Difference between revisions of "009A Sample Final 2, Problem 3"
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!Step 1: | !Step 1: | ||
|- | |- | ||
| − | |Let <math>f(x)=\sin(x)</math> and <math>g(x)=\sin^{-1}x.</math> | + | |Let <math style="vertical-align: -5px">f(x)=\sin(x)</math> and <math style="vertical-align: -5px">g(x)=\sin^{-1}x.</math> |
|- | |- | ||
|These functions are inverses of each other since | |These functions are inverses of each other since | ||
|- | |- | ||
| − | | <math>f(g(x))=x</math> and <math>g(f(x))=x.</math> | + | | <math style="vertical-align: -5px">f(g(x))=x</math> and <math style="vertical-align: -5px">g(f(x))=x.</math> |
|- | |- | ||
|Therefore, | |Therefore, | ||
| Line 102: | Line 102: | ||
\end{array}</math> | \end{array}</math> | ||
|- | |- | ||
| − | |Now, let <math>y=\sin^{-1}(x).</math> Then, <math>x=\sin(y).</math> | + | |Now, let <math style="vertical-align: -5px">y=\sin^{-1}(x).</math> Then, <math style="vertical-align: -5px">x=\sin(y).</math> |
|- | |- | ||
| − | |So, <math>\cos(\sin^{-1} x)=\cos(y).</math> | + | |So, <math style="vertical-align: -5px">\cos(\sin^{-1} x)=\cos(y).</math> |
|- | |- | ||
|Therefore, | |Therefore, | ||
| Line 117: | Line 117: | ||
|- | |- | ||
| <math>\cos^2 y+\sin^2 y =1,</math> | | <math>\cos^2 y+\sin^2 y =1,</math> | ||
| + | |- | ||
| + | |we have | ||
|- | |- | ||
| <math>\begin{array}{rcl} | | <math>\begin{array}{rcl} | ||
| Line 137: | Line 139: | ||
| '''(b)''' <math>\frac{-x\sin(\sqrt{x+1})}{2\sqrt{x+1}}+\cos(\sqrt{x+1})</math> | | '''(b)''' <math>\frac{-x\sin(\sqrt{x+1})}{2\sqrt{x+1}}+\cos(\sqrt{x+1})</math> | ||
|- | |- | ||
| − | | '''(c)''' <math> | + | | '''(c)''' <math>\frac{1}{\sqrt{1-x^2}}</math> |
|} | |} | ||
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 17:36, 7 March 2017
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}.}
(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 f(x)=\sin(x)} 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 g(x)=\sin^{-1}x.} |
| These functions are inverses of each other 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 f(g(x))=x} 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 g(f(x))=x.} |
| 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 \begin{array}{rcl} \displaystyle{g'(x)} & = & \displaystyle{\frac{1}{f'(g(x))}}\\ &&\\ & = & \displaystyle{\frac{1}{\cos(\sin^{-1} x)}.} \end{array}} |
| Now, 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=\sin(y).} |
| 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 \cos(\sin^{-1} x)=\cos(y).} |
| 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 g'(x)=\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 \cos^2 y+\sin^2 y =1,} |
| 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{\cos(y)} & = & \displaystyle{\sqrt{1-\sin^2 y}}\\ &&\\ & = & \displaystyle{\sqrt{1-x^2}.} \end{array}} |
| 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 g'(x)=\frac{1}{\sqrt{1-x^2}}.} |
| 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}}} |