Difference between revisions of "009A Sample Final 2, Problem 3"

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(Created page with "<span class="exam">Compute   <math>\frac{dy}{dx}.</math> <span class="exam">(a)  <math style="vertical-align: -15px">y=\bigg(\frac{x^2+3}{x^2-1}\bigg)^3</math> <sp...")
 
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|Let &nbsp;<math style="vertical-align: -5px">f(x)=\sin(x)</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">g(x)=\sin^{-1}x.</math>
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|Let &nbsp;<math style="vertical-align: -5px">y=\sin^{-1}(x).</math>&nbsp; Then,
 
|-
 
|-
−
|These functions are inverses of each other since
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\sin(y)=x</math>
 
|-
 
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">f(g(x))=x</math>&nbsp; and &nbsp; <math style="vertical-align: -5px">g(f(x))=x.</math>
+
|for &nbsp;<math>y</math>&nbsp; in the interval &nbsp;<math>\bigg[-\frac{\pi}{2},\frac{\pi}{2}\bigg].</math>
 
|-
 
|-
−
|Therefore,
+
|Using implicit differentiation, we have
 
|-
 
|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\cos(y) \frac{dy}{dx}=1.</math>
−
\displaystyle{g'(x)} & = & \displaystyle{\frac{1}{f'(g(x))}}\\
 
−
&&\\
 
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& = & \displaystyle{\frac{1}{\cos(\sin^{-1} x)}.}
 
−
\end{array}</math>
 
 
|-
 
|-
−
|Now, let &nbsp;<math style="vertical-align: -5px">y=\sin^{-1}(x).</math>&nbsp; Then, &nbsp;<math style="vertical-align: -5px">x=\sin(y).</math>
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|Solving for &nbsp;<math style="vertical-align: -15px">\frac{dy}{dx},</math>&nbsp; we get
 
|-
 
|-
−
|So, &nbsp;<math style="vertical-align: -5px">\cos(\sin^{-1} x)=\cos(y).</math>
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{dy}{dx}=\frac{1}{\cos(y)}.</math>
−
|-
 
−
|Therefore,
 
−
|-
 
−
|&nbsp; &nbsp; &nbsp; &nbsp;<math>g'(x)=\frac{1}{\cos(y)}.</math>
 
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|Now, since
+
|Now, since &nbsp;<math>\sin(y)=x,</math>&nbsp; we have the following diagram.
 
|-
 
|-
−
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\cos^2 y+\sin^2 y =1,</math>
+
|(Insert diagram)
 
|-
 
|-
−
|we have
+
|Therefore,
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\cos(y)=\sqrt{1-x^2}.</math>
 +
|-
 +
|Hence,
 
|-
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
−
\displaystyle{\cos(y)} & = & \displaystyle{\sqrt{1-\sin^2 y}}\\
+
\displaystyle{\frac{dy}{dx}} & = & \displaystyle{\frac{1}{\cos(y)}}\\
 
&&\\
 
&&\\
−
& = & \displaystyle{\sqrt{1-x^2}.}
+
& = & \displaystyle{\frac{1}{\sqrt{1-x^2}}.}
 
\end{array}</math>
 
\end{array}</math>
−
|-
 
−
|Hence,
 
−
|-
 
−
|&nbsp; &nbsp; &nbsp; &nbsp;<math>g'(x)=\frac{1}{\sqrt{1-x^2}}.</math>
 
 
|}
 
|}
  

Latest revision as of 17:11, 20 May 2017

Compute  

(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=\bigg(\frac{x^2+3}{x^2-1}\bigg)^3}

(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\cos(\sqrt{x+1})}

(c)  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=\sin^{-1} x}

Foundations:  
1. Product Rule
        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}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)}
2. Quotient Rule
        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}{dx}\bigg(\frac{f(x)}{g(x)}\bigg)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}}
3. Chain Rule
        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}{dx}(f(g(x)))=f'(g(x))g'(x)}


Solution:

(a)

Step 1:  
Using the Chain Rule, 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 \frac{dy}{dx}=3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{x^2+3}{x^2-1}\bigg)'.}
Step 2:  
Now, using the Quotient Rule, 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 \begin{array}{rcl} \displaystyle{\frac{dy}{dx}} & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{x^2+3}{x^2-1}\bigg)'}\\ &&\\ & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{(x^2-1)(x^2+3)'-(x^2+3)(x^2-1)'}{(x^2-1)^2}\bigg)}\\ &&\\ & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{(x^2-1)(2x)-(x^2+3)(2x)}{(x^2-1)^2}\bigg)}\\ &&\\ & = & \displaystyle{3\bigg(\frac{x^2+3}{x^2-1}\bigg)^2\bigg(\frac{2x^3-2x-2x^3-6x}{(x^2-1)^2}\bigg)}\\ &&\\ & = & \displaystyle{\frac{3(x^2+3)^2(-8x)}{(x^2-1)^4}.} \end{array}}

(b)

Step 1:  
Using the Product Rule, 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 \frac{dy}{dx}=x(\cos(\sqrt{x+1}))'+(x)'\cos(\sqrt{x+1}).}
Step 2:  
Now, using the Chain Rule, 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 \begin{array}{rcl} \displaystyle{\frac{dy}{dx}} & = & \displaystyle{x(\cos(\sqrt{x+1}))'+(x)'\cos(\sqrt{x+1})}\\ &&\\ & = & \displaystyle{x(-\sin(\sqrt{x+1}))(\sqrt{x+1})'+(1)\cos(\sqrt{x+1})}\\ &&\\ & = & \displaystyle{-x\sin(\sqrt{x+1})\frac{1}{2\sqrt{x+1}}(x+1)'+\cos(\sqrt{x+1})}\\ &&\\ & = & \displaystyle{\frac{-x\sin(\sqrt{x+1})}{2\sqrt{x+1}}+\cos(\sqrt{x+1}).} \end{array}}

(c)

Step 1:  
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=\sin^{-1}(x).}   Then,
       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 \sin(y)=x}
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 y}   in the interval  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 \bigg[-\frac{\pi}{2},\frac{\pi}{2}\bigg].}
Using implicit differentiation, 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 \cos(y) \frac{dy}{dx}=1.}
Solving 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},}   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}=\frac{1}{\cos(y)}.}
Step 2:  
Now, since  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 \sin(y)=x,}   we have the following diagram.
(Insert diagram)
Therefore,
        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 \cos(y)=\sqrt{1-x^2}.}
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 \begin{array}{rcl} \displaystyle{\frac{dy}{dx}} & = & \displaystyle{\frac{1}{\cos(y)}}\\ &&\\ & = & \displaystyle{\frac{1}{\sqrt{1-x^2}}.} \end{array}}


Final Answer:  
   (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 \frac{3(x^2+3)^2(-8x)}{(x^2-1)^4}}
   (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 \frac{-x\sin(\sqrt{x+1})}{2\sqrt{x+1}}+\cos(\sqrt{x+1})}
   (c)    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{1}{\sqrt{1-x^2}}}

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