Difference between revisions of "009A Sample Final 1, Problem 4"

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|First, we compute <math>\frac{dy}{dx}</math>. We get
 
|First, we compute <math>\frac{dy}{dx}</math>. We get
 
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|-
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|<math>\frac{dy}{dx}=2x-\sin(\pi(x^2+1))(2\pi x)</math>.
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::<math>\frac{dy}{dx}=2x-\sin(\pi(x^2+1))(2\pi x)</math>.
 
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|}
  
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|To find the equation of the tangent line, we first find the slope of the line.  
 
|To find the equation of the tangent line, we first find the slope of the line.  
 
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|Using <math>x_0=1</math> in the formula for <math>\frac{dy}{dx}</math> from Step 1, we get
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|Using <math style="vertical-align: -3px">x_0=1</math> in the formula for <math style="vertical-align: -12px">\frac{dy}{dx}</math> from Step 1, we get
 
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|<math>m=2(1)-\sin(2\pi)2\pi=2</math>.
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|
 +
::<math>m=2(1)-\sin(2\pi)2\pi=2</math>.
 
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|To get a point on the line, we plug in <math>x_0=1</math> into the equation given.  
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|To get a point on the line, we plug in <math style="vertical-align: -3px">x_0=1</math> into the equation given.  
 
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|So, we have <math>y=1^2+\cos(2\pi)=2</math>.
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|So, we have <math style="vertical-align: -5px">y=1^2+\cos(2\pi)=2</math>.
 
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|Thus, the equation of the tangent line is <math>y=2(x-1)+2</math>.
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|Thus, the equation of the tangent line is <math style="vertical-align: -5px">y=2(x-1)+2</math>.
 
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|}
  

Revision as of 14:07, 22 February 2016

If

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^2+\cos (\pi(x^2+1))}

compute 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}{dx}} and find the equation for the tangent line 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 x_0=1} . You may leave your answers in point-slope form.

Foundations:  

Solution:

Step 1:  
First, we compute 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}{dx}} . 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}{dx}=2x-\sin(\pi(x^2+1))(2\pi x)} .
Step 2:  
To find the equation of the tangent line, we first find the slope of the line.
Using 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_0=1} in the formula for 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}{dx}} from Step 1, 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 m=2(1)-\sin(2\pi)2\pi=2} .
To get a point on the line, we plug in 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_0=1} into the equation given.
So, 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 y=1^2+\cos(2\pi)=2} .
Thus, 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=2(x-1)+2} .
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 \frac{dy}{dx}=2x-\sin(\pi(x^2+1))(2\pi x)}
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=2(x-1)+2}

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