009A Sample Final 3, Problem 10

Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \tan(0.885).} Hint: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 (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\,=\,\sec^2 x\,dx.}

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

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

Final Answer:
(a)
(b)
(c)

Return to Sample Exam