11DIFFERENTIATION - Department of Mathematics
11DIFFERENTIATION - Department of Mathematics
11DIFFERENTIATION - Department of Mathematics
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Using Technology<br />
780<br />
EXAMPLE 1<br />
F INDING THE D IFFERENTIAL<br />
OF A F UNCTION<br />
The calculation <strong>of</strong> the differential <strong>of</strong> f at a given value <strong>of</strong> x involves the<br />
evaluation <strong>of</strong> the derivative <strong>of</strong> f at that point and can be facilitated through<br />
the use <strong>of</strong> the numerical derivative function.<br />
Use dy to approximate y if y x 2 (2x 2 x 1) 2/3 and x changes from 2 to 1.98.<br />
SOLUTION ✔ Let f(x) x2 (2x 2 x 1) 2/3 . Since dx 1.98 2 0.02, we find the<br />
required approximation to be<br />
dy f(2) (0.02)<br />
But using the numerical derivative operation, we find<br />
f(2) 30.5758132855<br />
and so<br />
dy (0.02)(30.5758132855) 0.611516266 <br />
EXAMPLE 2<br />
SOLUTION ✔<br />
The Meyers are considering the purchase <strong>of</strong> a house in the near future and<br />
estimate that they will need a loan <strong>of</strong> $120,000. Based on a 30-year conventional<br />
mortgage with an interest rate <strong>of</strong> r per year, their monthly repayment will be<br />
10,000r<br />
P <br />
1 1 r<br />
360<br />
12<br />
dollars. If the interest rate increases from the present rate <strong>of</strong> 10%/year to<br />
10.2% per year between now and the time the Meyers decide to secure the loan,<br />
approximately how much more per month will their mortgage payment be?<br />
Let’s write<br />
10,000r<br />
P f(r) <br />
1 1 r<br />
360<br />
12<br />
Then the increase in the mortgage payment will be approximately<br />
dP f(0.1)dr f(0.1)(0.002) (Since dr 0.102 0.1)<br />
(8867.59947979)(0.002) 17.7352 (Using the numerical<br />
derivative operation)<br />
or approximately $17.74 per month.
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