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Local Linear Approximation; Differentials Solutions To Selected ...

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4. Find formulas for dy and △y.<br />

(a)<br />

dy<br />

dx<br />

y = x 2 − 2x + 1<br />

= 2x − 2<br />

dy = (2x − 2)dx<br />

△y = f(x + △x) − f(x)<br />

(b) y = sin(x)<br />

dy<br />

dx<br />

= [(x + △x) 2 − 2(x + △x) + 1] − [x 2 − 2x + 1]<br />

= x 2 + 2x △ x + (△x) 2 − 2x − 2 △ x + 1 − x 2 + 2x − 1<br />

= 2x △ x + (△x) 2 − 2 △ x<br />

= cos(x)<br />

dy = cos(x)dx<br />

△y = f(x + △x) − f(x)<br />

= sin(x + △x) − sin(x)<br />

5

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