/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Let \(w=f(x, y, z)=2 x+3 y+4 z\)... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 69

Let \(w=f(x, y, z)=2 x+3 y+4 z\) which is defined for all \((x, y, z)\) in \(\mathbb{R}^{3}\). Suppose that we are interested in the partial derivative \(w_{x}\) on a subset of \(\mathbb{R}^{3}\), such as the plane \(P\) given by \(z=4 x-2 y .\) The point to be made is that the result is not unique unless we specify which variables are considered independent. a. We could proceed as follows. On the plane \(P\), consider \(x\) and \(y\) as the independent variables, which means \(z\) depends on \(x\) and \(y,\) so we write \(w=f(x, y, z(x, y)) .\) Differentiate with respect to \(x\) holding \(y\) fixed to show that \(\left(\frac{\partial w}{\partial x}\right)_{y}=18,\) where the subscript \(y\) indicates that \(y\) is held fixed. b. Alternatively, on the plane \(P,\) we could consider \(x\) and \(z\) as the independent variables, which means \(y\) depends on \(x\) and \(z,\) so we write \(w=f(x, y(x, z), z)\) and differentiate with respect to \(x\) holding \(z\) fixed. Show that \(\left(\frac{\partial w}{\partial x}\right)_{z}=8,\) where the subscript \(z\) indicates that \(z\) is held fixed. c. Make a sketch of the plane \(z=4 x-2 y\) and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find \(\left(\frac{\partial w}{\partial y}\right)_{x}\) \(\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},\) and \(\left(\frac{\partial w}{\partial z}\right)_{y}\)

Short Answer

Expert verified
Answer: In part a, the partial derivative with respect to x holding y constant, \(\left(\frac{\partial w}{\partial x}\right)_{y} = 18\), indicates that when y is held constant, w increases at a rate of 18 units per unit increase in x. In part b, the partial derivative with respect to x holding z constant, \(\left(\frac{\partial w}{\partial x}\right)_{z} = 8\), indicates that when z is held constant, w increases at a rate of 8 units per unit increase in x. Similarly, the remaining partial derivatives in part d have the following interpretations: - \(\left(\frac{\partial w}{\partial y}\right)_{x} = -5\): When x is held constant, w decreases at a rate of 5 units per unit increase in y. - \(\left(\frac{\partial w}{\partial y}\right)_{z} = 3\): When z is held constant, w increases at a rate of 3 units per unit increase in y. - \(\left(\frac{\partial w}{\partial z}\right)_{x} = \frac{5}{2}\): When x is held constant, w increases at a rate of 2.5 units per unit increase in z. - \(\left(\frac{\partial w}{\partial z}\right)_{y} = 4\): When y is held constant, w increases at a rate of 4 units per unit increase in z.

Step by step solution

01

Rewrite the function in terms of the independent variables

We are given that \(w = f(x, y, z) = 2x + 3y + 4z\) and the plane \(P\) is given by \(z = 4x - 2y\). So, we can write \(w\) as \(f(x, y, z(x, y)) = 2x + 3y + 4(4x - 2y)\).
02

Differentiate w with respect to x holding y fixed

Now we find the derivative of \(w\) with respect to x, holding y constant: \(\left(\frac{\partial w}{\partial x}\right)_{y} = 2 + 0 + 4(4) = 18\). b. Find \(\left(\frac{\partial w}{\partial x}\right)_{z}\)
03

Rewrite the function in terms of the independent variables

The function \(w\) is given as \(f(x, y, z) = 2x + 3y + 4z\). We know that on the plane \(P\), \(y(x,z) = 2x - \frac{1}{2}z\). So, we can write \(w\) as \(f(x, y(x, z), z) = 2x + 3(2x -\frac{1}{2}z) + 4z\).
04

Differentiate w with respect to x holding z fixed

Now we find the derivative of \(w\) with respect to x, holding z constant: \(\left(\frac{\partial w}{\partial x}\right)_{z} = 2 + 6 - 0 = 8\). c. Geometrical interpretation A sketch of plane \(P\) (z = 4x - 2y) can be made by plotting x, y, and z axes, and finding the intersections of the plane with these axes. The results from parts (a) and (b) indicate that the rate of change of \(w\) with respect to \(x\) depends on whether we hold \(y\) or \(z\) constant. In part (a), \(\left(\frac{\partial w}{\partial x}\right)_{y} = 18\), meaning that when \(y\) is held constant, w increases faster with respect to \(x\). In part (b), \(\left(\frac{\partial w}{\partial x}\right)_{z} = 8\), meaning that when \(z\) is held constant, the change of \(w\) with respect to \(x\) is slower. d. Find remaining partial derivatives 1. Find \(\left(\frac{\partial w}{\partial y}\right)_{x}\) and \(\left(\frac{\partial w}{\partial y}\right)_{z}\) - Holding x fixed: \(\left(\frac{\partial w}{\partial y}\right)_{x} = 0+3+4(-2)= -5\) - Holding z fixed: \(\left(\frac{\partial w}{\partial y}\right)_{z} = 0+3+0 = 3\) 2. Find \(\left(\frac{\partial w}{\partial z}\right)_{x}\) and \(\left(\frac{\partial w}{\partial z}\right)_{y}\) - Holding x fixed: \(\left(\frac{\partial w}{\partial z}\right)_{x} = 0+3(-\frac{1}{2})+4 = \frac{5}{2}\) - Holding y fixed: \(\left(\frac{\partial w}{\partial z}\right)_{y} = 0+0+4 = 4\)

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Key Concepts

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

Multivariable Calculus
Multivariable calculus extends the concepts of calculus to functions of multiple variables. Unlike single-variable calculus where we deal with functions of one variable like \( f(x) \), multivariable calculus involves functions like \( f(x, y, z) \), which depend on two or more variables. This makes the study of rates of change and accumulation much more complex, as we now have to consider how these functions change in different directions within space.

In our given exercise, the function \( w = f(x, y, z) \) represents a three-dimensional space where any change in \( w \) can be associated with changes along the \( x \)-, \( y \)-, or \( z \) axis, or any combination of these directions. The intricacies of dealing with such functions make multivariable calculus particularly useful in fields like engineering, economics, and the sciences, where multiple factors influence outcomes simultaneously.
Partial Differentiation
Partial differentiation is a technique used to explore how a multivariable function changes as one variable changes while keeping the others constant. For a function like \( w = f(x, y, z) \), the partial derivatives \( \frac{\partial w}{\partial x} \) tell us the rate at which \( w \) changes with \( x \) when \( y \) and \( z \) are held fixed, and similarly for \( \frac{\partial w}{\partial y} \) and \( \frac{\partial w}{\partial z}\).

In the exercise at hand, the computation of \( \frac{\partial w}{\partial x} \) on a plane, while holding \( y \) fixed, takes into account that \( z \) is a function of \( x \) and \( y \) on that plane. This consideration leads to the understanding of how \( w \) changes in different scenarios depending on the choice of independent variables.
Rate of Change
In calculus, the rate of change of a function is a measure of how quickly the value of the function changes as its input changes. When dealing with multivariable functions, the rate of change can differ when approaching from different directions. This directional dependence is captured by partial derivatives. Each partial derivative measures the rate of change of the function in terms of one of the independent variables while holding the others constant.

The results of the exercise show that \( \frac{\partial w}{\partial x} \) differs when calculated with \( y \) constant versus with \( z \) constant: \( 18 \) versus \( 8 \) respectively. This highlights that the choice of variables considered independent affects the calculated rate of change, emphasizing the importance of understanding the context within which variables interact.
Plane Equations
Equations of planes in multivariable calculus are algebraic expressions that represent flat, two-dimensional surfaces in three-dimensional space. The general form of a plane equation is \( Ax + By + Cz + D = 0 \), where \( A \) , \( B \) , and \( C \) determine the orientation of the plane, and \( D \) the plane's position relative to the origin. In our exercise, the plane equation is \( z = 4x - 2y \), which defines a specific

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Most popular questions from this chapter

Given positive numbers \(x_{1}, \ldots, x_{n},\) prove that the geometric mean \(\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n}\) is no greater than the arithmetic mean \(\left(x_{1}+\cdots+x_{n}\right) / n\) in the following cases. a. Find the maximum value of \(x y z,\) subject to \(x+y+z=k\) where \(k\) is a real number and \(x>0, y>0\), and \(z>0 .\) Use the result to prove that $$(x y z)^{1 / 3} \leq \frac{x+y+z}{3}.$$ b. Generalize part (a) and show that $$\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n} \leq \frac{x_{1}+\cdots+x_{n}}{n}.$$

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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 electric field due to a point charge of strength \(Q\) at the origin has a potential function \(\varphi=k Q / r,\) where \(r^{2}=x^{2}+y^{2}+z^{2}\) is the square of the distance between a variable point \(P(x, y, z)\) and the charge, and \(k>0\) is a physical constant. The electric field is given by \(\mathbf{E}=-\nabla \varphi,\) where \(\nabla \varphi\) is the gradient in three dimensions. a. Show that the three-dimensional electric field due to a point charge is given by $$ \mathbf{E}(x, y, z)=k Q\left\langle\frac{x}{r^{3}}, \frac{y}{r^{3}}, \frac{z}{r^{3}}\right\rangle $$ b. Show that the electric field at a point has a magnitude \(|\mathbf{E}|=k Q / r^{2} .\) Explain why this relationship is called an inverse square law.
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