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  $y=\tan x,$  we have
$dy\,=\,\sec ^{2}x\,dx.$

Step 2:
Now, we plug  $x={\frac {\pi }{4}}$  into the differential from Step 1.
So, we get
$dy\,=\,{\bigg (}\sec {\bigg (}{\frac {\pi }{4}}{\bigg )}{\bigg )}^{2}\,dx\,=\,2\,dx.$

(b)

Step 1:
First, we find  $dx.$  We have
$dx=0.885-{\frac {\pi }{4}}\approx 0.885-0.785=0.1.$
Then, we plug this into the differential from part (a).
So, we have
$dy\,=\,2(0.1)\,=\,0.2.$

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
$\tan(0.885)\,\approx \,\tan {\bigg (}{\frac {\pi }{4}}{\bigg )}+0.2\,=\,1.2.$

Final Answer:
(a)     $dy=2\,dx$
(b)     $1.2$

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