Difference between revisions of "009A Sample Midterm 3, Problem 4"
Kayla Murray (talk | contribs) |
Kayla Murray (talk | contribs) |
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|From Problem 3, we have | |From Problem 3, we have | ||
|- | |- | ||
| − | | <math>f'(x)=\frac{ | + | | <math>f'(x)=-\frac{3}{\sqrt{-2x+5}}.</math> |
|- | |- | ||
|Therefore, the slope of the tangent line is | |Therefore, the slope of the tangent line is | ||
| Line 32: | Line 32: | ||
\displaystyle{m} & = & \displaystyle{f'(-2)}\\ | \displaystyle{m} & = & \displaystyle{f'(-2)}\\ | ||
&&\\ | &&\\ | ||
| − | & = & \displaystyle{\frac{ | + | & = & \displaystyle{-\frac{3}{\sqrt{-2(-2)+5}}}\\ |
&&\\ | &&\\ | ||
| − | & = & \displaystyle{\frac{ | + | & = & \displaystyle{-\frac{3}{\sqrt{9}}}\\ |
&&\\ | &&\\ | ||
& = & \displaystyle{-1.} | & = & \displaystyle{-1.} | ||
Revision as of 09:21, 13 March 2017
Find the equation of the tangent line 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 y=3\sqrt{-2x+5}} at 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 (-2,9).}
| Foundations: |
|---|
| The equation of the tangent line 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 f(x)} at the point 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 (a,b)} 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 y=m(x-a)+b} where 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 m=f'(a).} |
Solution:
| Step 1: |
|---|
| First, we need to calculate the slope of the tangent line. |
| 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 f(x)=3\sqrt{-2x+5}.} |
| From Problem 3, we have |
| 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)=-\frac{3}{\sqrt{-2x+5}}.} |
| Therefore, the slope of the tangent line 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 \begin{array}{rcl} \displaystyle{m} & = & \displaystyle{f'(-2)}\\ &&\\ & = & \displaystyle{-\frac{3}{\sqrt{-2(-2)+5}}}\\ &&\\ & = & \displaystyle{-\frac{3}{\sqrt{9}}}\\ &&\\ & = & \displaystyle{-1.} \end{array}} |
| Step 2: |
|---|
| Now, the tangent line has slope 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 m=-1} |
| and passes through the point 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 (-2,9).} |
| Hence, the equation of the tangent line 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 y=-1(x+2)+9.} |
| Final Answer: |
|---|
| 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(x+2)+9} |