009A Sample Final 1, Problem 8

Let

$y=x^{3}.$

a) Find the differential $dy$ of $y=x^{3}$ at $x=2$.

b) Use differentials to find an approximate value for $1.9^{3}$.

1

Foundations:
What is the differential $dy$ of $y=x^{2}$ at $x=1?$
Since  $x=1,$  the differential is  $dy=2xdx=2dx.$

Solution:

2

(a)

Step 1:
First, we find the differential $dy.$
Since $y=x^{3},$ we have
$dy=3x^{2}dx.$

Step 2:
Now, we plug in 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=2} into the differential from Step 1.
So, we get
$dy=3(2)^{2}dx=12dx.$

3

(b)

Step 1:
First, we find $dx$. We have $dx=1.9-2=-0.1.$
Then, we plug this into the differential from part (a).
So, we have
$dy=12(-0.1)=-1.2.$

Step 2:
Now, we add the value for $dy$ to $2^{3}$ to get an
approximate value of $1.9^{3}.$
Hence, 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 1.9^3\approx 2^3+-1.2=6.8.}

4

Final Answer:
(a) 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=12dx}
(b) 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 6.8}

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