Difference between revisions of "Math 22 Related Rates"
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related. | 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>: | '''Solution''': Differentiate both sides of the equation with respect to <math>t</math>: | ||
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Substitute, we get <math>\frac{dy}{dt}=\frac{2(3)(1)(1)}{3(1)^2-(3)^2}=-1</math> | Substitute, we get <math>\frac{dy}{dt}=\frac{2(3)(1)(1)}{3(1)^2-(3)^2}=-1</math> | ||
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| + | '''Example 2''': The revenue <math>R</math> from selling <math>x</math> units of a product is given by <math>R=1200x-x^2</math>. The sales are increasing at a rate of <math>30</math> units per day. Find the rate of change of the revenue when <math>x=400</math> | ||
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| + | '''Solution''': The sales are increasing at a rate of <math>30</math> units per day. So, <math>\frac{dx}{dt}=30</math>. we want to find the rate of change of the revenue, this means we want to find <math>\frac{dR}{dt}</math>. So, differentiate both sides of the equation with respect to <math>t</math> to get: | ||
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| + | <math>\frac{d}{dt}[R]=\frac{d}{dt}[1200x-x^2]</math> | ||
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| + | <math>=\frac{dR}{dt}=1200\frac{dx}{dt}-2x\frac{dx}{dt}</math> | ||
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| + | Substitute to get <math>\frac{dR}{dt}=1200(30)-2(400)(30)=12000</math> | ||
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| + | Therefore, the rate of change of the revenue is $12000 per day. | ||
==Guidelines for Solving a Related-Rate Problem== | ==Guidelines for Solving a Related-Rate Problem== | ||
Latest revision as of 07:23, 27 July 2020
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 1: Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^3=x^2y+3} Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt}} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=3} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dt}=1}
Solution: Differentiate both sides of the equation with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} :
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}[y^3]=\frac{d}{dt}[x^2y+3]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =3y^2\frac{dy}{dt}=2x\frac{dx}{dt}y+x^2\frac{dy}{dt}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =3y^2\frac{dy}{dt}-x^2\frac{dy}{dt}=2xy\frac{dx}{dt}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(3y^2-x^2)\frac{dy}{dt}=2xy\frac{dx}{dt}}
Hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt}=\frac{2xy\frac{dx}{dt}}{3y^2-x^2}} .
Substitute, we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt}=\frac{2(3)(1)(1)}{3(1)^2-(3)^2}=-1}
Example 2: The revenue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} from selling Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} units of a product is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=1200x-x^2} . The sales are increasing at a rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 30} units per day. Find the rate of change of the revenue when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=400}
Solution: The sales are increasing at a rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 30} units per day. So, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dt}=30} . we want to find the rate of change of the revenue, this means we want to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dR}{dt}} . So, differentiate both sides of the equation with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} to get:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}[R]=\frac{d}{dt}[1200x-x^2]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{dR}{dt}=1200\frac{dx}{dt}-2x\frac{dx}{dt}}
Substitute to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dR}{dt}=1200(30)-2(400)(30)=12000}
Therefore, the rate of change of the revenue is $12000 per day.
Guidelines for Solving a Related-Rate Problem
1. Identify all given quantities and all quantities to be determined. If possible, make a sketch and label the quantities. 2. Identify all given quantities and all quantities to be determined. If possible, make a sketch and label the quantities. 3. Use the Chain Rule to implicitly differentiate both sides of the equation with respect to time. 4. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.
This page were made by Tri Phan