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 $1.9^{3}$.

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

Solution:

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

(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
$1.9^{3}\approx 2^{3}+-1.2=6.8$.

Final Answer:
(a) $dy=12dx$
(b) $6.8$

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