11DIFFERENTIATION - Department of Mathematics

S ELF-CHECK E XERCISES 11.7
11.7 Exercises
In Exercises 1–14, find the differential of the
given function.
1. f(x) 2x2 2. f(x) 3x2 1
3. f(x) x 3 x 4. f(x) 2x 3 x
5. f(x) x 1
6. f(x) 3
x
7. f(x) 2x 3/2 x 1/2 8. f(x) 3x 5/6 7x 2/3
11.7 DIFFERENTIALS 777
Therefore,
dV V 3dx 3(0.02) 0.06
x
and we see that the maximum percentage error in the measurement of the
volume of the cube is 6%.
Finally, we want to point out that if at some point in reading this section
you have a sense of déjà vu, do not be surprised, because the notion of the
differential was first used in Section 11.4 (see Example 1). There we took
x 1 since we were interested in finding the marginal cost when the level
of production was increased from x 250 to x 251. If we had used differentials,
we would have found
C(251) C(250) C(250)dx
so that taking dx x 1, we have C(251) C(250) C(250), which
agrees with the result obtained in Example 1. Thus, in Section 11.4, we touched
upon the notion of the differential, albeit in the special case in which dx 1.
1. Find the differential of f(x) x 1.
2. Acertain country’s government economists have determined that the demand equation
for corn in that country is given by
p f(x) 125
x2 1
where p is expressed in dollars per bushel and x, the quantity demanded per year,
is measured in billions of bushels. The economists are forecasting a harvest of 6
billion bushels for the year. If the actual production of corn were 6.2 billion bushels
for the year instead, what would be the approximate drop in the predicted price
of corn per bushel?
Solutions to Self-CheckExercises 11.7 can be found on page 782.
9. f(x) x 2
x
11. f(x)
x 1
x 2 1
10. f(x) 3
x 1
12. f(x) 2x2 1
x 1
13. f(x) 3x 2 x 14. f(x) (2x 2 3) 1/3
15. Let f be a function defined by
y f(x) x 2 1