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Chapter 12: Problem 64

Power functions and percent change Suppose that \(z=f(x, y)=x^{a} y^{b},\) where \(a\) and \(b\) are real numbers. Let \(d x / x, d y / y,\) and \(d z / z\) be the approximate relative (percent) changes in \(x, y,\) and \(z,\) respectively. Show that \(d z / z=a(d x) / x+b(d y) / y ;\) that is, the relative changes are additive when weighted by the exponents \(a\) and \(b.\)

Short Answer

Expert verified
Question: Prove that for a power function of the form \(z = f(x,y) = x^a y^b\), with \(a\) and \(b\) as real numbers, the relative changes in \(x\), \(y\), and \(z\), weighted by the exponents \(a\) and \(b\), are additive. In other words, show that \(dz/z = a(dx/x) + b(dy/y)\). Answer: To prove this, we first express \(dz\) using the chain rule as \(dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy\). Then, we calculate the partial derivatives of \(z\) with respect to \(x\) and \(y\) and substitute them into the expression for \(dz\). Next, we divide both sides of the equation by \(z = x^a y^b\) and simplify the expression to obtain the desired result: \(dz/z = a(dx/x) + b(dy/y)\). This shows that the relative changes are additive when weighted by the exponents \(a\) and \(b\).

Step by step solution

01

Express dz using the chain rule

Using the chain rule in multivariable calculus, we find the total derivative of \(z\) with respect to \(x\) and \(y\). If \(z = f(x,y) = x^a y^b\), then \(dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy\).
02

Calculate the partial derivatives of z with respect to x and y

To proceed with Step 1, we first need the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). Calculate these as follows: \(\frac{\partial z}{\partial x} = \frac{\partial}{\partial x}(x^a y^b) = ax^{a-1}y^b\) \(\frac{\partial z}{\partial y} = \frac{\partial}{\partial y}(x^a y^b) = bx^{a}y^{b-1}\)
03

Substitute the partial derivatives into the expression for dz

Now we substitute the partial derivatives found in Step 2 into the expression for \(dz\) obtained in Step 1: \(dz = ax^{a-1}y^b dx + bx^{a}y^{b-1}dy\)
04

Divide through by z

In order to obtain the expression for \(dz/z\), divide both sides of the equation in Step 3 by \(z = x^a y^b\): \(\frac{dz}{z} = \frac{ax^{a-1}y^b dx + bx^{a}y^{b-1}dy}{x^a y^b}\)
05

Simplify the expression

Simplify the expression obtained in Step 4: \(\frac{dz}{z} = \frac{a dx}{x} + \frac{b dy}{y}\) The simplified expression is \(dz/z = a(dx/x) + b(dy/y)\), which shows that the relative changes are additive when weighted by the exponents \(a\) and \(b\).

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Key Concepts

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

Power Functions
Power functions are mathematical expressions of the form f(x) = x^n, where n is a real number known as the exponent. In multivariable calculus, these functions can extend to multiple variables, such as f(x, y) = x^a y^b where a and b are the exponents for the variables x and y, respectively. Power functions with multiple variables are essential for modeling various phenomena where the relationship between the variables is not simply additive or proportional.

When working with power functions, it's important to understand how small changes in the variables (such as dx or dy) affect the overall function. This is especially crucial in applied fields like economics, where slight variations in inputs can result in significant changes in output.
Relative Percent Change
Relative percent change measures how much a quantity changes in proportion to its initial value, expressed as a percentage. It is calculated by the formula (Change in value) / (Original value). Relative percent change is a key concept in understanding the dynamics of systems described by power functions, as it captures the variability of the system's output in response to changes in its inputs.

In the context of power functions like f(x, y) = x^a y^b, the relative percent changes of the variables x and y with respect to changes in their respective values are denoted as dx/x and dy/y. This notation allows us to relate the percentage changes directly to the change in the overall function, dz/z, in a manner that highlights the proportional impact of each variable.
Partial Derivatives
Partial derivatives are a cornerstone of multivariable calculus. Instead of finding the rate of change of a function with respect to a single variable, as in regular derivatives, partial derivatives find how the function changes with respect to one variable while holding other variables constant. For a function like f(x, y) = x^a y^b, the partial derivatives with respect to x and y are found by treating y and x as constants, respectively.

The notation 鈭俩/鈭倄 and 鈭俩/鈭倅 signifies finding the partial derivatives of the function z with respect to x and y. When applied to power functions, the rules for derivatives are similarly employed; however, the power of the held-constant variable is treated as a coefficient.
Chain Rule
The chain rule is an essential rule in calculus for differentiating compositions of functions. In multivariable calculus, the chain rule can be used to take the derivative of a function with respect to multiple variables. When functions depend on more than one variable, the total derivative includes contributions from each partial derivative.

In the problem at hand, we apply the chain rule to find the derivative of z with respect to both x and y. This approach combines the effects of changing x and y on z into a single expression dz, which then allows us to explore how the overall function z changes. Understanding the chain rule is crucial for assessing multivariable systems where the output depends on multiple, interconnected factors. It enables one to quantify how changes in inputs collectively influence the output.

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Most popular questions from this chapter

Let \(h\) be continuous for all real numbers. a. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{x}^{y} h(s) d s\). b. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{1}^{x y} h(s) d s\).

The angle between two planes is the angle \(\theta\) between the normal vectors of the planes, where the directions of the normal vectors are chosen so that \(0 \leq \theta<\pi\) Find the angle between the planes \(5 x+2 y-z=0\) and \(-3 x+y+2 z=0\)

Find an equation of the plane passing through (0,-2,4) that is orthogonal to the planes \(2 x+5 y-3 z=0\) and \(-x+5 y+2 z=8\)

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14 .) The gravitational potential associated with two objects of mass \(M\) and \(m\) is \(\varphi=-G M m / r,\) where \(G\) is the gravitational constant. If one of the objects is at the origin and the other object is at \(P(x, y, z),\) then \(r^{2}=x^{2}+y^{2}+z^{2}\) is the square of the distance between the objects. The gravitational field at \(P\) is given by \(\mathbf{F}=-\nabla \varphi,\) where \(\nabla \varphi\) is the gradient in three dimensions. Show that the force has a magnitude \(|\mathbf{F}|=G M m / r^{2}\) Explain why this relationship is called an inverse square law.

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) The data may be plotted as a scatterplot in the \(x y\) -plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Generalize the procedure in Exercise 70 by assuming that \(n\) data points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) are given. Write the function \(E(m, b)\) (summation notation allows for a more compact calculation). Show that the coefficients of the best-fit line are $$ \begin{aligned} m &=\frac{\left(\sum x_{k}\right)\left(\sum y_{k}\right)-n \sum x_{k} y_{k}}{\left(\sum x_{k}\right)^{2}-n \sum x_{k}^{2}} \text { and } \\ b &=\frac{1}{n}\left(\sum y_{k}-m \Sigma x_{k}\right) \end{aligned}, $$ where all sums run from \(k=1\) to \(k=n\).
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