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Chapter 10: Problem 6

Find the first partial derivatives of \(f(x, y, z)=z \arctan \left(\frac{y}{x}\right)\) at the point (4,4,-3) A. \(\frac{\partial f}{\partial x}(4,4,-3)=\) ____________. B. \(\frac{\partial f}{\partial y}(4,4,-3)=\) ____________. C. \(\frac{\partial f}{\partial z}(4,4,-3)=\) ____________.

Short Answer

Expert verified
A. \(\frac{\partial f}{\partial x}(4,4,-3)=\frac{3}{8}\) B. \(\frac{\partial f}{\partial y}(4,4,-3)=-\frac{3}{8}\) C. \(\frac{\partial f}{\partial z}(4,4,-3)=\arctan(1)\)

Step by step solution

01

Find the general expression for \(\frac{\partial f}{\partial x}\)

Using the chain rule, we have: \[\frac{\partial f}{\partial x}=z\frac{\partial}{\partial x} \arctan\left(\frac{y}{x}\right)\] Now, compute the derivative of the inside function, \(\frac{y}{x}\), with respect to x: \[\frac{\partial}{\partial x}\left(\frac{y}{x}\right) = \frac{-y}{x^2}\] Now, find the derivative of the arctan function with respect to its argument: \[\frac{\partial}{\partial u}\arctan(u)=\frac{1}{1+u^2}\] Substitute \(u=\frac{y}{x}\) and use the chain rule: \[\frac{\partial f}{\partial x} = z \cdot \frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{-y}{x^2}\]
02

Evaluate \(\frac{\partial f}{\partial x}(4,4,-3)\)

Plug in the values x=4, y=4, z=-3 into the expression: \[\frac{\partial f}{\partial x}(4,4,-3)=-3 \cdot \frac{1}{1+\left(\frac{4}{4}\right)^2} \cdot \frac{-4}{(4)^2} = -3\cdot\frac{1}{1+1}\cdot\frac{-1}{4}=\frac{3}{8}\] A. \(\frac{\partial f}{\partial x}(4,4,-3)=\frac{3}{8}\)
03

Find the general expression for \(\frac{\partial f}{\partial y}\)

Again, using the chain rule, we obtain: \[\frac{\partial f}{\partial y}=z\frac{\partial}{\partial y} \arctan\left(\frac{y}{x}\right)\] Now, compute the derivative of the inside function, \(\frac{y}{x}\), with respect to y: \[\frac{\partial}{\partial y}\left(\frac{y}{x}\right) = \frac{1}{x}\] Now, substitute \(u=\frac{y}{x}\) and use the chain rule: \[\frac{\partial f}{\partial y} = z \cdot \frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{1}{x}\]
04

Evaluate \(\frac{\partial f}{\partial y}(4,4,-3)\)

Plug in the values x=4, y=4, z=-3 into the expression: \[\frac{\partial f}{\partial y}(4,4,-3)=-3 \cdot \frac{1}{1+\left(\frac{4}{4}\right)^2} \cdot \frac{1}{4} = -3\cdot\frac{1}{1+1}\cdot\frac{1}{4}=-\frac{3}{8}\] B. \(\frac{\partial f}{\partial y}(4,4,-3)=-\frac{3}{8}\)
05

Find the general expression for \(\frac{\partial f}{\partial z}\)

This derivative is straightforward. Since \(z \arctan\left(\frac{y}{x}\right)\) only contains z in a single term, we can simply differentiate that term with respect to z: \[\frac{\partial f}{\partial z}=\arctan\left(\frac{y}{x}\right)\]
06

Evaluate \(\frac{\partial f}{\partial z}(4,4,-3)\)

Plug in the values x=4, y=4, z=-3 into the expression: \[\frac{\partial f}{\partial z}(4,4,-3)=\arctan\left(\frac{4}{4}\right)=\arctan(1)\] C. \(\frac{\partial f}{\partial z}(4,4,-3)=\arctan(1)\)

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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 concepts from single-variable calculus to functions of several variables. It involves partial derivatives, which measure how a function changes as each variable is varied while others are held constant.

In our exercise, we're dealing with a function of three variables, namely \(f(x, y, z) = z \arctan\left(\frac{y}{x}\right)\). To explore the behavior of this function with respect to each variable, we look at its partial derivatives.

When finding the partial derivative with respect to \(x\), for instance, we treat \(y\) and \(z\) as constants. In doing so, we can gain insight into how the function's value adjusts to small changes in \(x\), which is crucial for applications in fields such as physics, economics, and engineering where multiple factors come into play.
Chain Rule
The chain rule in calculus is a fundamental formula used to compute the derivative of a composite function. When functions are combined, the rate at which they change together depends on both the outer function's sensitivity and the inner function's rate of change.

The chain rule tells us that if we have a composed function \(h(g(x))\), the derivative \(h'(g(x))\) is the product of the derivative of \(h\) with respect to \(g\), and the derivative of \(g\) with respect to \(x\): \(h'(g(x)) = h'(g) \cdot g'(x)\).

In the given exercise, we used the chain rule to differentiate the function \(z\arctan\left(\frac{y}{x}\right)\) with respect to \(x\) and \(y\). We broke it down into the product of the derivative of \(\arctan\) and the derivative of the ratio \(\frac{y}{x}\), treating \(z\) as a constant during the differentiation process. The chain rule makes such complex derivatives manageable and is invaluable in multivariable calculus.
Arctan Derivative
The derivative of the arctangent function, or \(\arctan\), is significant in calculus because it appears frequently in problems involving inverse trigonometric functions. The derivative of \(\arctan(u)\) with respect to \(u\) is given by \(\frac{1}{1 + u^2}\).

This formula arises from the inverse relationship between \(\arctan\) and \(\tan\), and it鈥檚 used whenever \(\arctan\) is part of a function we鈥檙e differentiating. In our case, we applied it within the context of the chain rule. Understanding the derivative of \(\arctan\) and how it integrates into the chain rule was essential for solving the exercise and finding the partial derivatives of \(f(x, y, z) = z \arctan\left(\frac{y}{x}\right)\) with respect to \(x\) and \(y\).

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Most popular questions from this chapter

The Heat Index, \(I,\) (measured in apparent degrees \(F)\) is a function of the actual temperature \(T\) outside (in degrees \(\mathrm{F}\) ) and the relative humidity \(H\) (measured as a percentage). A portion of the table which gives values for this function, \(I=I(T, H),\) is reproduced in Table \(10.2 .10 .\) $$\begin{array}{ccccc}\hline T \downarrow \backslash H \rightarrow & 70 & 75 & 80 & 85 \\ \hline 90 & 106 & 109 & 112 & 115 \\\\\hline 92 & 112 & 115 & 119 & 123 \\\\\hline 94 & 118 & 122 & 127 & 132 \\\\\hline 96 & 125 & 130 & 135 & 141 \\\\\hline\end{array}$$ a. State the limit definition of the value \(I_{T}(94,75)\). Then, estimate \(I_{T}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including its units. b. State the limit definition of the value \(I_{H}(94,75)\). Then, estimate \(I_{H}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including its units. c. Suppose you are given that \(I_{T}(92,80)=3.75\) and \(I_{H}(92,80)=0.8\). Estimate the values of \(I(91,80)\) and \(I(92,78)\). Explain how the partial derivatives are relevant to your thinking. d. On a certain day, at 1 p.m. the temperature is 92 degrees and the relative humidity is \(85 \%\). At 3 p.m., the temperature is 96 degrees and the relative humidity \(75 \% .\) What is the average rate of change of the heat index over this time period, and what are the units on your answer? Write a sentence to explain your thinking.

Apply the Method of Lagrange Multipliers solve each of the following constrained optimization problems. a. Determine the absolute maximum and absolute minimum values of \(f(x, y)=(x-1)^{2}+(y-2)^{2}\) subject to the constraint that \(x^{2}+y^{2}=\) 16 b. Determine the points on the sphere \(x^{2}+y^{2}+z^{2}=4\) that are closest to and farthest from the point (3,1,-1) . (As in the preceding exercise, you may find it simpler to work with the square of the distance formula, rather than the distance formula itself.) c. Find the absolute maximum and minimum of \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) subject to the constraint that \((x-3)^{2}+(y+2)^{2}+(z-5)^{2} \leq 16\). (Hint: here the constraint is a closed, bounded region. Use the boundary of that region for applying Lagrange Multipliers, but don't forget to also test any critical values of the function that lie in the interior of the region.)

Use the chain rule to find \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t},\) where $$z=e^{x y} \tan y, x=4 s+5 t, y=\frac{3 s}{3 t}$$ First the pieces: \(\frac{\partial z}{\partial x}=\) ________ \(\frac{\partial z}{\partial y}=\) \(\frac{\partial x}{\partial s}=\) ________ \(\frac{\partial x}{\partial t}=\) \(\frac{\partial y}{\partial s}=\) ________ \(\frac{\partial y}{\partial t}=\)

Calculate all four second-order partial derivatives of \(f(x, y)=4 x^{2} y+8 x y^{3}\). \(f_{x x}(x, y)=\) _________. \(f_{x y}(x, y)=\) _________. \(f_{y x}(x, y)=\) _________. \(f_{y y}(x, y)=\) _________.

Find the maximum and minimum volumes of a rectangular box whose surface area equals 7000 square \(\mathrm{cm}\) and whose edge length (sum of lengths of all edges) is \(440 \mathrm{~cm}\). Maximum value is ________________. occuring at ( ____________, ______________) Manimum value is ________________. occuring at ( ____________, ______________)
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