# 4.E: Functions of Two Variables (Exercises)

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## 4.1 Exercises

##### Exercise $$\PageIndex{1}$$

$$F(x,y) = x^2-y^2$$. 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 $$\PageIndex{2}$$

$$g(s,t)=\sqrt{st^2}$$. Find

a) $$g(1,9)$$

b) $$g(9,1)$$

c) $$g(1,t)$$

d) $$g(s,9)$$

e) $$g(w,z+1)$$

##### Exercise $$\PageIndex{3}$$

Let $$f(x,y,z,w) = x^2 - \frac{1}{zw}+xyz^2$$. Evaluate $$f(1,2,3,4)$$.

##### Exercise $$\PageIndex{4}$$

Let $$f(x,y,z,w) = \sqrt{xy} - w^2 + 102yz$$. Evaluate $$f(1,2,3,4)$$.

##### Exercise $$\PageIndex{5}$$

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

 $$\overset{t}{\downarrow} \ r\rightarrow$$ 0.03 0.04 0.05 0.06 0.07 1 30.45 40.81 51.27 61.84 72.51 2 61.84 83.29 105.17 127.5 150.27 3 94.17 127.5 161.83 197.22 233.68

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 $$\PageIndex{6}$$

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

 $$\overset{t}{\downarrow} \ h\rightarrow$$ 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 $$\PageIndex{7}-\PageIndex{10}$$

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 $$\PageIndex{11}-\PageIndex{14}$$

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 $$\PageIndex{15}-\PageIndex{18}$$

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 $$\PageIndex{19}-\PageIndex{22}$$

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 $$\PageIndex{23}-\PageIndex{24}$$

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

 23. $$(x–3)^2 + (y+4)^2 + (z–1)^2 = 16$$ 24. $$(x+2)^2 + y^2 + (z–4)^2 = 25$$
##### Exercise $$\PageIndex{25}-\PageIndex{30}$$

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

##### Exercise $$\PageIndex{31}-\PageIndex{36}$$

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

 a. $$f(x,y) = y-x$$ b. $$f(x,y) = xy^2$$ c. $$f(x,y) = \sqrt{25-x^2-y^2}$$ d. $$f(x,y) = 5-x-y$$ e. $$f(x,y) = 0.01x^2y^2$$ f. $$f(x,y) = x^2-y^2$$
##### Exercise $$\PageIndex{37}$$

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

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 $$\PageIndex{38}$$

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

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 $$\PageIndex{39}$$

The demand functions for two products are given below. $$p_1$$, $$p_2$$, $$q_1$$, and $$q_2$$ are the prices (in dollars) and quantities for products 1 and 2. Are these two products complementary goods or substitute goods

$q_1 = 200 -3p_1 + p_2 \nonumber$

$q_2 = 150 + p_1 -2p_2 \nonumber$

##### Exercise $$\PageIndex{40}$$

The demand functions for two products are given below. $$p_1$$, $$p_2$$, $$q_1$$, and $$q_2$$ are the prices (in dollars) and quantities for products 1 and 2. Are these two products complementary goods or substitute goods

$q_1 = 350 + p_1 + 2p_2 \nonumber$

$q_2 = 225 + p_1 + p_2 \nonumber$

##### Exercise $$\PageIndex{41}$$

Consider the Cobb-Douglas Production function: $$P(L,K) = 11 L^{0.3} K^{0.7}$$. Find the total units of production when 19 units of labor and 12 units of capital are invested.

##### Exercise $$\PageIndex{42}$$

Consider the Cobb-Douglas Production function: $$P(L,K) = 6 L^{0.6} K^{0.4}$$. Find the total units of production when 24 units of labor and 8 units of capital are invested.

## 4.2 Exercises

##### Exercise $$\PageIndex{1}-\PageIndex{16}$$

For problems 1 through 16, find $$f_x$$ and $$f_y$$ for the function given

 1. $$f(x,y) = x^2-5y^2$$ 2. $$f(x,y) = \frac{x^2-5y^2}{x+4}$$ 3. $$f(x,y) = e^{x+6y}$$ 4. $$f(x,y) = (x^2-5y^2) e^x$$ 5. $$f(x,y) = (x^2 - 5y^2) \left(\frac{1}{3y}+4\right)$$ 6. $$f(x,y) = x$$ 7. $$f(x,y) = 6$$ 8. $$f(x,y) = \ln (xy+2x-6y)$$ 9. $$f(x,y) = \frac{x^2-5y^2}{y^4-5x^4}$$ 10. $$f(x,y) = e^{\sqrt{x-4y}} (x-4y)$$ 11. $$f(x,y) = y^5 e^x$$ 12. $$f(x,y) = \frac{1}{16xy}$$ 13. $$f(x,y) = (x+e^y)^7$$ 14. $$f(x,y) = x^4 + 4x^3y - 6x^2y^2 - 4xy^3 + y^4$$ 15. $$f(x,y) = \sqrt{x+\sqrt{y}}$$ 16. $$f(x,y) = x^2y^3-4x^3$$
##### Exercise $$\PageIndex{17}$$

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

 $$\overset{t}{\downarrow} \ r\rightarrow$$ 0.03 0.04 0.05 0.06 0.07 1 30.45 40.81 51.27 61.84 72.51 2 61.84 83.29 105.17 127.5 150.27 3 94.17 127.5 161.83 197.22 233.68

a. Estimate $$A_t (2, .05)$$.

b. Estimate $$A_r (2, .05)$$

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 $$A(t,r)=1000 (e^{rt} - 1)$$, 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 $$\PageIndex{18}$$

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

 $$\overset{t}{\downarrow} \ h\rightarrow$$ 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 $$\frac{\partial H}{dt}$$ at (3, 150).

b. Estimate the value of $$\frac{\partial H}{dh}$$ 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 $$H (t,h) = h + 15t - 4.9t^2$$, 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 $$\PageIndex{19}$$

Given the function $$f(x,y) = x^2\sqrt{y}$$

a. Calculate $$f(2,4)$$, $$f_x (2,4)$$, and $$f_y(2,4)$$

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

##### Exercise $$\PageIndex{20}$$

Given the function $$f(x,y) = \ln (10 - x^2 - y)$$

a. Calculate $$f(2,5)$$, $$f_x (2,5)$$, and $$f_y(2,5)$$

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

##### Exercise $$\PageIndex{21}-\PageIndex{26}$$

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

 21. $$f_x(1,-5)$$ 22. $$f_x(-5,2)$$ 23. $$f_x(5,5)$$ 24. $$f_x(0,0)$$ 25. $$f_y(1,-5)$$ 26. $$f_y(-5,2)$$

## 4.3 Exercises

##### Exercise $$\PageIndex{1}-\PageIndex{6}$$

For problems 1 through 6, find $$f_{xx}$$, $$f_{yy}$$, $$f_{xy}$$ and $$f_{yx}$$ for the function given. Confirm that $$f_{xy} = f_{yx}$$.

 1. $$f(x,y) = x^2-5y^2$$ 2. $$f(x,y) = x^4+4x^3y-6x^2y^y-4xy^3+y^4$$ 3. $$f(x,y) = 5x^2y^2$$ 4. $$f(x,y) = e^{x+6y}$$ 5. $$f(x,y) = \ln (xy + 2x - 6y)$$ 6. $$f(x,y) = \frac{x^2}{y^4-5}$$
##### Exercise $$\PageIndex{7}$$

Find the critical points of $$f(x,y) = y^3 - x^3 + 15x^2 - 12y + 12$$ and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

##### Exercise $$\PageIndex{8}$$

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

##### Exercise $$\PageIndex{9}$$

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

##### Exercise $$\PageIndex{10}$$

Find the critical points of $$f(x,y)= xy - 6x^2 + 3x -y+2$$ and use the Second Derivative Test to classify them. If the test fails, say “the test fails.”

##### Exercise $$\PageIndex{11}$$

The origin is a critical point for the function $$f(x,y) = x^3+y^3$$, 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 $$\PageIndex{12}$$

The origin is a critical point for the function $$f(x,y) = 15 - x^2y^2$$, 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 $$\PageIndex{13}-\PageIndex{18}$$

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

 13. $$f(x,y) = xy -5x^2 - 5y^2 + 33y$$ 14. $$f(x,y) = 10xy-x^2-y^2+3x$$ 15. $$f(x,y) = x^3+y^3-3xy$$ 16. $$f(x,y) = 5x^2-4xy+2y^2+4x-4y+10$$ 17. $$f(x,y) = y^2e^x+x^2$$ 18. $$f(x,y) = xy+2x-\ln (x^2y)$$, for $$x>0$$ and $$y>0$$.
##### Exercise $$\PageIndex{19}$$

The demand functions for two products are given below. $$p_1$$, $$p_2$$, $$q_1$$, and $$q_2$$ are the prices (in dollars) and quantities for products 1 and 2.

$q_1 = 200+3p_1+p_2 \nonumber$

$q_2=150+p_1+2p_2\nonumber$

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 $$R(p_1,p_2)$$ that expresses the total revenue from these two products.

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

##### Exercise $$\PageIndex{20}$$

The demand functions for two products are given below. $$p_1$$, $$p_2$$, $$q_1$$, and $$q_2$$ are the prices (in dollars) and quantities for products 1 and 2.

$q_1 = 350+p_1+2p_2 \nonumber$

$q_2=225+p_1+p_2\nonumber$

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 $$R(p_1,p_2)$$ that expresses the total revenue from these two products.

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

##### Exercise $$\PageIndex{21}$$

Suppose the demand functions for two products are $$q_1 = f(p_1, p_2)$$ and $$q_2 = g(p_1, p_2)$$, where $$p_1$$, $$p_2$$, $$q_1$$, and $$q_2$$ are the prices (in dollars) and quantities for products 1 and 2. Consider the four partial derivatives $$\frac{\partial q_1}{\partial p_1}$$, $$\frac{\partial q_1}{\partial p_2}$$, $$\frac{\partial q_2}{\partial p_1}$$, and $$\frac{\partial q_2}{\partial p_2}$$. Tell the sign of each of these partial derivatives if

a. the products are complementary goods.

b. the products are substitute goods.

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