11DIFFERENTIATION - Department of Mathematics

11.7 Differentials
EXAMPLE 1
SOLUTION ✔
11.7 DIFFERENTIALS 771
The Millers are planning to buy a house in the near future and estimate that
they will need a 30-year fixed-rate mortgage for $120,000. If the interest rate
increases from the present rate of 9% per year to 9.4% per year between now
and the time the Millers decide to secure the loan, approximately how much
more per month will their mortgage be? (You will be asked to answer this
question in Exercise 44, page 781.)
Questions such as this, in which one wishes to estimate the change in the
dependent variable (monthly mortgage payment) corresponding to a small
change in the independent variable (interest rate per year), occur in many
real-life applications. For example:
■ An economist would like to know how a small increase in a country’s capital
expenditure will affect the country’s gross domestic output.
■ Asociologist would like to know how a small increase in the amount of
capital investment in a housing project will affect the crime rate.
■ Abusinesswoman would like to know how raising a product’s unit price
by a small amount will affect her profit.
■ Abacteriologist would like to know how a small increase in the amount
of a bactericide will affect a population of bacteria.
To calculate these changes and estimate their effects, we use the differential
of a function, a concept that will be introduced shortly.
I NCREMENTS
Let x denote a variable quantity and suppose x changes from x1 to x2. This
change in x is called the increment in x and is denoted by the symbol x
(read ‘‘delta x’’). Thus,
Find the increment in x:
x x2 x1 (Final value minus initial value) (9)
a. As x changes from 3 to 3.2 b. As x changes from 3 to 2.7
a. Here, x1 3 and x2 3.2, so
x x2 x 1 3.2 3 0.2
b. Here, x1 3 and x2 2.7. Therefore,
x x2 x1 2.7 3 0.3
Observe that x plays the same role that h played in Section 10.4.
Now, suppose two quantities, x and y, are related by an equation
y f(x), where f is a function. If x changes from x to x x, then the