/*! 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 14 Use Theorem 12.7 to find the fol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 14

Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$\begin{aligned} &d Q / d t, \text { where } Q=\sqrt{x^{2}+y^{2}+z^{2}}, x=\sin t, y=\cos t, \text { and }\\\ &z=\cos t \end{aligned}$$

Short Answer

Expert verified
Question: Find the derivative of \(Q=\sqrt{x^{2}+y^{2}+z^{2}}\) with respect to t, where \(x=\sin t\), \(y=\cos t\), and \(z=\cos t\). Answer: \(\frac{dQ}{dt} = \left(\frac{\sin t}{\sqrt{1+(\cos t)^{2}}}\right)(\cos t) - \left(\frac{2\cos t\sin t}{\sqrt{1+(\cos t)^{2}}}\right)\)

Step by step solution

01

Find the partial derivatives of Q with respect to x, y, and z

First, we will calculate the partial derivatives of Q with respect to its variables x, y, and z. Given the function \(Q=\sqrt{x^{2}+y^{2}+z^{2}}\), we can rewrite it as \(Q=(x^{2}+y^{2}+z^{2})^{\frac{1}{2}}\) to differentiate more easily. \(\frac{\partial Q}{\partial x}=\frac{1}{2}(x^{2}+y^{2}+z^{2})^{-\frac{1}{2}}(2x)=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\) \(\frac{\partial Q}{\partial y}=\frac{1}{2}(x^{2}+y^{2}+z^{2})^{-\frac{1}{2}}(2y)=\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}\) \(\frac{\partial Q}{\partial z}=\frac{1}{2}(x^{2}+y^{2}+z^{2})^{-\frac{1}{2}}(2z)=\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\)
02

Find the derivatives of x, y, and z with respect to t

Next, we need to find the derivatives of x, y, and z with respect to t: \(\frac{dx}{dt}=\frac{d(\sin t)}{dt}=\cos t\) \(\frac{dy}{dt}=\frac{d(\cos t)}{dt}=-\sin t\) \(\frac{dz}{dt}=\frac{d(\cos t)}{dt}=-\sin t\)
03

Apply Theorem 12.7 (Multivariable Chain Rule)

Using the multivariable chain rule (Theorem 12.7), we can now find the total derivative of Q with respect to t: \(\frac{dQ}{dt} = \frac{\partial Q}{\partial x} \frac{dx}{dt} + \frac{\partial Q}{\partial y} \frac{dy}{dt} + \frac{\partial Q}{\partial z} \frac{dz}{dt}\) Substitute the partial derivatives and the derivative of x, y, and z with respect to t: \(\frac{dQ}{dt} = \left(\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)(\cos t) + \left(\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)(-\sin t) + \left(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)(-\sin t)\)
04

Replace x, y, and z with their expressions in terms of t

Now we will express the answer in terms of the independent variable t by replacing x, y, and z with their given expressions: \(\frac{dQ}{dt} = \left(\frac{\sin t}{\sqrt{(\sin t)^{2}+(\cos t)^{2}+(\cos t)^{2}}}\right)(\cos t) + \left(\frac{\cos t}{\sqrt{(\sin t)^{2}+(\cos t)^{2}+(\cos t)^{2}}}\right)(-\sin t) + \left(\frac{\cos t}{\sqrt{(\sin t)^{2}+(\cos t)^{2}+(\cos t)^{2}}}\right)(-\sin t)\)
05

Simplify the expression

Simplify the expression by noticing that \((\sin t)^{2}+(\cos t)^{2}=1\). \(\frac{dQ}{dt} = \left(\frac{\sin t}{\sqrt{1+(\cos t)^{2}}}\right)(\cos t) - \left(\frac{\cos t}{\sqrt{1+(\cos t)^{2}}}\right)(\sin t) - \left(\frac{\cos t}{\sqrt{1+(\cos t)^{2}}}\right)(\sin t)\) Combine the last two terms: \(\frac{dQ}{dt} = \left(\frac{\sin t}{\sqrt{1+(\cos t)^{2}}}\right)(\cos t) - \left(\frac{2\cos t\sin t}{\sqrt{1+(\cos t)^{2}}}\right)\) This is the final expression for \(\frac{dQ}{dt}\) in terms of the independent variable t.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Partial Derivatives
Partial derivatives are a fundamental aspect of studying functions with more than one variable. They measure how a function changes as only one of its variables varies, while the other variables are held constant. For instance, given a multivariate function like the distance from the origin to a point in a three-dimensional space, expressed by the equation \( Q = \sqrt{x^{2} + y^{2} + z^{2}} \), we might want to understand how the distance 'Q' changes only with respect to 'x' while 'y' and 'z' stay the same.

Calculating the partial derivative of 'Q' with respect to 'x' involves treating 'y' and 'z' as constants and only differentiating with respect to 'x'. Thus we apply the usual rules of differentiation to obtain \( \frac{\partial Q}{\partial x} = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} \), which represents the rate of change of 'Q' in the direction of the 'x' axis. Similarly, we compute the partial derivatives \( \frac{\partial Q}{\partial y} \) and \( \frac{\partial Q}{\partial z} \) to understand changes in the 'y' and 'z' directions.

The concept of a partial derivative is a cornerstone of vector calculus and is critical for analyzing functions across multiple dimensions, which is notably useful in fields such as physics, engineering, and economics.
Total Derivative
The total derivative of a function takes an inclusive look at how the function changes when all of its variables are allowed to vary. Unlike partial derivatives, which consider one variable at a time, the total derivative integrates the effect of changes in all independent variables. It is particularly relevant when the variables themselves are functions of another variable, say time.

For example, re-examining \( Q = \sqrt{x^{2} + y^{2} + z^{2}} \), suppose 'x', 'y', and 'z' are all functions of time, 't'. To find the rate at which 'Q' changes with time, we need to apply the chain rule from calculus, which seamlessly blends each independent variable's contribution.

Applying the multivariable chain rule is like knitting together the partial derivatives with the derivatives of 'x', 'y', and 'z' with respect to time. This results in the total derivative \( \frac{dQ}{dt} \), which provides a comprehensive picture of how 'Q' evolves as time advances. We calculate it by adding up each product of the function's partial derivatives and the derivatives of 'x', 'y', and 'z' with respect to 't'. The total derivative is essential for understanding dynamic systems where multiple factors simultaneously affect the behaviour or state of the system.
Independent Variable
An independent variable is a critical concept in calculus and the broader world of mathematics and science. It is the variable that influencers or controls the changes in a function or equation. Primarily, we see independent variables as the input upon which the output depends. In many functions, time 't' is chosen as an independent variable to track how other variables or states change as time progresses.

When solving for \( \frac{dQ}{dt} \), we treated 't' as the independent variable and explored how 'x', 'y', and 'z' are functions of 't'. Doing so allows us to understand the behavior of 'Q' over time, as 'x', 'y', and 'z' change accordingly. Expressing a function solely in terms of its independent variables, as we did in the final step of our solution, simplifies its interpretation and application. This focus on the independent variable is especially illuminating for mathematical modelling, predicting phenomena, and analyzing experiments across various scientific domains.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K,\) is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b} .\) Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=1\). a. Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\). b. Suppose \(L=10\) is fixed and \(K\) increases from \(K=20\) to \(K=20.5 .\) Use linear approximation to estimate the change in \(Q\). c. Suppose \(K=20\) is fixed and \(L\) decreases from \(L=10\) to \(L=9.5 .\) Use linear approximation to estimate the change in \(\bar{Q}\). d. Graph the level curves of the production function in the first quadrant of the \(L K\) -plane for \(Q=1,2,\) and 3. e. Use the graph of part (d). If you move along the vertical line \(L=2\) in the positive \(K\) -direction, how does \(Q\) change? Is this consistent with \(Q_{K}\) computed in part (a)? f. Use the graph of part (d). If you move along the horizontal line \(K=2\) in the positive \(L\) -direction, how does \(Q\) change? Is this consistent with \(Q_{L}\) computed in part (a)?

Extending Exercise \(62,\) when three electrical resistors with resistance \(R_{1}>0, R_{2}>0,\) and \(R_{3}>0\) are wired in parallel in a circuit (see figure), the combined resistance \(R,\) measured in ohms \((\Omega),\) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) Estimate the change in \(R\) if \(R_{1}\) increases from \(2 \Omega\) to \(2.05 \Omega, R_{2}\) decreases from \(3 \Omega\) to \(2.95 \Omega,\) and \(R_{3}\) increases from \(1.5 \Omega\) to \(1.55 \Omega.\)

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius \(R\) and length \(L,\) the velocity of the fluid \(r \leq R\) units from the center-line of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right),\) where \(P\) is the difference in the pressure between the ends of the cylinder and \(\nu\) is the viscosity of the fluid (see figure). Assuming that \(P\) and \(\nu\) are constant, the velocity \(V\) along the center line of the cylinder \((r=0)\) is \(V=k R^{2} / L,\) where \(k\) is a constant that we will take to be \(k=1.\) a. Estimate the change in the centerline velocity \((r=0)\) if the radius of the flow cylinder increases from \(R=3 \mathrm{cm}\) to \(R=3.05 \mathrm{cm}\) and the length increases from \(L=50 \mathrm{cm}\) to \(L=50.5 \mathrm{cm}.\) b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder \(R\) decreases by \(1 \%\) and the length \(L\) increases by \(2 \%.\) c. Complete the following sentence: If the radius of the cylinder increases by \(p \%,\) then the length of the cylinder must increase by approximately __________ \(\%\) in order for the velocity to remain constant.

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 \(f(x, y)=\int_{1}^{x y} h(s) d s\).

a. Show that the point in the plane \(a x+b y+c z=d\) nearest the origin is \(P\left(a d / D^{2}, b d / D^{2}, c d / D^{2}\right),\) where \(D^{2}=a^{2}+b^{2}+c^{2} .\) Conclude that the least distance from the plane to the origin is \(|d| / D\). (Hint: The least distance is along a normal to the plane.) b. Show that the least distance from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) to the plane \(a x+b y+c z=d\) is \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\) (Hint: Find the point \(P\) on the plane closest to \(P_{0}\).)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.