022 Sample Final A, Problem 13

Use differentials to find $dy$ given $y=x^{2}-6x,~x=4,~dx=-0.5.$

Foundations:
When we use differentials, we are approximating a value for a function by using the slope of the derivative. The idea is that given a distance $dx$ from a point $x$, we can use $f'(x)$, the slope of the tangent line, to find the rise, $dy$. Recalling that we can write
$f'(x)\,=\,{\frac {dy}{dx}},$
the relation is
$dy\,=\,f'(x)\cdot dx,$
where we use the given specific $x$-value to evaluate $f'(x)$.

 Solution:

Step 1:
By the power rule, we have
$f'(x)\,=\,2x-6.$
We need to evaluate this at the given value $x=4$, so
$f'(4)\,=\,2(4)-6\,=\,2.$

Step 2:
We use the values given and the result from step 1 to find
$dy\,=\,f'(x)\cdot dx\,=\,2(-0.5)\,=\,-1.$

Final Answer:
$dy\,=\,-1.$

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