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Q1E

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Essential Calculus: Early Transcendentals

Found in: Page 132

Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

If \(V\) is the volume of a cube with edge length \(x\) and the cube expands as time passes, find \(\frac{{dV}}{{dt}}\) in terms of \(\frac{{dx}}{{dt}}\).

The value of \(\frac{{dV}}{{dt}}\) in terms of \(\frac{{dx}}{{dt}}\) is \(\left( {{\rm{3}}{x^{\rm{2}}}} \right)\frac{{{\rm{d}}x}}{{{\rm{dt}}}}\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of the Chain rule

If \(y = f\left( u \right)\) and \(u = g\left( x \right)\) are differentiable functions, then the Chain rule is given by \(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}}\)

Step 2: Find \(\frac{{dV}}{{dt}}\) in terms of \(\frac{{dx}}{{dt}}\).

It is given that a cube of a volume V with edge length \(x\).

Then, the volume of the cube is given by, \(V\left( x \right) = {x^3}\).

Differentiate both sides of \(V\left( x \right) = {x^3}\) with respect to \(t\) as follows:

\(\frac{{dV}}{{dt}} = \frac{d}{{dt}}\left( {{x^3}} \right)\)

Apply the chain rule of differentiation as follows:

\(\begin{array}{c}\frac{{dV}}{{dt}} = \frac{d}{{dx}}\left( {{x^3}} \right)\frac{{dx}}{{dt}}\\\frac{{dV}}{{dt}} = 3{x^2}\frac{{dx}}{{dt}}\end{array}\)

Therefore, \(\frac{{dV}}{{dt}}\) in terms of \(\frac{{dx}}{{dt}}\) is equal to \(3{x^2}\frac{{dx}}{{dt}}\).

Most popular questions for Math Textbooks

(a) Find \(y'\) by implicit differentiation.

(b) Solve the equation explicitly for\(y\)and differentiate to get \(y'\) in terms of \(x\).

(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for\(y\) into your solution for part (a).

1.\(9{x^2} - {y^2} = 1\)

Water is leaking out of an inverted conical tank at a rate of \(10,000c{m^3}/min\)at the same time that water is being pumped into the tank at a constant rate. The tank has height \(6m\) and the diameter at the top is\(4m\). If the water level is rising at a rate of \(20cm/min\) when the height of the water is \(2m\), find the rate at which water is being pumped into the tank

Find \({dy}/{dx}\;\) by implicit differentiation.

8. \(\cos \left( {xy} \right) = 1 + \sin y\)

(a) What quantities are given in the problem?

(b) What is the unknown?

(c) Draw a picture of the situation for any time\(t\).

(d) Write an equation that relates the quantities.

(e) Finish solving the problem.

12. At noon, ship A is \(150{\rm{ km}}\) west of ship B. Ship A is sailing east at\(35{\rm{ km/h}}\)and ship B is sailing north at\(25{\rm{ km/h}}\). How fast is the distance between the ships changing at\(4:00{\rm{ PM}}\)?

A spotlight on the ground shines on a wall \(12\)\(m\) away. If a man \(2\)\(m\) tall walks from the spotlight toward the building at a speed of \(1.6\)\(m/s\), how fast is the length of his shadow on the building decreasing when he is \(4\)\(m\) from the building?

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