Difference between revisions of "Math 22 Differentials and Marginal Analysis"
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is any nonzero real number. The differential of <math>y</math> (denoted by ) is <math>dy=f'(x) dx</math>. | is any nonzero real number. The differential of <math>y</math> (denoted by ) is <math>dy=f'(x) dx</math>. | ||
| − | '''Example''': Consider the function <math>f(x)=3x^3</math>. Find <math>dy</math> when <math>x=1</math> and <math>dx=0.01</math> | + | '''Example''': '''1)''' Consider the function <math>f(x)=3x^3</math>. Find <math>dy</math> when <math>x=1</math> and <math>dx=0.01</math> |
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
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|Notice: <math>f(x)=3x^3</math>, so <math>dy=f'(x)dx=9x^2 dx=9(1)^2.(0.01)=0.09</math> | |Notice: <math>f(x)=3x^3</math>, so <math>dy=f'(x)dx=9x^2 dx=9(1)^2.(0.01)=0.09</math> | ||
|} | |} | ||
| + | |||
| + | '''2)''' Find <math>dy</math> of each function below: | ||
| + | |||
| + | '''a)''' <math>y=\frac{5x+7}{15}</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |Notice: <math>y=\frac{5x+7}{15}=\frac{5x}{15}+\frac{7}{15}</math>, so | ||
| + | |- | ||
| + | |<math>dy=f'(x)dx=\frac{5}{15}=\frac{1}{3} dx</math> | ||
| + | |} | ||
| + | |||
| + | '''b)''' <math>y=x(1.25+0.02\sqrt{x})</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |Notice: <math>y=x(1.25+0.02\sqrt{x})=1.25x+0.02x\sqrt{x}=1.25x+0.02x^{\frac{3}{2}}</math>, so <math>dy=f'(x)dx=[1.25+0.02(\frac{3}{2})x^{\frac{1}{2}}]dx=[1.25+0.03\sqrt{x}]dx</math> | ||
| + | |} | ||
| + | |||
[[Math_22| '''Return to Topics Page''']] | [[Math_22| '''Return to Topics Page''']] | ||
'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' | ||
Latest revision as of 07:00, 10 August 2020
Differentials
Let 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=f(x)}
represent a differentiable function. The differential 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 x}
(denoted 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 dx}
)
is any nonzero real number. The differential 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 y}
(denoted by ) is 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 dy=f'(x) dx}
.
Example: 1) Consider the function 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 f(x)=3x^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 dy} 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=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 dx=0.01}
| Solution: |
|---|
| Notice: 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 f(x)=3x^3} , 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 dy=f'(x)dx=9x^2 dx=9(1)^2.(0.01)=0.09} |
2) 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 dy} of each function below:
a) 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=\frac{5x+7}{15}}
| Solution: |
|---|
| Notice: 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=\frac{5x+7}{15}=\frac{5x}{15}+\frac{7}{15}} , 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 dy=f'(x)dx=\frac{5}{15}=\frac{1}{3} dx} |
b) 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=x(1.25+0.02\sqrt{x})}
| Solution: |
|---|
| Notice: 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=x(1.25+0.02\sqrt{x})=1.25x+0.02x\sqrt{x}=1.25x+0.02x^{\frac{3}{2}}} , 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 dy=f'(x)dx=[1.25+0.02(\frac{3}{2})x^{\frac{1}{2}}]dx=[1.25+0.03\sqrt{x}]dx} |
This page were made by Tri Phan