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2021-11-08

evaluate dy and delta y

let $y = t a n x$

evaluate dy and delta y if $x = p i / 4$ and dx=-1

Answer & Explanation

Jeffrey Jordon

Expert2021-11-12Added 2605 answers

Consider the function $y = \tan x$ .
Let $f (x) = y = \tan x$
a)
Find the differential $d y$ .
Recollect:
The differential $d y$ is then defined in terms of $d x$ by the equation

$d y = f^{'} (x) d x$

Given $f (x) = \tan x$

$f^{'} (x) = \sec^{2} x$

Thus, $d y = \sec^{2} x d x$

Evaluate $d y$ at $x = \frac{π}{4}$ and $d x = - 1$

$d y = \sec^{2} x d x$

$= \sec^{2} (\frac{π}{4}) (- 1)$

$= (\sqrt{2})^{2} (- 1)$ Since $\sec (\frac{π}{4}) = \sqrt{2}$

$= 2 (- 1)$

$= - 2$

Find $Δ y$ .

Recollect $Δ y = f (x + Δ x) - f (x)$

Let $Δ x = d x = - 1$

$Δ y = f (x + Δ x) - f (x)$

$= f (\frac{π}{4} - 1) - f (\frac{π}{4})$

$= \tan (\frac{π}{4} - 1) - \tan (\frac{π}{4})$

$= (1 - 1) - 1$

$= - 1$

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