348 CHAPTER 6 Applications of the Derivative
6.6 EXERCISES
For Exercises 1–8, find dy for the given values of x and D x. 21. Material Requirement A cube 4 in. on an edge is given a pro-
tective coating 0.1 in. thick. About how much coating should a
1. y 5 2x3 2 5x; x 5 22, Dx 5 0.1 production manager order for 1000 such cubes?
3
2. y 5 4x 2 3x; x 5 3, Dx 5 0.2 22. Material Requirement Beach balls 1 ft in diameter have a
3 2
3. y 5 x 2 2x 1 3; x 5 1, Dx 5 20.1 thickness of 0.03 in. How much material would be needed to
4. y 5 2x3 1 x2 2 4x; x 5 2, Dx 5 20.2 make 5000 beach balls?
5. y 5 "3x 1 2 x 5 4, Dx 5 0.15
L i fe S c i en c e s
6. y 5 "4x 2 1; x 5 5, Dx 5 0.08
23. Alcohol Concentration In Exercise 55 in the section on Poly-
2x 2 5
7. y 5 ; x 5 2, Dx 5 20.03 nomial and Rational Functions, we gave the function defined by
x11
A 1 x 2 5 0.003631x3 2 0.03746x2 1 0.1012x 1 0.009
6x 2 3
8. y 5 ; x 5 3, Dx 5 20.04
2x 1 1 as the approximate blood alcohol concentration in a 170-lb
woman x hours after drinking 2 oz of alcohol on an empty
Use the differential to approximate each quantity. Then use a stomach, for x in the interval [0, 5]. Source: Medicolegal
calculator to approximate the quantity, and give the absolute Aspects of Alcohol Determination in Biological Specimens.
value of the difference in the two results to 4 decimal places.
a. Approximate the change in alcohol level from 1 to
9. "145 10. "23 1.2 hours.
11. "0.99 12. "17.02 b. Approximate the change in alcohol level from 3 to
13. e0.01 14. e20.002 3.2 hours.
15. ln 1.05 16. ln 0.98 24. Drug Concentration The concentration of a certain drug in
the bloodstream x hours after being administered is approxi-
mately
APPLICATIONS 5x
C1 x 2 5 .
Business and Economics 9 1 x2
17. Demand The demand for grass seed (in thousands of pounds) Use the differential to approximate the changes in concentra-
at a price of p dollars is tion for the following changes in x.
D 1 p 2 5 23p3 2 2p2 1 1500. a. 1 to 1.5 b. 2 to 2.25
Use the differential to approximate the changes in demand for 25. Bacteria Population The population of bacteria (in millions)
the following changes in p. in a certain culture x hours after an experimental nutrient is
introduced into the culture is
a. $2 to $2.10 b. $6 to $6.15
18. Average Cost The average cost (in dollars) to manufacture 25x
P1 x 2 5 .
x dozen marking pencils is 8 1 x2
A 1 x 2 5 0.04x3 1 0.1x2 1 0.5x 1 6. Use the differential to approximate the changes in population
Use the differential to approximate the changes in the average for the following changes in x.
cost for the following changes in x. a. 2 to 2.5 b. 3 to 3.25
a. 3 to 4 b. 5 to 6 26. Area of a Blood Vessel The radius of a blood vessel is 1.7 mm.
A drug causes the radius to change to 1.6 mm. Find the approxi-
19. Revenue A company estimates that the revenue (in dollars)
mate change in the area of a cross section of the vessel.
from the sale of x doghouses is given by
R 1 x 2 5 12,000 ln 1 0.01x 1 1 2 . 27. Volume of a Tumor A tumor is approximately spherical in
shape. If the radius of the tumor changes from 14 mm to 16 mm,
Use the differential to approximate the change in revenue from the find the approximate change in volume.
sale of one more doghouse when 100 doghouses are sold.
28. Area of an Oil Slick An oil slick is in the shape of a circle.
20. Profit The cost function for the company in Exercise 19 is Find the approximate increase in the area of the slick if its
C 1 x 2 5 150 1 75x, radius increases from 1.2 miles to 1.4 miles.
where x represents the demand for the product. Find the approx- 29. Area of a Bacteria Colony The shape of a colony of bacteria
imate change in profit for a 1-unit change in demand when on a Petri dish is circular. Find the approximate increase in its
demand is at a level of 100 doghouses. Use the differential. area if the radius increases from 20 mm to 22 mm.
6.6 EXERCISES
For Exercises 1–8, find dy for the given values of x and D x. 21. Material Requirement A cube 4 in. on an edge is given a pro-
tective coating 0.1 in. thick. About how much coating should a
1. y 5 2x3 2 5x; x 5 22, Dx 5 0.1 production manager order for 1000 such cubes?
3
2. y 5 4x 2 3x; x 5 3, Dx 5 0.2 22. Material Requirement Beach balls 1 ft in diameter have a
3 2
3. y 5 x 2 2x 1 3; x 5 1, Dx 5 20.1 thickness of 0.03 in. How much material would be needed to
4. y 5 2x3 1 x2 2 4x; x 5 2, Dx 5 20.2 make 5000 beach balls?
5. y 5 "3x 1 2 x 5 4, Dx 5 0.15
L i fe S c i en c e s
6. y 5 "4x 2 1; x 5 5, Dx 5 0.08
23. Alcohol Concentration In Exercise 55 in the section on Poly-
2x 2 5
7. y 5 ; x 5 2, Dx 5 20.03 nomial and Rational Functions, we gave the function defined by
x11
A 1 x 2 5 0.003631x3 2 0.03746x2 1 0.1012x 1 0.009
6x 2 3
8. y 5 ; x 5 3, Dx 5 20.04
2x 1 1 as the approximate blood alcohol concentration in a 170-lb
woman x hours after drinking 2 oz of alcohol on an empty
Use the differential to approximate each quantity. Then use a stomach, for x in the interval [0, 5]. Source: Medicolegal
calculator to approximate the quantity, and give the absolute Aspects of Alcohol Determination in Biological Specimens.
value of the difference in the two results to 4 decimal places.
a. Approximate the change in alcohol level from 1 to
9. "145 10. "23 1.2 hours.
11. "0.99 12. "17.02 b. Approximate the change in alcohol level from 3 to
13. e0.01 14. e20.002 3.2 hours.
15. ln 1.05 16. ln 0.98 24. Drug Concentration The concentration of a certain drug in
the bloodstream x hours after being administered is approxi-
mately
APPLICATIONS 5x
C1 x 2 5 .
Business and Economics 9 1 x2
17. Demand The demand for grass seed (in thousands of pounds) Use the differential to approximate the changes in concentra-
at a price of p dollars is tion for the following changes in x.
D 1 p 2 5 23p3 2 2p2 1 1500. a. 1 to 1.5 b. 2 to 2.25
Use the differential to approximate the changes in demand for 25. Bacteria Population The population of bacteria (in millions)
the following changes in p. in a certain culture x hours after an experimental nutrient is
introduced into the culture is
a. $2 to $2.10 b. $6 to $6.15
18. Average Cost The average cost (in dollars) to manufacture 25x
P1 x 2 5 .
x dozen marking pencils is 8 1 x2
A 1 x 2 5 0.04x3 1 0.1x2 1 0.5x 1 6. Use the differential to approximate the changes in population
Use the differential to approximate the changes in the average for the following changes in x.
cost for the following changes in x. a. 2 to 2.5 b. 3 to 3.25
a. 3 to 4 b. 5 to 6 26. Area of a Blood Vessel The radius of a blood vessel is 1.7 mm.
A drug causes the radius to change to 1.6 mm. Find the approxi-
19. Revenue A company estimates that the revenue (in dollars)
mate change in the area of a cross section of the vessel.
from the sale of x doghouses is given by
R 1 x 2 5 12,000 ln 1 0.01x 1 1 2 . 27. Volume of a Tumor A tumor is approximately spherical in
shape. If the radius of the tumor changes from 14 mm to 16 mm,
Use the differential to approximate the change in revenue from the find the approximate change in volume.
sale of one more doghouse when 100 doghouses are sold.
28. Area of an Oil Slick An oil slick is in the shape of a circle.
20. Profit The cost function for the company in Exercise 19 is Find the approximate increase in the area of the slick if its
C 1 x 2 5 150 1 75x, radius increases from 1.2 miles to 1.4 miles.
where x represents the demand for the product. Find the approx- 29. Area of a Bacteria Colony The shape of a colony of bacteria
imate change in profit for a 1-unit change in demand when on a Petri dish is circular. Find the approximate increase in its
demand is at a level of 100 doghouses. Use the differential. area if the radius increases from 20 mm to 22 mm.
