11DIFFERENTIATION - Department of Mathematics

778 11 DIFFERENTIATION
a. Find the differential of f.
b. Use your result from part (a) to find the approximate
change in y if x changes from 1 to 1.02.
c. Find the actual change in y if x changes from 1 to 1.02
and compare your result with that obtained in part (b).
16. Let f be a function defined by
y f(x) 3x 2 2x 6
a. Find the differential of f.
b. Use your result from part (a) to find the approximate
change in y if x changes from 2 to 1.97.
c. Find the actual change in y if x changes from 2 to 1.97
and compare your result with that obtained in part (b).
17. Let f be a function defined by
y f(x) 1
x
a. Find the differential of f.
b. Use your result from part (a) to find the approximate
change in y if x changes from 1 to0.95.
c. Find the actual change in y if x changes from 1 to
0.95 and compare your result with that obtained in
part (b).
18. Let f be a function defined by
y f(x) 2x 1
a. Find the differential of f.
b. Use your result from part (a) to find the approximate
change in y if x changes from 4 to 4.1.
c. Find the actual change in y if x changes from 4 to 4.1
and compare your result with that obtained in part (b).
In Exercises 19–26, use differentials to approximate
the given quantity.
19. 10 20. 17 21. 49.5
22. 99.7 23. 3 7.8 24. 4 81.6
25. 0.089 26. 3 0.00096
27. Use a differential to approximate 4.02 1
4.02 .
Hint: Let f(x) x 1
and compute dy with x 4 and
x
dx 0.02.
28. Use a differential to approximate
Hint: Study the hint for Exercise 27.
2(4.98)
(4.98) 2 1 .
A calculator is recommended for the remainder
of this exercise set.
29. E RROR E STIMATION The length of each edge of a cube is
12 cm, with a possible error in measurement of 0.02 cm.
Use differentials to estimate the error that might occur
when the volume of the cube is calculated.
30. E STIMATING THE A MOUNT OF P AINT R EQUIRED Acoat of paint
of thickness 0.05 cm is to be applied uniformly to the
faces of a cube of edge 30 cm. Use differentials to find
the approximate amount of paint required for the job.
31. E RROR E STIMATION Ahemisphere-shaped dome of radius
60 ft is to be coated with a layer of rust-proofer before
painting. Use differentials to estimate the amount of
rust-proofer needed if the coat is to be 0.01 in. thick.
Hint: The volume of a hemisphere of radius r is V r 3 .
32. G ROWTH OF A C ANCEROUS T UMOR The volume of a spherical
cancer tumor is given by
V(r) 4 3 r
3
If the radius of a tumor is estimated at 1.1 cm, with a
maximum error in measurement of 0.005 cm, determine
the error that might occur when the volume of the tumor
is calculated.
33. U NCLOGGING A RTERIES Research done in the 1930s by the
French physiologist Jean Poiseuille showed that the resistance
R of a blood vessel of length l and radius r is
R kl/r 4 , where k is a constant. Suppose a dose of the
drug TPAincreases r by 10%. How will this affect the
resistance R? Assume that l is constant.
34. G ROSS D OMESTIC P RODUCT An economist has determined
that a certain country’s gross domestic product (GDP)
is approximated by the function f(x) 640x 1/5 , where
f(x) is measured in billions of dollars and x is the capital
outlay in billions of dollars. Use differentials to estimate
the change in the country’s GDP if the country’s capital
expenditure changes from $243 billion to $248 billion.
35. L EARNING C URVES The length of time (in seconds) a certain
individual takes to learn a list of n items is approximated
by
f(n) 4n n 4
Use differentials to approximate the additional time it
takes the individual to learn the items on a list when n
is increased from 85 to 90 items.