Deck 14: Multivariate Calculus

Question 1

Evaluate the function.

-Find $f(5,-4)$ when $f(x, y)=9 x+8 y-8$ .

A) -3
B) -4
C) 5
D) 13

Answer 1

5

Question 2

Evaluate the function.

-Find $g(4,-3)$ when $g(x, y)=4 y^{2}-2 x y$ .

A) 85
B) 88
C) 60
D) 56

Answer 2

60

Question 3

Evaluate the function.

-Find $h(3,6)$ when $h(x, y)=\sqrt{3 x+y^{2}}$ .

A) 9
B) 10
C) $3 \sqrt{5}$
D) $5 \sqrt{3}$

Answer 3

$3 \sqrt{5}$

Question 4

Evaluate the function.

-Find $g(3,4)$ when $g(x, y)=\frac{x-6 y}{x^{2}+y^{2}}$ .

A) $-\frac{5}{21}$
B) $-\frac{21}{5}$
C) $-\frac{21}{25}$
D) $-\frac{25}{21}$

Question 5

Evaluate the function.

-Find $h(7,3)$ when $h(x, y)=(x+y)^{3}$ .

A) 30
B) 370
C) 100
D) 1000

Question 6

Evaluate the function.

-Find $f(3,9)$ when $f(x, y)=e^{3 y} \ln x$ .

A) $17,804.30$
B) $7.62 \mathrm{e}+11$
C) $5.85 \mathrm{e}+11$
D) $1.93 e+12$

Question 7

Sketch the indicated plane.

- $f(x, y)=2-2 x+2 y$

A)
B)
C)
D)

Question 8

Sketch the indicated plane.

- $f(x, y)=x+\frac{y}{2}-1$

A)
B)
C)
D)

Question 9

Sketch the indicated plane.

- $f(x, y)=\frac{x}{2}+y-1$

A)
B)
C)
D)

Question 10

Sketch the indicated plane.

- $f(x, y)=1-\frac{x}{2}+y$

A)
B)
C)
D)

Question 11

Sketch the indicated plane.

- $f(x, y)=x+y-1$

A)
B)
C)
D)

Question 12

Sketch the indicated plane.

- $f(x, y)=1-x+y$

A)
B)
C)
D)

Question 13

Sketch the indicated plane.

- $f(x, y)=1+x-y$

A)
B)
C)
D)

Question 14

Sketch the indicated plane.

- $f(x, y)=x-y-1$

A)
B)
C)
D)

Question 15

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $5 x+3 y+z=30$

A)
B)
C)
D)

Question 16

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $5 x+3 y+2 z=8$

A)
B)
C)
D)

Question 17

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $x^{2}+y^{2}=16$

A)
B)
C)
D)

Question 18

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $x^{2}+z^{2}=16$

A)
B)
C)
D)

Question 19

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $y^{2}+z^{2}=16$

A)
B)
C)
D)

Question 20

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $x^{2}+y^{2}+z^{2}=16$

A)
B)
C)
D)

Question 21

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $z=2 x-y^{2}$

A)
B)
C)
D)

Question 22

Graph the level curves in the first octant at heights of $z=0, z=2$ , and $z=4$ for the given equation.

- $z=2 y-x^{2}$

A)
B)
C)
D)

Question 23

Solve the problem.

-The following table shows the number (in thousands) of albums and t-shirts sold for three different months in the same year.

Model the number of $\mathrm{t}$ -shirts sold as a function of the album sales and time using a linear function of the form $s(a, t)=p a+q t+r(p, q, r$ constants) where $s$ represents thousands of $t$ -shirts sold to date, a represents thousands of albums sold to date, and $t$ represents time in months since April of that year.

A) $p=0.02, q=10, r=40$
B) $p=10, q=0.02, r=-40$
C) $p=0.02, q=10, r=-40$
D) $p=0.2, q=-10, r=-40$

Question 24

Solve the problem.

-Production of television sets is given by $P(x, y)=100\left[\frac{2}{3} x^{-2 / 3}+\frac{2}{5} y^{-1 / 3}\right]^{-4}$ , where $x$ is work hours and $y$ is the amount of capital. If 64 work hours and 125 units of capital are used, what is the production output?

A) 456,356
B) 4630
C) 4564
D) 463,030

Question 25

Solve the problem.

-The number of cows that can graze on a ranch is approximated by $C(x, y)=9 x+5 y-4$ , where $x$ is the number of acres of grass and y the number of acres of alfalfa. If the ranch has 25 acres of alfalfa and 70 acres of grass, how many cows may graze?

A) 755 cows
B) 751 cows
C) 575 cows
D) 571 cows

Question 26

Solve the problem.

-The surface area of a human body (in square meters) is approximated by $\mathrm{A}=0.202 \mathrm{~W}(.425) \mathrm{H}^{(.725)}$ , where $\mathrm{W}$ is the weight of the person in kilograms and $\mathrm{H}$ is the height in meters. Find $\mathrm{A}$ if $\mathrm{W}=49$ and $\mathrm{H}=1.72$ .

A) $1.25 \mathrm{~m}^{2}$
B) $1.56 \mathrm{~m}^{2}$
C) $1.82 \mathrm{~m}^{2}$
D) $1.42 \mathrm{~m}^{2}$

Question 27

Choose the graph of the equation.

- $z^{2}-y^{2}-x^{2}=1$

A)
B)
C)
D)

Question 28

Choose the graph of the equation.

- $x^{2}-y^{2}-z^{2}=1$

A)
B)
C)
D)

Question 29

Choose the graph of the equation.

- $y^{2}-x^{2}-z^{2}=1$

A)
B)
C)
D)

Question 30

Choose the graph of the equation.

- $z=x^{2}+y^{2}$

A)
B)
C)
D)

Question 31

Choose the graph of the equation.

- $z=y^{2}-x^{2}$

A)
B)
C)
D)

Question 32

Choose the graph of the equation.

- $z^{2}=y^{2}+x^{2}$

A)
B)
C)
D)

Question 33

Choose the graph of the equation.

- $\frac{x^{2}}{4}+\frac{y^{2}}{16}+\frac{z^{2}}{25}=1$

A)
B)
C)
D)

Question 34

Choose the graph of the equation.

- $z=5\left(x^{2}+y^{2}\right)^{-1 / 2}$

A)
B)
C)
D)

Question 35

Choose the graph of the equation.

- $z=5\left(x^{2}+y^{2}\right)^{1 / 2}$

A)
B)
C)
D)

Question 36

Choose the graph of the equation.

- $z=x^{2}-y^{2}$

A)
B)
C)
D)

Question 37

Find the partial derivative.

-Let $z=f(x, y)=5 x^{2}-11 x y+2 y^{3}$ . Find $\frac{\partial z}{\partial x}$ .

A) $-11 x+6 y^{2}$
B) $-11 x-6 y$
C) $10 x+11 y^{2}$
D) $10 x-11 y$

Question 38

Find the partial derivative.

-Let $z=g(x, y)=4 x+2 x^{2} y^{2}-4 y^{2}$ . Find $\frac{\partial z}{\partial x}$ .

A) $4+4 x^{2} y$
B) $4+4 x y^{2}$
C) $4 x y^{2}-8 y$
D) $4 x^{2} y-8 y$

Question 39

Find the partial derivative.

-Let $z=g(x, y)=8 x+4 x^{2} y^{2}-6 y^{2}$ . Find $\frac{\partial z}{\partial y}$ .

A) $8 y x^{2}-12 y$
B) $8 y x-8 y$
C) $8+8 y^{2}$
D) $8 y x-12 y$

Question 40

Find the partial derivative.

-Let $z=g(x, y)=2 x+3 x^{2} y^{2}-2 y^{2}$ . Find $\frac{\partial z}{\partial y}$ .

A) $2+6 x^{2} y$
B) $6 x y^{2}-4 y$
C) $6 x^{2} y-4 y$
D) $2+6 x y^{2}$

Question 41

Find the partial derivative.

-Find $f_{x}(9,-9)$ when $f(x, y)=7 x^{2}-9 x y$ .

A) 45
B) -45
C) -162
D) 207

Question 42

Find the partial derivative.

-Find $f(5,-6)$ when $f(x, y)=7 x^{2}-9 x y$ .

A) 129
B) -54
C) -45
D) 39

Question 43

Find the partial derivative.

-Find $f_{x}(2,-5)$ when $f(x, y)=e^{3 x}+4 y$ .

A) $3 e^{6}$
B) $3 e^{-14}$
C) $4 \mathrm{e}^{6}$
D) $4 e^{-14}$

Question 44

Find the partial derivative.

-Find $g_{y}(3,4)$ when $g(x, y)=x e^{x y}$ .

A) $9 e^{3}$
B) $9 \mathrm{e}^{12}$
C) $3 e^{3}$
D) $3 \mathrm{e}^{12}$

Question 45

Find the partial derivative.

-Find $g_{x}(2,5)$ when $g(x, y)=\frac{7 x}{9 y}$ .

A) $\frac{14}{9}$
B) $\frac{7}{45}$
C) $\frac{14}{45}$
D) $\frac{7}{9}$

Question 46

Find the partial derivative.

-Find $g_{x}(2,1)$ when $g(x, y)=\ln \left|1+5 x^{2} y^{3}\right|$ .

A) $\frac{21}{11}$
B) $\frac{11}{21}$
C) $\frac{21}{20}$
D) $\frac{20}{21}$

Question 47

Find the second-order partial derivative.

-Find $f_{x y}$ when $f(x, y)=8 x^{3} y-7 y^{2}+2 x$ .

A) -14
B) $48 x y$
C) -28
D) $24 x^{2}$

Question 48

Find the second-order partial derivative.

-Find $f_{y x}$ when $f(x, y)=8 x^{3} y-7 y^{2}+2 x$ .

A) $48 x y$
B) $24 x^{2}$
C) -28
D) -14

Question 49

Find the second-order partial derivative.

-Find $f_{x x}$ when $f(x, y)=8 x^{3} y-7 y^{2}+2 x$ .

A) $24 x^{2}$
B) -28
C) -14
D) $48 x y$

Question 50

Find the second-order partial derivative.

-Find $f_{y y}$ when $f(x, y)=8 x^{3} y-7 y^{2}+2 x$ .

A) 48xy
B) -28
C) -14
D) $24 x^{2}$

Question 51

Find the second-order partial derivative.

-Find gyy when $g(x, y)=\frac{x}{y}$ .

A) $-\frac{3 x}{y^{2}}$
B) $\frac{3 x}{y^{2}}$
C) $-\frac{2 x}{y^{3}}$
D) $\frac{2 x}{y^{3}}$

Question 52

Find the second-order partial derivative.

-Find $\frac{\partial^{2} z}{\partial y^{2}}$ when $z=8 x e^{x y}$ .

A) $8 x^{3} e^{x}$
B) $8 x^{3} e^{x y}$
C) $16 x^{2} e^{x y}$
D) $16 x^{2} e^{x}$

Question 53

Find the second-order partial derivative.

-Find $\frac{\partial^{2} z}{\partial x \partial y}$ when $z=\ln |2 x+9 y|$ .

A) $\frac{9}{(2 x+9 y)^{2}}$
B) $\frac{-18}{(2 x+9 y)^{2}}$
C) $\frac{18}{(2 x+9 y)^{2}}$
D) $\frac{-9}{(2 x+9 y)^{2}}$

Question 54

Find the second-order partial derivative.

-Find $\frac{\partial^{2} z}{\partial x^{2}}$ when $z=e^{x} \ln |y|$ .

A) $e^{x} \ln |y|$
B) $-\frac{e^{x}}{y^{2}}$
C) $\frac{e^{x}}{y^{2}}$
D) $-e^{x} \ln |y|$

Question 55

Find the second-order partial derivative.

-Find $\frac{\partial^{2} z}{\partial y^{2}}$ when $z=e^{x} \ln |y|$ .

A) $\frac{e^{x}}{y^{2}}$
B) $\frac{e^{x}}{y}$
C) $-\frac{e^{x}}{y^{2}}$
D) $-\frac{e^{x}}{y}$

Question 56

Find the second-order partial derivative.

-Find $f_{x y}$ when $f(x, y)=10 x^{2} y^{4}-7 x^{3} y^{5}$ .

A) $160 x y^{3}-21 x^{2} y^{4}$
B) $160 x y^{3}-105 x^{2} y^{4}$
C) $80 x y^{3}-105 x^{2} y^{4}$
D) $80 x y^{3}-21 x^{2} y^{4}$

Question 57

Evaluate the required second order partial derivative.

-Find $f_{x y}(5,-2)$ , if $f(x, y)=-5 x^{3} y^{2}+4 x^{4}-3 x y^{3}$ .

A) 542
B) -633
C) 1464
D) -825

Question 58

Evaluate the required second order partial derivative.

-Find $f_{y x}(2,-4)$ , if $f(x, y)=4 x^{3} y^{2}+4 x^{4}+2 x y^{3}$ .

A) -96
B) 752
C) -192
D) -288

Question 59

Evaluate the required second order partial derivative.

-Find $f_{y y}(5,2)$ , if $f(x, y)=\frac{-5 y}{5 x+y}$ .

A) $\frac{100}{19683}$
B) $\frac{350}{19683}$
C) $\frac{1490}{19683}$
D) $\frac{250}{19683}$

Question 60

Evaluate the required second order partial derivative.

-Find $f_{x x}(-3,1)$ , if $f(x, y)=\frac{3 y}{-4 x+y}$ .

A) $\frac{96}{2197}$
B) $\frac{24}{2197}$
C) $\frac{244}{2197}$
D) $-\frac{72}{2197}$

Question 61

Evaluate the required second order partial derivative.

-Find $f_{x y}(5,-1)$ , if $f(x, y)=-2 x e^{5 y}$ .

A) -0.404
B) -1.819
C) 0
D) -0.067

Question 62

Evaluate the required second order partial derivative.

-Find $f_{y x}(1,3)$ , if $f(x, y)=2 x e^{4 y}$ .

A) $1,302,038.17$
B) $2,604,076.34$
C) $7,812,229.01$
D) $-81,376.739$

Question 63

Evaluate the required second order partial derivative.

-Find $f_{x x}(5,-1)$ , if $f(x, y)=\ln (5 x-3 y)$ .

A) -0.032
B) -0.011
C) 0.179
D) -0.043

Question 64

Evaluate the required second order partial derivative.

-Find $f_{y y}(2,-1)$ , if $f(x, y)=\ln (4 x-3 y)$ .

A) -0.132
B) 0.364
C) -0.207
D) -0.074

Question 65

Evaluate the required second order partial derivative.

-Find $f_{x y}(-2,-5)$ , if $f(x, y)=(x-4) \ln \left(x y^{5}\right)$ .

A) 7.5
B) -2.5
C) -1.25
D) -1

Question 66

Evaluate the required second order partial derivative.

-Find $f_{y x}(-4,-2)$ , if $f(x, y)=(x+4) \ln \left(x y^{-1}\right)$ .

A) 0.063
B) 3.75
C) 0.25
D) 0.5

Question 67

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=8 x^{2}+8 y^{2}-2 x y$

A) $x=0, y=0$
B) $x=0, y=1$
C) $x=1, y=0$
D) $x=1, y=1$

Question 68

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=3 x^{2}+3 y^{2}-6 x-6 y+2$

A) $x=0, y=0$
B) $x=6, y=6$
C) $x=3, y=3$
D) $x=1, y=1$

Question 69

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=12 x y-4 x-4 y+9$

A) $x=-\frac{1}{3}, y=-\frac{1}{3}$
B) $x=3, y=3$
C) $\mathrm{x}=\frac{1}{3}, \mathrm{y}=\frac{1}{3}$
D) $x=-3, y=-3$

Question 70

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=3200+x y-9 x+8 y$

A) $x=8, y=-9$
B) $x=-8, y=9$
C) $x=-9, y=8$
D) $x=9, y=-8$

Question 71

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=4 x^{3}-2 x y+8 y$

A) $x=4, y=96$
B) $x=4, y=48$
C) $x=2, y=48$
D) $x=2, y=96$

Question 72

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=12 x y-6 x^{2}+8 x$

A) $x=0, y=-\frac{2}{3}$
B) $\mathrm{x}=-\frac{2}{3}, \mathrm{y}=0$
C) $x=\frac{2}{3}, y=0$
D) $x=0, y=\frac{3}{2}$

Question 73

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=x^{2}+x y+y^{2}+2 x+3 y$

A) $\mathrm{x}=\frac{1}{3}, \mathrm{y}=\frac{4}{3}$
B) $\mathrm{x}=-\frac{1}{3}, \mathrm{y}=-\frac{4}{3}$
C) $\mathrm{x}=-\frac{4}{3}, \mathrm{y}=-\frac{1}{3}$
D) $\mathrm{x}=\frac{4}{3}, \mathrm{y}=\frac{1}{3}$

Question 74

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=x^{2}+y^{2}+6 x-7 y$

A) $x=\frac{3}{2}, y=-7$
B) $x=3, y=-\frac{7}{2}$
C) $x=-3, y=\frac{7}{2}$
D) $x=-\frac{3}{2}, y=7$

Question 75

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=x+y+8 x y+3$

A) $\mathrm{x}=-\frac{1}{8}, \mathrm{y}=-\frac{1}{8}$
B) $\mathrm{x}=-\frac{1}{8}, \mathrm{y}=-\frac{3}{8}$
C) $\mathrm{x}=-\frac{3}{8}, \mathrm{y}=-\frac{1}{8}$
D) $\mathrm{x}=-\frac{3}{8}, \mathrm{y}=-\frac{3}{8}$

Question 76

Find values of $x$ and $y$ such that both $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ .

- $f(x, y)=3 x^{2}+12 x y+y^{2}$

A) $x=0, y=0$
B) $x=-4, y=-4$
C) $x=0, y=-4$
D) $x=-4, y=0$

Question 77

Find the indicated derivative.

-Find $f_{x} f(x, y, z)=10 x^{8} y^{8}+3 x^{7} z^{4}+5 y^{3}$

A) $80 x^{7}+21 x^{6}$
B) $640 x^{7} y^{7}+84 x^{6} z^{3}$
C) $80 x^{7} y^{8}+21 x^{6} z^{4}$
D) $80 x^{8} y^{7}+12 x^{7} z^{3}$

Question 78

Find the indicated derivative.

-Find $f_{y} f(x, y, z)=9 x^{6} y^{7}+10 x^{8} z^{8}+10 y 10$

A) $63 y^{6}+100 y^{9}$
B) $54 x^{5} y^{7}+80 x^{7} z^{8}$
C) $63 x^{6} y^{6}+100 y^{9}$
D) $378 x^{5} y^{6}+640 x^{7} z^{7}+100 y^{9}$

Question 79

Find the indicated derivative.

-Find $f_{z} f(x, y, z)=6 x^{8} y^{3}+8 x^{5} z^{7}+6 y^{4}$

A) $8 x^{5}$
B) $280 x^{4} z^{6}$
C) $56 x^{5} z^{6}$
D) $7 z^{6}$

Question 80

Find the indicated derivative.

-Find $f_{x} f(x, y, z)=6 x^{5} e^{\left(8 y^{4}+10 z^{9}\right)}$

A) $30 x^{4} e^{\left(8 y^{4}+10 z^{9}\right)}$
B) $30 x^{4}$
C) $6 x^{5}$
D) $30 x^{4}\left(32 y^{3}+90 z^{8}\right)$