Question

Ley y = tan x. (a) Find the differential dy.(b) Evaluate dy and Δy if x = π/4 and dx = −0.1. (Round Δy to four decimal places.)

Answer 1

Answer:

Required differential (a) $dy=-0.0002\times10^{3}$ and (b) $\Delta y=-0.3370$.

Step-by-step explanation:

Given,

$y=\tan x$

Let, $y=f(x)=\tsn x\hfill (1)$

(a) To find differential,

$dy$ differentiate (1) we get,

$dy=\sec^2 x dx$

when, $x=\frac{\pi}{4}$ and dx=-0.1, then

$dy=\sec^2(\frac{\pi}{4})\times (-0.1)=-0.2$

$dy=-0.0002\times 10^{3}$    (Correct upto four decimal places)

(b) To find,

$\Delta y=\Delta f(x)=f(x+h)-f(x)$    ( where h=dy)

when $x=\frac{\pi}{4}, dx=h=-0.1$

$\therefore \Delta y=\tan (\frac{\pi}{4}-0.2)-\tan \frac{\pi}{4}$

$=-0.3370$

Correct upto four decimal places.