009A Sample Final 1, Problem 8 Detailed Solution

Let

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=x^{3}.}

(a) Find the differential  $dy$  of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=x^{3}}   at  $x=2$.

(b) Use differentials to find an approximate value for  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1.9^{3}} .

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Background Information:
What is the differential  $dy$  of  $y=x^{2}$  at  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=1?}
Since  $x=1,$  the differential is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy=2xdx=2dx.}

Solution:

(a)

Step 1:
First, we find the differential  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy.}
Since  $y=x^{3},$  we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy\,=\,3x^{2}\,dx.}

Step 2:
Now, we plug  $x=2$  into the differential from Step 1.
So, we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy\,=\,3(2)^{2}\,dx\,=\,12\,dx.}

(b)

Step 1:
First, we 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 dx.}   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 dx=1.9-2=-0.1.}
Then, we plug this into the differential from part (a).
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 dy\,=\,12(-0.1)\,=\,-1.2.}

Step 2:
Now, we add the value 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 dy}   to  $2^{3}$  to get an
approximate value of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1.9^{3}.}
Hence, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1.9^{3}\,\approx \,2^{3}+-1.2\,=\,6.8.}

Final Answer:
(a)     $dy=12\,dx$
(b)     $6.8$

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