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Calculus II Homework 08: Partial Derivatives, Multiple Integrals, and Applications, Assignments of Calculus

The University of Texas at AustinCalculus

This homework assignment covers key concepts in calculus ii, focusing on partial derivatives, multiple integrals, and their applications. It includes exercises on finding partial derivatives, evaluating definite integrals, and calculating volumes of solids of revolution. The problems are designed to reinforce understanding of these concepts and their practical applications.

Typology: Assignments

2024/2025

Uploaded on 03/05/2025

oscar-cruces
oscar-cruces 🇺🇸

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cruces (oc5592) – HW 08 – cabrera – (53925) 1
This print-out should have 18 questions.
Multiple-choice questions may continue on
the next column or page – find all choices
before answering.
001 10.0 points
Determine fx−fywhen
f(x, y) = 3x2+ 4xy −y2−x+ 3y .
1. fx−fy= 10x+ 2y−4
2. fx−fy= 10x+ 6y+ 2
3. fx−fy= 10x+ 2y+ 2
4. fx−fy= 2x+ 2y−4
5. fx−fy= 2x+ 6y−4
6. fx−fy= 2x+ 6y+ 2
002 10.0 points
Determine whether the partial derivatives
fx, fyof fare positive, negative or zero at the
point Pon the graph of fshown in
P
x
z
y
1. fx<0, fy<0
2. fx>0, fy= 0
3. fx= 0 , fy= 0
4. fx<0, fy= 0
5. fx= 0 , fy>0
6. fx>0, fy>0
7. fx<0, fy>0
8. fx= 0 , fy<0
003 10.0 points
Determine fxwhen
f(x, y) = 2x−y
2x+y.
1. fx=−3y
(2x+y)2
2. fx=3x
(2x+y)2
3. fx=−5x
(2x+y)2
4. fx=−4x
(2x+y)2
5. fx=5y
(2x+y)2
6. fx=4y
(2x+y)2
004 10.0 points
Determine fxwhen
f(x, y) = xsin(x+ 2y) + cos(x+ 2y).
1. fx= 2 cos(x+ 2y) + xsin(x+ 2y)
2. fx= 2 sin(x+ 2y) + xcos(x+ 2y)
3. fx= 2 cos(x+ 2y)−xsin(x+ 2y)
4. fx= 2 sin(x+ 2y)−xcos(x+ 2y)
5. fx=−2xsin(x+ 2y)
pf3
pf4
pf5

Partial preview of the text

Download Calculus II Homework 08: Partial Derivatives, Multiple Integrals, and Applications and more Assignments Calculus in PDF only on Docsity!

This print-out should have 18 questions.

Multiple-choice questions may continue on

the next column or page – find all choices

before answering.

001 10.0 points

Determine fx − fy when

f (x, y) = 3x

2

  • 4xy − y

2 − x + 3y.

  1. fx − fy = 10x + 2y − 4
  2. fx − fy = 10x + 6y + 2
  3. fx − fy = 10x + 2y + 2
  4. fx − fy = 2x + 2y − 4
  5. fx − fy = 2x + 6y − 4
  6. fx − fy = 2x + 6y + 2

002 10.0 points

Determine whether the partial derivatives

fx, fy of f are positive, negative or zero at the

point P on the graph of f shown in

P

x

z

y

  1. fx < 0 , fy < 0
  2. fx > 0 , fy = 0
  3. fx = 0 , fy = 0
    1. fx < 0 , fy = 0
    2. fx = 0 , fy > 0
    3. fx > 0 , fy > 0
    4. fx < 0 , fy > 0
    5. fx = 0 , fy < 0

003 10.0 points

Determine fx when

f (x, y) =

2 x − y

2 x + y

  1. fx = −

3 y

(2x + y)

2

  1. fx =

3 x

(2x + y)^2

  1. fx = −

5 x

(2x + y)^2

  1. fx = −

4 x

(2x + y)^2

  1. fx =

5 y

(2x + y)

2

  1. fx =

4 y

(2x + y)

2

004 10.0 points

Determine fx when

f (x, y) = x sin(x + 2y) + cos(x + 2y).

  1. fx = 2 cos(x + 2y) + x sin(x + 2y)
  2. fx = 2 sin(x + 2y) + x cos(x + 2y)
  3. fx = 2 cos(x + 2y) − x sin(x + 2y)
  4. fx = 2 sin(x + 2y) − x cos(x + 2y)
  5. fx = − 2 x sin(x + 2y)
  1. fx = x cos(x + 2y)
  2. fx = 2x cos(x + 2y)
  3. fx = −x sin(x + 2y)

005 10.0 points

Determine fxy when

f (x, y) =

x tan

− 1

y

x

  1. fxy =

x

2 y

2(x

2

  • y

2 )

  1. fxy =

x

2 y

(x

2

  • y

2 )

2

  1. fxy = −

xy

2

(x

2

  • y

2 )

2

  1. fxy = −

xy

2

2(x

2

  • y

2 )

  1. fxy =

xy

2

(x

2

  • y

2 )

2

  1. fxy = −

x

2 y

2(x

2

  • y

2 )

006 10.0 points

Determine fyx when

f (x, y) = x

2 sin xy.

  1. fyx = 2x

2 (3 sin xy − xy cos xy)

  1. fyx = 2y

2 (3 sin xy − xy cos xy)

  1. fyx = −x

2 (3 sin xy + xy cos xy)

  1. fyx = − 2 x

2 (3 cos xy + xy sin xy)

  1. fyx = −y

2 (3 sin xy + xy cos xy)

  1. fyx = x

2 (3 cos xy − xy sin xy)

  1. fyx = y

2 (3 cos xy − xy sin xy)

  1. fyx = − 2 y

2 (3 cos xy + xy sin xy)

007 10.0 points

Determine

∂z

∂x

when

z =

y

x

f

x

y

∂z

∂x

y

2

yf

x

y

  • xf

′

x

y

∂z

∂x

= x

f

x

y

−xy f

′

x

y

∂z

∂x

x

y^2

yf

x

y

− y f

′

x

y

∂z

∂x

x

2

yf

x

y

− xf

′

x

y

∂z

∂x

x

f

x

y

+xyf

′

x

y

∂z

∂x

= yf

x

y

  • xf

′

x

y

008 10.0 points

Find fx when

f (x, y) =

∫ (^) x

y

cos

t

4

dt.

  1. fx = 4x

3 sin

x

4

  1. fx = sin

x

4

  1. fx = 4x

3 cos

x

4

  1. fx = 0
  2. fx = cos

x

4

009 10.0 points

1. V =

(e

3 − 1) cu. units

  1. V = 1 cu. unit

3. V =

cu. units

  1. V = 2(e

3 − 1) cu. units

5. V =

cu. units

014 10.0 points

Evaluate the integral

I =

0

4 − x^2 ) dx.

  1. I = 5 − π
  2. I = 10 + 2π
  3. I = 10 − π
  4. I = 10 − 2 π
  5. I = 5 + 2π
  6. I = 5 + π

015 10.0 points

Find the value of the integral

I =

3

4 + (x − 3)^2

dx.

1. I =

2. I =

π

3. I =

π

  1. I = π
  2. I = 2π

6. I = 2

016 10.0 points

Rewrite the expression

f (x) =

x

2

  • x − 20

using partial fractions.

  1. f (x) =

x − 4

x + 5

  1. None of these
  2. f (x) =

x − 4

x + 5

  1. f (x) =

x − 4

x + 5

  1. f (x) =

x − 4

x + 5

017 10.0 points

Determine the integral

I =

dx

(x − 1)(x + 3)

1. I =

ln

x − 1

x + 3

+ C

2. I =

ln

2 x + 2

(x − 1)(x + 3)

+ C

  1. I = ln (|(x − 1)(x + 3)|) + C

4. I =

ln

x + 3

x − 1

+ C

5. I =

ln (|(x − 1)(x + 3)|) + C

018 10.0 points

Evaluate the integral

I =

3

(x − 2)(6 − x)

dx.

1. I =

ln

2. I =

ln

  1. I = ln (3)

4. I =

ln (3)

5. I =

ln (3)

  1. I = ln