009A Sample Final 3, Problem 10

Let  $y=\tan(x).$

(a) Find the differential  $dy$  of  $y=\tan(x)$  at  $x={\frac {\pi }{4}}.$

(b) Use differentials to find an approximate value for  $\tan(0.885).$  Hint:  ${\frac {\pi }{4}}\approx 0.785.$

Foundations:
What is the differential  $dy$  of  $y=x^{2}$  at  $x=1?$
Since  $x=1,$  the differential is  $dy=2xdx=2dx.$

Solution:

(a)

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

Step 2:
Now, we plug  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x={\frac {\pi }{4}}}   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\,=\,{\bigg (}\sec {\bigg (}{\frac {\pi }{4}}{\bigg )}{\bigg )}^{2}\,dx\,=\,2\,dx.}

(b)

Step 1:
First, we find  $dx.$  We have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dx=0.885-{\frac {\pi }{4}}\approx 0.885-0.785=0.1.}
Then, we plug this into the differential from part (a).
So, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dy\,=\,2(0.1)\,=\,0.2.}

Step 2:
Now, we add the value for  $dy$  to  $\tan {\bigg (}{\frac {\pi }{4}}{\bigg )}$  to get an
approximate value of  $\tan(0.885).$
Hence, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \tan(0.885)\,\approx \,\tan {\bigg (}{\frac {\pi }{4}}{\bigg )}+0.2\,=\,1.2.}

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 dy=2\,dx}
(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 1.2}

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