
02

Find Partial Derivatives for Part (a)

For the function given in part (a), \( z=3x^2+xy-2y^3 \), we first need to find the partial derivatives with respect to \( x \) and \( y \). The partial derivative with respect to \( x \) is \( \frac{\partial z}{\partial x} = 6x + y \). The partial derivative with respect to \( y \) is \( \frac{\partial z}{\partial y} = x - 6y^2 \).
03

Write the Total Differential for Part (a)

The total differential \( dz \) is given by: \[ dz = \frac{\partial z}{\partial x} \cdot dx + \frac{\partial z}{\partial y} \cdot dy \] Substituting the partial derivatives: \( dz = (6x + y) \, dx + (x - 6y^2) \, dy \).
04

Find Partial Derivatives for Part (b)

For the function given in part (b), \( U = 2x_1 + 9x_1x_2 + x_2^2 \), we need to find the partial derivatives with respect to \( x_1 \) and \( x_2 \). The partial derivative with respect to \( x_1 \) is \( \frac{\partial U}{\partial x_1} = 2 + 9x_2 \). The partial derivative with respect to \( x_2 \) is \( \frac{\partial U}{\partial x_2} = 9x_1 + 2x_2 \).
05

Write the Total Differential for Part (b)

The total differential \( dU \) is given by: \[ dU = \frac{\partial U}{\partial x_1} \cdot dx_1 + \frac{\partial U}{\partial x_2} \cdot dx_2 \] Substituting the partial derivatives: \( dU = (2 + 9x_2) \, dx_1 + (9x_1 + 2x_2) \, dx_2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives are a cornerstone concept in multivariable calculus, especially when dealing with functions of more than one variable. They represent how a function changes with respect to one of its variables, while the other variables are held constant.
  • For a function of two variables, such as the one given in part (a) \( z = 3x^2 + xy - 2y^3 \), partial derivatives are calculated with respect to \( x \) and \( y \).
  • The process involves treating one variable as constant while differentiating with respect to the other.
  • In step 2 of the solution, the partial derivative of \( z \) with respect to \( x \) is \( \frac{\partial z}{\partial x} = 6x + y \) and with respect to \( y \) is \( \frac{\partial z}{\partial y} = x - 6y^2 \).
  • This shows how \( z \) changes if just \( x \) or \( y \) varies independently.
Partial derivatives help in constructing the total differential, showing the cumulative effect of independent changes in each variable.
Multivariable Calculus
Multivariable calculus is a branch of mathematics that deals with functions of more than one variable. It extends the concepts of differential calculus to higher dimensions.
  • Functions in multivariable calculus can describe complex systems involving multiple interacting variables.
  • In this exercise, functions with two variables like \( z = 3x^2 + xy - 2y^3 \) and \( U = 2x_1 + 9x_1x_2 + x_2^2 \) demonstrate multivariable functions.
  • To study these functions, we explore properties like partial derivatives and total differentials.
  • Using these tools, we determine how small changes in multiple variables can affect the overall outcome of the function.
This multivariable approach allows us to analyze and solve problems in science and engineering, where phenomena often depend on multiple factors.
Differential Approximation
Differential approximation is a method used to estimate the change in a function due to small changes in its variables. Using total differentials provides a powerful tool for this approximation.
  • The total differential for a function gives an approximation of the change in the function as the variables undergo small changes.
  • In part (a), the total differential \( dz \) is expressed as \( dz = (6x + y) \, dx + (x - 6y^2) \, dy \).
  • This expression combines changes in both \( x \) and \( y \) to approximate the resultant change in \( z \).
  • Similarly, in part (b), the total differential \( dU = (2 + 9x_2) \, dx_1 + (9x_1 + 2x_2) \, dx_2 \) provides an approximation for changes in \( U \).
This method is particularly useful in scientific and engineering contexts, where it helps in predicting how systems respond to small perturbations.

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Most popular questions from this chapter

Given \(Q=100-2 P+0.02 Y\), where \(Q\) is quantity demanded, \(P\) is price, and \(Y\) is income, and given \(P=20\) and \(Y=5,000\), find the (a) Price elasticity of demand. (b) Income elasticity of demand.

For each \(F(x, y, z)=0\) use the implicit-function rule to find \(3 y / \partial x\) and \(\partial y / \partial z\) $$(a) F(x, y, z)=x^{2} y^{3}+z^{2}+x y z=0$$ $$(b) F(x, y, z)=x^{3} z^{2}+y^{3}+4 x y z=0$$ $$(c) F(x, y, z)=3 x^{2} y^{3}+x z^{2} y^{2}+y^{3} z x^{4}+y^{2} z=0$$

Let the demand and supply functions for a commodity be $$\begin{array}{ll}Q_{d}=D\left(P, Y_{0}\right) & \left(D_{p}<0 ; \quad D_{Y_{0}}>0\right) \\ Q_{s}=S\left(P, T_{0}\right) & \left(S_{p}>0: \quad S_{T_{0}}<0\right)\end{array}$$ where \(Y_{0}\) is income and \(T_{0}\) is the tax on the commodity. All derivatives are continuous. (a) Write the equilibrium condition in a single equation. (b) Check whether the implicit-function theorem is applicable. If so, write the equilibrium identity. (c) Find \(\left(\partial P^{*} / \partial Y_{0}\right)\) and \(\left(\partial P^{*} / \partial T_{0}\right),\) and discuss their economic implications. (d) Using a procedure similar to \((8.37),\) find \(\left(\partial Q^{*} / \partial Y_{0}\right)\) from the supply function and \(\left(\partial Q^{*} / \partial T_{0}\right)\) from the demand function. (Why not use the demand function for the former, and the supply function for the latter?)

For each \(F(x, y)=0\) use the implicit-function rule to find \(d y / d x\) $$(a) F(x, y)=3 x^{2}+2 x y+4 y^{3}=0$$ $$(b) F(x, y)=12 x^{5}-2 y=0$$ $$(c) F(x, y)=7 x^{2}+2 x y^{2}+9 y^{4}=0$$ $$(d) F(x, y)=6 x^{3}-3 y=0$$

Find the point elasticity of demand, given \(Q=k / \rho^{n},\) where \(k\) and \(n\) are positive constants. (a) Does the elasticity depend on the price in this case? (b) In the special case where \(n=1\), what is the shape of the demand curve? What is the point elasticity of demand?
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