Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

4.E: Functions of Two Variables (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

4.1 Exercises

Exercise 4.E.1

F(x,y)=x2y2. Find

a) F(0,4)

b) F(4,0)

c) F(x,4)

d) F(4,y)

e) F(800,800)

f) F(x,x)

g) F(x,x)

Exercise 4.E.2

g(s,t)=st2. Find

a) g(1,9)

b) g(9,1)

c) g(1,t)

d) g(s,9)

e) g(w,z+1)

Exercise 4.E.3

Let f(x,y,z,w)=x21zw+xyz2. Evaluate f(1,2,3,4).

Exercise 4.E.4

Let f(x,y,z,w)=xyw2+102yz. Evaluate f(1,2,3,4).

Exercise 4.E.5

Here is a table showing the function A(t,r)

t r

.03

.04

.05

.06

.07

1

30.45

40.81

51.27

61.84

72.51

2

61.84

83.29

105.17

127.50

150.27

3

94.17

127.50

161.83

197.22

233.68

a) Find A(2,.05)

b) Find A(.05,.2)

c) Is A(t,.06) an increasing or decreasing function of t?

d) Is A(3,r) an increasing or decreasing function of r?

Exercise 4.E.6

Here is a table showing values for the function H(t,h).

t h

100

150

200

0

100

150

200

1

110.1

160.1

210.1

2

110.4

160.4

210.4

3

100.9

150.9

200.9

4

81.6

131.6

181.6

5

52.5

102.5

152.5

a) Is H(t,150) an increasing or decreasing function of t?

b) Is H(4,h) an increasing or decreasing function of h?

c) Fill in the blanks: The maximum value shown on this table is H(___,___) =____.

d) Fill in the blanks: The minimum value shown on this table is H(___,___) =____.

Exercise 4.E.74.E.10

In problems 7 – 10, plot the given points.

7. A=(0,3,4),B=(1,4,0),C=(1,3,4),D=(1,4,2)

8. E=(4,3,0),F=(3,0,1),G=(0,4,1),H=(3,3,1)

9. P=(2,3,4),Q=(1,2,3),R=(4,1,2),S=(2,1,3)

10. T=(2,3,4),U=(2,0,3),V=(2,0,0),W=(3,1,2)

Exercise 4.E.114.E.14

In problems 11 – 14, calculate the distances between the given points

11. A=(5,3,4),B=(3,4,4)

12. A=(6,2,1),B=(3,2,1)

13. A=(3,4,2),B=(1,6,2)

14. A=(1,5,0),B=(1,3,2)

Exercise 4.E.154.E.18

In problems 15 – 18, graph the given planes.

15. y=1 and z=2 16. x=4 and y=2
17. x=1 and y=0 18. x=2 and z=0
Exercise 4.E.194.E.22

In problems 19 – 22, the center and radius of a sphere are given. Find an equation for the sphere.

19. Center = (4, 3, 5), radius = 3 20. Center = (0, 3, 6), radius = 2
21. Center = (5, 1, 0), radius = 5 22. Center = (1, 2, 3), radius = 4
Exercise 4.E.234.E.24

In problems 23 – 24, the equation of a sphere is given. Find the center and radius of the sphere.

23. (x3)2+(y+4)2+(z1)2=16 24. (x+2)2+y2+(z4)2=25
Exercise 4.E.254.E.30

For problems 25 through 30. Match the contour diagram to the computer-generated, perspective drawing (a through f) it matches. Briefly explain your answer.

clipboard_efe2d1cdfbc09745912c5ce1dc90e6040.png
25.
clipboard_e8c98ddfe0de5e6ed53881b4feb36cb0b.png
26.
clipboard_e4eec730dcc5a9e78954d8911c35f5f19.png
27.
clipboard_e10885a15178e06fbea1fb11b5e8636d9.png
28.
clipboard_eca4cbc0940a558e97dfd5c968687299a.png
29.
clipboard_e312bb04675b5be57146ed602fd50a73d.png
30.
clipboard_ea68328bd1b4aefb30cb2c56a053cd138.png
a.
clipboard_e7f48ec1902bace287ad789502a7f98a5.png
b.
clipboard_ea7939840222bb25f245c3b99b100d611.png
c.
clipboard_ed32ecd956693ac9b21e450c7d0fee6d7.png
d.
clipboard_edd8564361e43a17663487a749c96c554.png
e.
clipboard_e718f3428628c60d5d932e65c1ec4a6e0.png
f.
Exercise 4.E.314.E.36

For problems 31 through 36. Match the contour diagram to the equation (a through f) it matches. Briefly explain your answer.

clipboard_ed340079b274839cc609797f811d9e173.png
31.
clipboard_e42539ec6b98d1e66a36c46d9278a49d9.png
32.
clipboard_ed3c63fc41c538bc1750a9379328cea58.png
33.
clipboard_e28090318afdd6f72e911159f948b1ba4.png
34.
clipboard_ea6b99ab7182789e503ceb06f50204059.png
35.
clipboard_e052a26b3c45f5a549b51bb26ae359855.png
36.
a. f(x,y)=yx b. f(x,y)=xy2
c. f(x,y)=25x2y2 d. f(x,y)=5xy
e. f(x,y)=0.01x2y2 f. f(x,y)=x2y2
Exercise 4.E.37

The contour diagram shown is for a function M(x,y).

clipboard_ec5afce2e5112612b273eb9ab6684f186.png

Use the diagram to answer the following:

a) Estimate M(1,3)

b) Estimate M(3,1)

c) Is M(x,3) an increasing or decreasing function of x?

d) Is M(3,y) an increasing or decreasing function of y?

e) Find a value of c so that M(c,y) is a constant function of y.

Exercise 4.E.38

The contour diagram shown is for a function G(x,y).

clipboard_e9202d1c7d64e083145dc1cc74450d482.png

Use the diagram to answer the following:

a) Estimate G(2,3)

b) Suppose you travel north (in the direction of increasing y) along the surface, starting above (2, 3). Describe your journey.

c) Suppose you travel east (in the direction of increasing x) along the surface, starting above (2, 3). Describe your journey.

Exercise 4.E.39

The demand functions for two products are given below. p1, p2, q1, and q2 are the prices (in dollars) and quantities for products 1 and 2. Are these two products complementary goods or substitute goods

q1=2003p1+p2

q2=150+p12p2

Exercise 4.E.40

The demand functions for two products are given below. p1, p2, q1, and q2 are the prices (in dollars) and quantities for products 1 and 2. Are these two products complementary goods or substitute goods

q1=350+p1+2p2

q2=225+p1+p2

Exercise 4.E.41

Consider the Cobb-Douglas Production function: P(L,K)=11L0.3K0.7. Find the total units of production when 19 units of labor and 12 units of capital are invested.

Exercise 4.E.42

Consider the Cobb-Douglas Production function: P(L,K)=6L0.6K0.4. Find the total units of production when 24 units of labor and 8 units of capital are invested.

4.2 Exercises

Exercise 4.E.14.E.16

For problems 1 through 16, find fx and fy for the function given

1. f(x,y)=x25y2
2. f(x,y)=x25y2x+4
3. f(x,y)=ex+6y
4. f(x,y)=(x25y2)ex
5. f(x,y)=(x25y2)(13y+4)
6. f(x,y)=x
7. f(x,y)=6
8. f(x,y)=ln(xy+2x6y)
9. f(x,y)=x25y2y45x4
10. f(x,y)=ex4y(x4y)
11. f(x,y)=y5ex
12. f(x,y)=116xy
13. f(x,y)=(x+ey)7
14. f(x,y)=x4+4x3y6x2y24xy3+y4
15. f(x,y)=x+y
16. f(x,y)=x2y34x3
Exercise 4.E.17

Here is a table showing the function A(t,r)

t r

.03

.04

.05

.06

.07

1

30.45

40.81

51.27

61.84

72.51

2

61.84

83.29

105.17

127.50

150.27

3

94.17

127.50

161.83

197.22

233.68

a. Estimate At(2,.05).

b. Estimate Ar(2,.05)

c. Use your answers to parts a and b to estimate the value of A(2.5,.054)

d. The values in the table came from A(t,r)=1000(ert1), which shows the interest earned if 1000 dollars is deposited in an account earning r annual interest, compounded continuously, and left there for t years. How close are your estimates from parts a, b, and c?

Exercise 4.E.18

18. Here is a table showing values for the function H(t,h).

t h

100

150

200

0

100

150

200

1

110.1

160.1

210.1

2

110.4

160.4

210.4

3

100.9

150.9

200.9

4

81.6

131.6

181.6

5

52.5

102.5

152.5

a. Estimate the value of Hdt at (3, 150).

b. Estimate the value of Hdh at (3, 150).

c. Use your answers to parts a and b to estimate the value of H(2.6,156).

d. The values in the table came from H(t,h)=h+15t4.9t2, which gives the height in meters above the ground after t seconds of an object that is thrown upward from an initial height of h meters with an initial velocity of 15 meters per second. How close are your estimates from parts a, b, and c?

Exercise 4.E.19

Given the function f(x,y)=x2y

a. Calculate f(2,4), fx(2,4), and fy(2,4)

b. Use your answers from part a to estimate f(1.9,4.1)

Exercise 4.E.20

Given the function f(x,y)=ln(10x2y)

a. Calculate f(2,5), fx(2,5), and fy(2,5)

b. Use your answers from part a to estimate f(1.8,4.8)

Exercise 4.E.214.E.26

In problems 21 - 26, use the contour plot shown to estimate the desired value.

clipboard_e36d9cc5d2669270612b725571cb500b0.png
21. fx(1,5)
22. fx(5,2)
23. fx(5,5)
24. fx(0,0)
25. fy(1,5)
26. fy(5,2)

4.3 Exercises

Exercise 4.E.14.E.6

For problems 1 through 6, find fxx, fyy, fxy and fyx for the function given. Confirm that fxy=fyx.

1. f(x,y)=x25y2
2. f(x,y)=x4+4x3y6x2yy4xy3+y4
3. f(x,y)=5x2y2
4. f(x,y)=ex+6y
5. f(x,y)=ln(xy+2x6y)
6. f(x,y)=x2y45
Exercise 4.E.7

Find the critical points of f(x,y)=y3x3+15x212y+12 and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

Exercise 4.E.8

Find the critical points of f(x,y)=2xyx22y2+6x+4 and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

Exercise 4.E.9

Find the critical points of f(x,y)=y24ln(x)+4x and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

Exercise 4.E.10

Find the critical points of f(x,y)=xy6x2+3xy+2 and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

Exercise 4.E.11

The origin is a critical point for the function f(x,y)=x3+y3, and D=0 there. That is, the Second Derivative Test fails. Use what you know about shapes of functions to decide if there is a local minimum, local maximum, or saddle point for this function at (0, 0).

Exercise 4.E.12

The origin is a critical point for the function f(x,y)=15x2y2, and D=0 there. That is, the Second Derivative Test fails. Use what you know about shapes of functions to decide if there is a local minimum, local maximum, or saddle point for this function at (0, 0).

Exercise 4.E.134.E.18

For problems 13 through 18, find all local maxima, minima, and saddle points for the function.

13. f(x,y)=xy5x25y2+33y
14. f(x,y)=10xyx2y2+3x
15. f(x,y)=x3+y33xy
16. f(x,y)=5x24xy+2y2+4x4y+10
17. f(x,y)=y2ex+x2
18. f(x,y)=xy+2xln(x2y), for x>0 and y>0.
Exercise 4.E.19

The demand functions for two products are given below. p1, p2, q1, and q2 are the prices (in dollars) and quantities for products 1 and 2.

q1=200+3p1+p2

q2=150+p1+2p2

a. Are these two products complementary goods or substitute goods?

b. What is the quantity demanded for each when the price for product 1 is $20 per item and the price for product 2 is $30 per item?

c. Write a function R(p1,p2) that expresses the total revenue from these two products.

d. Find the price and quantity for each product that maximizes the total revenue.

Exercise 4.E.20

The demand functions for two products are given below. p1, p2, q1, and q2 are the prices (in dollars) and quantities for products 1 and 2.

q1=350+p1+2p2

q2=225+p1+p2

a. Are these two products complementary goods or substitute goods?

b. What is the quantity demanded for each when the price for product 1 is $20 per item and the price for product 2 is $30 per item?

c. Write a function R(p1,p2) that expresses the total revenue from these two products.

d. Find the price and quantity for each product that maximizes the total revenue.

Exercise 4.E.21

Suppose the demand functions for two products are q1=f(p1,p2) and q2=g(p1,p2), where p1, p2, q1, and q2 are the prices (in dollars) and quantities for products 1 and 2. Consider the four partial derivatives q1p1, q1p2, q2p1, and q2p2. Tell the sign of each of these partial derivatives if

a. the products are complementary goods.

b. the products are substitute goods.


4.E: Functions of Two Variables (Exercises) is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

Support Center

How can we help?