Math 22 Related Rates - Revision history

Tphan046: /* Related Variables */

2020-07-27T15:23:33Z

Related Variables

Tphan046 at 15:01, 27 July 2020

2020-07-27T15:01:14Z

Tphan046: Created page with "==Related Variables== We will study problems involving variables that are changing with respect to time. If two or more such variables are related to each other, then their r..."

2020-07-27T14:51:05Z

Created page with "==Related Variables== We will study problems involving variables that are changing with respect to time. If two or more such variables are related to each other, then their r..."

New page

==Related Variables==
We will study problems involving variables that are changing with respect to time. If two or more
such variables are related to each other, then their rates of change with respect to time are also
related.

'''Example''': Given <math>y^3=x^2y+3</math> Find <math>\frac{dy}{dt}</math> when <math>x=3</math>, <math>y=1</math> and <math>\frac{dx}{dt}=1</math>

'''Solution''': Differentiate both sides of the equation with respect to <math>t</math>:

<math>\frac{d}{dt}[y^3]=\frac{d}{dt}[x^2y+3]</math>

<math>=3y^2\frac{dy}{dt}=2x\frac{dx}{dt}y+x^2\frac{dy}{dt}</math>

<math>=3y^2\frac{dy}{dt}-x^2\frac{dy}{dt}=2xy\frac{dx}{dt}</math>

<math>=(3y^2-x^2)\frac{dy}{dt}=2xy\frac{dx}{dt}</math>

Hence, <math>\frac{dy}{dt}=\frac{2xy\frac{dx}{dt}}{3y^2-x^2}</math>.

Substitute, we get <math>\frac{dy}{dt}=\frac{2(3)(1)(1)}{3(1)^2-(3)^2}=-1</math>

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

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 related.
-'''Example''': Given <math>y^3=x^2y+3</math> Find <math>\frac{dy}{dt}</math> when <math>x=3</math>, <math>y=1</math> and <math>\frac{dx}{dt}=1</math>
+'''Example 1''': Given <math>y^3=x^2y+3</math> Find <math>\frac{dy}{dt}</math> when <math>x=3</math>, <math>y=1</math> and <math>\frac{dx}{dt}=1</math>
 '''Solution''': Differentiate both sides of the equation with respect to <math>t</math>:
@@ Line 19: / Line 19: @@
 Substitute, we get <math>\frac{dy}{dt}=\frac{2(3)(1)(1)}{3(1)^2-(3)^2}=-1</math>
 ==Guidelines for Solving a Related-Rate Problem==