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Multivariable Calculus

Multivariable Calculus

Introduction

Multivariable Calculus extends single-variable calculus to functions depending on two or more variables. It explains how quantities change in multiple directions and how to integrate over curves, regions and surfaces.

Recommended Background

A working knowledge of single-variable calculus and analytic geometry.

Topic-Wise Course Outline

Functions of Several Variables

Domains, ranges, level curves and surfaces; limits and continuity in more than one variable.

Partial Differentiation

Partial derivatives, higher-order partial derivatives, the chain rule and directional ideas.

Multiple Integrals

Double and triple integrals; iterated integration; applications to area, volume and mass.

Line and Surface Integrals

Integrals along curves and across surfaces; parametrization and applications.

Green’s and Stokes’ Theorems

Connections between integrals over boundaries and their enclosed regions or surfaces; appropriate orientation.

Fourier Series

Periodic functions and period 2L; even and odd functions; half-range sine and cosine expansions.

Integral and Discrete Transforms

An introduction to the Fourier transform, Laplace transform and Z-transform, with their different domains and uses.

How to Study This Subject

Start with the definitions and core concepts in each topic. Work through examples, check every step of a solution, and then practise questions to test understanding.

Begin with the MCQs and worked examples below, then review each topic in the outline. More practice material will be added progressively.

Foundational MCQs and Step-by-Step Practice

Ten introductory topic-wise questions with explanations, followed by two fully worked examples. This is a starter set; further coverage is planned.

Several Variables

Q1. For f(x,y)=x²+2y, find f(2,3).

  • A. 7
  • B. 10
  • C. 11
  • D. 12
Show correct answer and step-by-step explanation

Correct answer: B. 10

Substitute x=2 and y=3 to obtain 4+6=10.

Partial Derivatives

Q2. For f=x²y+3y, find ∂f/∂x.

  • A. 2xy
  • B. x²+3
  • C. 2x+3
  • D. 2y
Show correct answer and step-by-step explanation

Correct answer: A. 2xy

Treat y as constant, so derivative is 2xy.

Q3. For f=x²y+3y, find ∂f/∂y.

  • A. 2xy
  • B. x²y
  • C. x²+3
  • D. 3y
Show correct answer and step-by-step explanation

Correct answer: C. x²+3

Treat x as constant: derivative is x²+3.

Gradient

Q4. Find ∇f for f=x²+y².

  • A. (x,y)
  • B. (2x,2y)
  • C. (2,2)
  • D. (x²,y²)
Show correct answer and step-by-step explanation

Correct answer: B. (2x,2y)

Gradient collects first partial derivatives.

Directional Derivatives

Q5. For a unit vector u, what equals the directional derivative of differentiable f?

  • A. f·u
  • B. ∇f+u
  • C. ∇f×u
  • D. ∇f·u
Show correct answer and step-by-step explanation

Correct answer: D. ∇f·u

A directional derivative equals gradient dot unit direction.

Double Integrals

Q6. Evaluate ∫₀¹ ∫₀¹ 1 dy dx.

  • A. 1
  • B. 2
  • C. 0
  • D. 1/2
Show correct answer and step-by-step explanation

Correct answer: A. 1

Integrating 1 over the unit square gives area 1.

Triple Integrals

Q7. What is the integral of 1 over the unit cube?

  • A. 0
  • B. 3
  • C. 1
  • D. 6
Show correct answer and step-by-step explanation

Correct answer: C. 1

Its value equals the cube’s volume, 1.

Line Integrals

Q8. For a scalar field equal to 2 along a curve of length 3, find ∫C 2 ds.

  • A. 3
  • B. 6
  • C. 2
  • D. 9
Show correct answer and step-by-step explanation

Correct answer: B. 6

Multiply the constant scalar field by the arc length: 2×3=6.

Fourier Series

Q9. A 2Ï€-periodic odd integrable function has which Fourier terms?

  • A. Only cosine terms
  • B. Constant and cosine terms
  • C. Only sine terms
  • D. No terms at all
Show correct answer and step-by-step explanation

Correct answer: C. Only sine terms

Oddness forces the constant and cosine coefficients to vanish.

Transforms

Q10. What is the Laplace transform of the constant function 1 for Re(s)>0?

  • A. s
  • B. 1/s
  • C. s²
  • D. 1/s²
Show correct answer and step-by-step explanation

Correct answer: B. 1/s

∫₀∞ e^(−st)dt = 1/s for Re(s)>0.

Solved Examples with Step-by-Step Answers

Example 1: Partial derivatives

Problem: For f(x,y)=x²y+3y², find both first partial derivatives.

  1. To find f_x, keep y constant: derivative of x²y is 2xy and of 3y² is 0.
  2. To find f_y, keep x constant: derivative of x²y is x² and of 3y² is 6y.

Final answer: f_x=2xy and f_y=x²+6y.

Example 2: Double integral

Problem: Evaluate ∫₀² ∫₀¹ (x+y) dy dx.

  1. Integrate with respect to y: [xy+y²/2]₀¹=x+1/2.
  2. Integrate from x=0 to x=2: [x²/2+x/2]₀²=2+1.

Final answer: The integral equals 3.

Additional Topic-Wise MCQs — Questions 11–20

Additional concept and calculation questions. Select a question’s expandable answer to review the reasoning.

Limits in Two Variables

Q11. What is lim((x,y)→(0,0)) (x²+y²)?

  • A. 0
  • B. 1
  • C. 2
  • D. Does not exist
Show correct answer and explanation

Correct answer: A. 0

Both squared terms approach 0.

Partial Derivatives

Q12. Find ∂²f/∂x∂y for f=x²y.

  • A. x²
  • B. 2x
  • C. 2y
  • D. 0
Show correct answer and explanation

Correct answer: B. 2x

Differentiate first with respect to y to get x², then x to get 2x.

Chain Rule

Q13. If z=x²+y² with x=t and y=2t, find dz/dt.

  • A. 2t
  • B. 4t
  • C. 5t
  • D. 10t
Show correct answer and explanation

Correct answer: D. 10t

Substitute: z=t²+4t²=5t², hence dz/dt=10t.

Critical Points

Q14. At a differentiable interior local extremum of f(x,y), what is necessarily true?

  • A. ∇f=0
  • B. f=0
  • C. ∇f=(1,1)
  • D. f is linear
Show correct answer and explanation

Correct answer: A. ∇f=0

At an interior local extremum of a differentiable function, both first partial derivatives vanish.

Double Integrals

Q15. Evaluate ∫₀² ∫₀³ 1 dy dx.

  • A. 3
  • B. 5
  • C. 6
  • D. 12
Show correct answer and explanation

Correct answer: C. 6

The integral gives the area of a rectangle of width 2 and height 3.

Polar Coordinates

Q16. What is x²+y² in polar coordinates?

  • A. r
  • B. r²
  • C. r sinθ
  • D. r cosθ
Show correct answer and explanation

Correct answer: B. r²

With x=r cosθ and y=r sinθ, x²+y²=r².

Vector Fields

Q17. For F=(P,Q) in the plane, what is the scalar curl expression?

  • A. ∂P/∂x+∂Q/∂y
  • B. ∂P/∂y+∂Q/∂x
  • C. P+Q
  • D. ∂Q/∂x−∂P/∂y
Show correct answer and explanation

Correct answer: D. ∂Q/∂x−∂P/∂y

The planar curl’s out-of-plane component is ∂Q/∂x−∂P/∂y.

Green’s Theorem

Q18. For a positively oriented, piecewise-smooth simple closed curve C, with P and Q having continuous first partial derivatives on a region containing C and its interior, which area integral equals ∮C P dx+Q dy?

  • A. ∬(∂Q/∂x−∂P/∂y)dA
  • B. ∬(P+Q)dA
  • C. ∫P dx only
  • D. ∫Q dy only
Show correct answer and explanation

Correct answer: A. ∬(∂Q/∂x−∂P/∂y)dA

Green’s theorem relates circulation to the scalar curl over the enclosed region.

Fourier Series

Q19. For an even integrable function on [−π,π], which Fourier coefficients vanish?

  • A. Cosine coefficients
  • B. Constant term
  • C. Sine coefficients
  • D. All coefficients
Show correct answer and explanation

Correct answer: C. Sine coefficients

The product of an even function and sin(nx) is odd; its integral over symmetric bounds is zero.

Laplace Transforms

Q20. For a real constant a, what is L{e^(at)}(s) for Re(s)>a?

  • A. s−a
  • B. 1/(s−a)
  • C. 1/(s+a)
  • D. a/s
Show correct answer and explanation

Correct answer: B. 1/(s−a)

Integrate e^{−(s−a)t} from 0 to infinity.

Additional Practice MCQs — Questions 21–30

Further foundational and intermediate practice, with four answer options and explanations.

Domains

Q21. What is the domain of f(x,y)=√(9−x²−y²)?

  • A. All R²
  • B. x²+y²<9 only
  • C. x²+y²≤9
  • D. x²+y²≥9
Show correct answer and explanation

Correct answer: C. x²+y²≤9

The expression under the square root must be nonnegative.

Two-Variable Limits

Q22. Does lim((x,y)→(0,0)) xy/(x²+y²) exist?

  • A. Yes, 0
  • B. Yes, 1/2
  • C. Yes, −1/2
  • D. No
Show correct answer and explanation

Correct answer: D. No

Along y=x it tends to 1/2 and along y=−x to −1/2.

Gradient

Q23. Find ∇f for f=3x+4y.

  • A. (3,4)
  • B. (x,y)
  • C. (4,3)
  • D. (7,0)
Show correct answer and explanation

Correct answer: A. (3,4)

The two partial derivatives are 3 and 4.

Hessian

Q24. Find the Hessian of f=x²+3y².

  • A. diag(1,3)
  • B. diag(2,6)
  • C. diag(2,3)
  • D. diag(x,y)
Show correct answer and explanation

Correct answer: B. diag(2,6)

The diagonal second derivatives are 2 and 6; mixed partials vanish.

Double Integrals

Q25. Find ∫₀¹∫₀² x dy dx.

  • A. 1/2
  • B. 1
  • C. 2
  • D. 4
Show correct answer and explanation

Correct answer: B. 1

The inner integral is 2x and its integral from 0 to 1 equals 1.

Polar Coordinates

Q26. Which factor appears in the polar area element dA?

  • A. 1
  • B. r²
  • C. sin θ
  • D. r
Show correct answer and explanation

Correct answer: D. r

The Jacobian gives dA=r dr dθ.

Vector Calculus

Q27. Find div F for F=(x,y,z).

  • A. 3
  • B. x+y+z
  • C. 1
  • D. 0
Show correct answer and explanation

Correct answer: A. 3

Each of the three diagonal partial derivatives is 1.

Line Integrals

Q28. For F=(1,0), find ∫C F·dr along a straight segment from (0,0) to (2,0).

  • A. 0
  • B. 1
  • C. 2
  • D. 4
Show correct answer and explanation

Correct answer: C. 2

For constant F, the integral is F dotted with the displacement (2,0).

Stokes’ Theorem

Q29. Stokes’ theorem equates boundary circulation with which quantity?

  • A. Time derivative
  • B. Flux of curl through an oriented surface
  • C. Surface area alone
  • D. Volume alone
Show correct answer and explanation

Correct answer: B. Flux of curl through an oriented surface

Stokes links the boundary integral to the surface integral of curl.

Z-Transform

Q30. Find the unilateral Z-transform of a_n=1 for n≥0 and |z|>1.

  • A. z/(z−1)
  • B. 1/z
  • C. z−1
  • D. 1/(1−z)
Show correct answer and explanation

Correct answer: A. z/(z−1)

Sum the geometric series Σz^(−n)=1/(1−z^(−1)).

Additional Step-by-Step Solved Example

Path-dependent limits

Problem: Find lim((x,y)→(0,0)) xy/(x²+y²).

  1. Along y=x, the expression is 1/2 away from the origin.
  2. Along y=−x, the expression is −1/2 away from the origin.
  3. Since the path limits differ, the full two-variable limit cannot exist.

Final answer: The limit does not exist.

Additional Practice MCQs — Questions 31–150

Q31. For f(x,y)=x²+3y, evaluate f(1,2).

  • A. 6
  • B. 10
  • C. 7
  • D. 8
Show correct answer and explanation

Correct answer: C. 7

Substitution gives 1²+3×2=7.

Q32. For f(x,y)=x²+4y, evaluate f(2,3).

  • A. 17
  • B. 15
  • C. 19
  • D. 16
Show correct answer and explanation

Correct answer: D. 16

Substitution gives 2²+4×3=16.

Q33. For f(x,y)=x²+5y, evaluate f(3,4).

  • A. 29
  • B. 30
  • C. 28
  • D. 32
Show correct answer and explanation

Correct answer: A. 29

Substitution gives 3²+5×4=29.

Q34. For f(x,y)=x²+6y, evaluate f(4,5).

  • A. 49
  • B. 46
  • C. 47
  • D. 45
Show correct answer and explanation

Correct answer: B. 46

Substitution gives 4²+6×5=46.

Q35. For f(x,y)=x²+7y, evaluate f(5,6).

  • A. 66
  • B. 70
  • C. 67
  • D. 68
Show correct answer and explanation

Correct answer: C. 67

Substitution gives 5²+7×6=67.

Q36. For f(x,y)=x²+8y, evaluate f(6,7).

  • A. 93
  • B. 91
  • C. 95
  • D. 92
Show correct answer and explanation

Correct answer: D. 92

Substitution gives 6²+8×7=92.

Q37. For f(x,y)=x²+9y, evaluate f(7,8).

  • A. 121
  • B. 122
  • C. 120
  • D. 124
Show correct answer and explanation

Correct answer: A. 121

Substitution gives 7²+9×8=121.

Q38. For f(x,y)=x²+10y, evaluate f(8,9).

  • A. 157
  • B. 154
  • C. 155
  • D. 153
Show correct answer and explanation

Correct answer: B. 154

Substitution gives 8²+10×9=154.

Q39. For f(x,y)=x²+11y, evaluate f(9,10).

  • A. 190
  • B. 194
  • C. 191
  • D. 192
Show correct answer and explanation

Correct answer: C. 191

Substitution gives 9²+11×10=191.

Q40. For f(x,y)=x²+12y, evaluate f(10,11).

  • A. 233
  • B. 231
  • C. 235
  • D. 232
Show correct answer and explanation

Correct answer: D. 232

Substitution gives 10²+12×11=232.

Q41. For f(x,y)=2x²y, evaluate ∂f/∂x at (1,2).

  • A. 8
  • B. 9
  • C. 7
  • D. 11
Show correct answer and explanation

Correct answer: A. 8

The derivative is 4xy; at (1,2) it equals 8.

Q42. For f(x,y)=3x²y, evaluate ∂f/∂x at (2,2).

  • A. 27
  • B. 24
  • C. 25
  • D. 23
Show correct answer and explanation

Correct answer: B. 24

The derivative is 6xy; at (2,2) it equals 24.

Q43. For f(x,y)=4x²y, evaluate ∂f/∂x at (3,2).

  • A. 47
  • B. 51
  • C. 48
  • D. 49
Show correct answer and explanation

Correct answer: C. 48

The derivative is 8xy; at (3,2) it equals 48.

Q44. For f(x,y)=5x²y, evaluate ∂f/∂x at (4,2).

  • A. 81
  • B. 79
  • C. 83
  • D. 80
Show correct answer and explanation

Correct answer: D. 80

The derivative is 10xy; at (4,2) it equals 80.

Q45. For f(x,y)=6x²y, evaluate ∂f/∂x at (5,2).

  • A. 120
  • B. 121
  • C. 119
  • D. 123
Show correct answer and explanation

Correct answer: A. 120

The derivative is 12xy; at (5,2) it equals 120.

Q46. For f(x,y)=7x²y, evaluate ∂f/∂x at (6,2).

  • A. 171
  • B. 168
  • C. 169
  • D. 167
Show correct answer and explanation

Correct answer: B. 168

The derivative is 14xy; at (6,2) it equals 168.

Q47. For f(x,y)=8x²y, evaluate ∂f/∂x at (7,2).

  • A. 223
  • B. 227
  • C. 224
  • D. 225
Show correct answer and explanation

Correct answer: C. 224

The derivative is 16xy; at (7,2) it equals 224.

Q48. For f(x,y)=9x²y, evaluate ∂f/∂x at (8,2).

  • A. 289
  • B. 287
  • C. 291
  • D. 288
Show correct answer and explanation

Correct answer: D. 288

The derivative is 18xy; at (8,2) it equals 288.

Q49. For f(x,y)=10x²y, evaluate ∂f/∂x at (9,2).

  • A. 360
  • B. 361
  • C. 359
  • D. 363
Show correct answer and explanation

Correct answer: A. 360

The derivative is 20xy; at (9,2) it equals 360.

Q50. For f(x,y)=11x²y, evaluate ∂f/∂x at (10,2).

  • A. 443
  • B. 440
  • C. 441
  • D. 439
Show correct answer and explanation

Correct answer: B. 440

The derivative is 22xy; at (10,2) it equals 440.

Q51. For f(x,y)=2xy², evaluate ∂f/∂y at (1,3).

  • A. 11
  • B. 15
  • C. 12
  • D. 13
Show correct answer and explanation

Correct answer: C. 12

The derivative is 4xy; at (1,3) it equals 12.

Q52. For f(x,y)=3xy², evaluate ∂f/∂y at (2,3).

  • A. 37
  • B. 35
  • C. 39
  • D. 36
Show correct answer and explanation

Correct answer: D. 36

The derivative is 6xy; at (2,3) it equals 36.

Q53. For f(x,y)=4xy², evaluate ∂f/∂y at (3,3).

  • A. 72
  • B. 73
  • C. 71
  • D. 75
Show correct answer and explanation

Correct answer: A. 72

The derivative is 8xy; at (3,3) it equals 72.

Q54. For f(x,y)=5xy², evaluate ∂f/∂y at (4,3).

  • A. 123
  • B. 120
  • C. 121
  • D. 119
Show correct answer and explanation

Correct answer: B. 120

The derivative is 10xy; at (4,3) it equals 120.

Q55. For f(x,y)=6xy², evaluate ∂f/∂y at (5,3).

  • A. 179
  • B. 183
  • C. 180
  • D. 181
Show correct answer and explanation

Correct answer: C. 180

The derivative is 12xy; at (5,3) it equals 180.

Q56. For f(x,y)=7xy², evaluate ∂f/∂y at (6,3).

  • A. 253
  • B. 251
  • C. 255
  • D. 252
Show correct answer and explanation

Correct answer: D. 252

The derivative is 14xy; at (6,3) it equals 252.

Q57. For f(x,y)=8xy², evaluate ∂f/∂y at (7,3).

  • A. 336
  • B. 337
  • C. 335
  • D. 339
Show correct answer and explanation

Correct answer: A. 336

The derivative is 16xy; at (7,3) it equals 336.

Q58. For f(x,y)=9xy², evaluate ∂f/∂y at (8,3).

  • A. 435
  • B. 432
  • C. 433
  • D. 431
Show correct answer and explanation

Correct answer: B. 432

The derivative is 18xy; at (8,3) it equals 432.

Q59. For f(x,y)=10xy², evaluate ∂f/∂y at (9,3).

  • A. 539
  • B. 543
  • C. 540
  • D. 541
Show correct answer and explanation

Correct answer: C. 540

The derivative is 20xy; at (9,3) it equals 540.

Q60. For f(x,y)=11xy², evaluate ∂f/∂y at (10,3).

  • A. 661
  • B. 659
  • C. 663
  • D. 660
Show correct answer and explanation

Correct answer: D. 660

The derivative is 22xy; at (10,3) it equals 660.

Q61. For f(x,y)=2x+3y, find the x-component of ∇f.

  • A. 2
  • B. 3
  • C. 1
  • D. 5
Show correct answer and explanation

Correct answer: A. 2

The x-component is ∂f/∂x=2.

Q62. For f(x,y)=3x+4y, find the x-component of ∇f.

  • A. 6
  • B. 3
  • C. 4
  • D. 2
Show correct answer and explanation

Correct answer: B. 3

The x-component is ∂f/∂x=3.

Q63. For f(x,y)=4x+5y, find the x-component of ∇f.

  • A. 3
  • B. 7
  • C. 4
  • D. 5
Show correct answer and explanation

Correct answer: C. 4

The x-component is ∂f/∂x=4.

Q64. For f(x,y)=5x+6y, find the x-component of ∇f.

  • A. 6
  • B. 4
  • C. 8
  • D. 5
Show correct answer and explanation

Correct answer: D. 5

The x-component is ∂f/∂x=5.

Q65. For f(x,y)=6x+7y, find the x-component of ∇f.

  • A. 6
  • B. 7
  • C. 5
  • D. 9
Show correct answer and explanation

Correct answer: A. 6

The x-component is ∂f/∂x=6.

Q66. For f(x,y)=7x+8y, find the x-component of ∇f.

  • A. 10
  • B. 7
  • C. 8
  • D. 6
Show correct answer and explanation

Correct answer: B. 7

The x-component is ∂f/∂x=7.

Q67. For f(x,y)=8x+9y, find the x-component of ∇f.

  • A. 7
  • B. 11
  • C. 8
  • D. 9
Show correct answer and explanation

Correct answer: C. 8

The x-component is ∂f/∂x=8.

Q68. For f(x,y)=9x+10y, find the x-component of ∇f.

  • A. 10
  • B. 8
  • C. 12
  • D. 9
Show correct answer and explanation

Correct answer: D. 9

The x-component is ∂f/∂x=9.

Q69. For f(x,y)=10x+11y, find the x-component of ∇f.

  • A. 10
  • B. 11
  • C. 9
  • D. 13
Show correct answer and explanation

Correct answer: A. 10

The x-component is ∂f/∂x=10.

Q70. For f(x,y)=11x+12y, find the x-component of ∇f.

  • A. 14
  • B. 11
  • C. 12
  • D. 10
Show correct answer and explanation

Correct answer: B. 11

The x-component is ∂f/∂x=11.

Q71. Find the divergence of F=(2x,3y,z).

  • A. 5
  • B. 9
  • C. 6
  • D. 7
Show correct answer and explanation

Correct answer: C. 6

The sum of diagonal partials is 2+3+1=6.

Q72. Find the divergence of F=(3x,4y,z).

  • A. 9
  • B. 7
  • C. 11
  • D. 8
Show correct answer and explanation

Correct answer: D. 8

The sum of diagonal partials is 3+4+1=8.

Q73. Find the divergence of F=(4x,5y,z).

  • A. 10
  • B. 11
  • C. 9
  • D. 13
Show correct answer and explanation

Correct answer: A. 10

The sum of diagonal partials is 4+5+1=10.

Q74. Find the divergence of F=(5x,6y,z).

  • A. 15
  • B. 12
  • C. 13
  • D. 11
Show correct answer and explanation

Correct answer: B. 12

The sum of diagonal partials is 5+6+1=12.

Q75. Find the divergence of F=(6x,7y,z).

  • A. 13
  • B. 17
  • C. 14
  • D. 15
Show correct answer and explanation

Correct answer: C. 14

The sum of diagonal partials is 6+7+1=14.

Q76. Find the divergence of F=(7x,8y,z).

  • A. 17
  • B. 15
  • C. 19
  • D. 16
Show correct answer and explanation

Correct answer: D. 16

The sum of diagonal partials is 7+8+1=16.

Q77. Find the divergence of F=(8x,9y,z).

  • A. 18
  • B. 19
  • C. 17
  • D. 21
Show correct answer and explanation

Correct answer: A. 18

The sum of diagonal partials is 8+9+1=18.

Q78. Find the divergence of F=(9x,10y,z).

  • A. 23
  • B. 20
  • C. 21
  • D. 19
Show correct answer and explanation

Correct answer: B. 20

The sum of diagonal partials is 9+10+1=20.

Q79. Find the divergence of F=(10x,11y,z).

  • A. 21
  • B. 25
  • C. 22
  • D. 23
Show correct answer and explanation

Correct answer: C. 22

The sum of diagonal partials is 10+11+1=22.

Q80. Find the divergence of F=(11x,12y,z).

  • A. 25
  • B. 23
  • C. 27
  • D. 24
Show correct answer and explanation

Correct answer: D. 24

The sum of diagonal partials is 11+12+1=24.

Q81. Evaluate ∫₀^2∫₀^3 1 dy dx.

  • A. 6
  • B. 7
  • C. 5
  • D. 9
Show correct answer and explanation

Correct answer: A. 6

The integral equals rectangle area 2×3=6.

Q82. Evaluate ∫₀^3∫₀^4 1 dy dx.

  • A. 15
  • B. 12
  • C. 13
  • D. 11
Show correct answer and explanation

Correct answer: B. 12

The integral equals rectangle area 3×4=12.

Q83. Evaluate ∫₀^4∫₀^5 1 dy dx.

  • A. 19
  • B. 23
  • C. 20
  • D. 21
Show correct answer and explanation

Correct answer: C. 20

The integral equals rectangle area 4×5=20.

Q84. Evaluate ∫₀^5∫₀^6 1 dy dx.

  • A. 31
  • B. 29
  • C. 33
  • D. 30
Show correct answer and explanation

Correct answer: D. 30

The integral equals rectangle area 5×6=30.

Q85. Evaluate ∫₀^6∫₀^7 1 dy dx.

  • A. 42
  • B. 43
  • C. 41
  • D. 45
Show correct answer and explanation

Correct answer: A. 42

The integral equals rectangle area 6×7=42.

Q86. Evaluate ∫₀^7∫₀^8 1 dy dx.

  • A. 59
  • B. 56
  • C. 57
  • D. 55
Show correct answer and explanation

Correct answer: B. 56

The integral equals rectangle area 7×8=56.

Q87. Evaluate ∫₀^8∫₀^9 1 dy dx.

  • A. 71
  • B. 75
  • C. 72
  • D. 73
Show correct answer and explanation

Correct answer: C. 72

The integral equals rectangle area 8×9=72.

Q88. Evaluate ∫₀^9∫₀^10 1 dy dx.

  • A. 91
  • B. 89
  • C. 93
  • D. 90
Show correct answer and explanation

Correct answer: D. 90

The integral equals rectangle area 9×10=90.

Q89. Evaluate ∫₀^10∫₀^11 1 dy dx.

  • A. 110
  • B. 111
  • C. 109
  • D. 113
Show correct answer and explanation

Correct answer: A. 110

The integral equals rectangle area 10×11=110.

Q90. Evaluate ∫₀^11∫₀^12 1 dy dx.

  • A. 135
  • B. 132
  • C. 133
  • D. 131
Show correct answer and explanation

Correct answer: B. 132

The integral equals rectangle area 11×12=132.

Q91. Evaluate ∫₀¹∫₀^3 2x dy dx.

  • A. 2
  • B. 6
  • C. 3
  • D. 4
Show correct answer and explanation

Correct answer: C. 3

Integrating first in y yields 6x; ∫₀¹ x dx=1/2.

Q92. Evaluate ∫₀¹∫₀^4 3x dy dx.

  • A. 7
  • B. 5
  • C. 9
  • D. 6
Show correct answer and explanation

Correct answer: D. 6

Integrating first in y yields 12x; ∫₀¹ x dx=1/2.

Q93. Evaluate ∫₀¹∫₀^5 4x dy dx.

  • A. 10
  • B. 11
  • C. 9
  • D. 13
Show correct answer and explanation

Correct answer: A. 10

Integrating first in y yields 20x; ∫₀¹ x dx=1/2.

Q94. Evaluate ∫₀¹∫₀^6 5x dy dx.

  • A. 18
  • B. 15
  • C. 16
  • D. 14
Show correct answer and explanation

Correct answer: B. 15

Integrating first in y yields 30x; ∫₀¹ x dx=1/2.

Q95. Evaluate ∫₀¹∫₀^7 6x dy dx.

  • A. 20
  • B. 24
  • C. 21
  • D. 22
Show correct answer and explanation

Correct answer: C. 21

Integrating first in y yields 42x; ∫₀¹ x dx=1/2.

Q96. Evaluate ∫₀¹∫₀^8 7x dy dx.

  • A. 29
  • B. 27
  • C. 31
  • D. 28
Show correct answer and explanation

Correct answer: D. 28

Integrating first in y yields 56x; ∫₀¹ x dx=1/2.

Q97. Evaluate ∫₀¹∫₀^9 8x dy dx.

  • A. 36
  • B. 37
  • C. 35
  • D. 39
Show correct answer and explanation

Correct answer: A. 36

Integrating first in y yields 72x; ∫₀¹ x dx=1/2.

Q98. Evaluate ∫₀¹∫₀^10 9x dy dx.

  • A. 48
  • B. 45
  • C. 46
  • D. 44
Show correct answer and explanation

Correct answer: B. 45

Integrating first in y yields 90x; ∫₀¹ x dx=1/2.

Q99. Evaluate ∫₀¹∫₀^11 10x dy dx.

  • A. 54
  • B. 58
  • C. 55
  • D. 56
Show correct answer and explanation

Correct answer: C. 55

Integrating first in y yields 110x; ∫₀¹ x dx=1/2.

Q100. Evaluate ∫₀¹∫₀^12 11x dy dx.

  • A. 67
  • B. 65
  • C. 69
  • D. 66
Show correct answer and explanation

Correct answer: D. 66

Integrating first in y yields 132x; ∫₀¹ x dx=1/2.

Q101. If z=x²+y², x=2t and y=3t, find dz/dt at t=1.

  • A. 26
  • B. 27
  • C. 25
  • D. 29
Show correct answer and explanation

Correct answer: A. 26

Substitute z=(13)t², then dz/dt=2(13)t.

Q102. If z=x²+y², x=3t and y=4t, find dz/dt at t=1.

  • A. 53
  • B. 50
  • C. 51
  • D. 49
Show correct answer and explanation

Correct answer: B. 50

Substitute z=(25)t², then dz/dt=2(25)t.

Q103. If z=x²+y², x=4t and y=5t, find dz/dt at t=1.

  • A. 81
  • B. 85
  • C. 82
  • D. 83
Show correct answer and explanation

Correct answer: C. 82

Substitute z=(41)t², then dz/dt=2(41)t.

Q104. If z=x²+y², x=5t and y=6t, find dz/dt at t=1.

  • A. 123
  • B. 121
  • C. 125
  • D. 122
Show correct answer and explanation

Correct answer: D. 122

Substitute z=(61)t², then dz/dt=2(61)t.

Q105. If z=x²+y², x=6t and y=7t, find dz/dt at t=1.

  • A. 170
  • B. 171
  • C. 169
  • D. 173
Show correct answer and explanation

Correct answer: A. 170

Substitute z=(85)t², then dz/dt=2(85)t.

Q106. If z=x²+y², x=7t and y=8t, find dz/dt at t=1.

  • A. 229
  • B. 226
  • C. 227
  • D. 225
Show correct answer and explanation

Correct answer: B. 226

Substitute z=(113)t², then dz/dt=2(113)t.

Q107. If z=x²+y², x=8t and y=9t, find dz/dt at t=1.

  • A. 289
  • B. 293
  • C. 290
  • D. 291
Show correct answer and explanation

Correct answer: C. 290

Substitute z=(145)t², then dz/dt=2(145)t.

Q108. If z=x²+y², x=9t and y=10t, find dz/dt at t=1.

  • A. 363
  • B. 361
  • C. 365
  • D. 362
Show correct answer and explanation

Correct answer: D. 362

Substitute z=(181)t², then dz/dt=2(181)t.

Q109. If z=x²+y², x=10t and y=11t, find dz/dt at t=1.

  • A. 442
  • B. 443
  • C. 441
  • D. 445
Show correct answer and explanation

Correct answer: A. 442

Substitute z=(221)t², then dz/dt=2(221)t.

Q110. If z=x²+y², x=11t and y=12t, find dz/dt at t=1.

  • A. 533
  • B. 530
  • C. 531
  • D. 529
Show correct answer and explanation

Correct answer: B. 530

Substitute z=(265)t², then dz/dt=2(265)t.

Q111. For f(x,y)=x²+2y², find the trace of its Hessian.

  • A. 5
  • B. 9
  • C. 6
  • D. 7
Show correct answer and explanation

Correct answer: C. 6

The Hessian is diag(2,4); its trace is 6.

Q112. For f(x,y)=x²+3y², find the trace of its Hessian.

  • A. 9
  • B. 7
  • C. 11
  • D. 8
Show correct answer and explanation

Correct answer: D. 8

The Hessian is diag(2,6); its trace is 8.

Q113. For f(x,y)=x²+4y², find the trace of its Hessian.

  • A. 10
  • B. 11
  • C. 9
  • D. 13
Show correct answer and explanation

Correct answer: A. 10

The Hessian is diag(2,8); its trace is 10.

Q114. For f(x,y)=x²+5y², find the trace of its Hessian.

  • A. 15
  • B. 12
  • C. 13
  • D. 11
Show correct answer and explanation

Correct answer: B. 12

The Hessian is diag(2,10); its trace is 12.

Q115. For f(x,y)=x²+6y², find the trace of its Hessian.

  • A. 13
  • B. 17
  • C. 14
  • D. 15
Show correct answer and explanation

Correct answer: C. 14

The Hessian is diag(2,12); its trace is 14.

Q116. For f(x,y)=x²+7y², find the trace of its Hessian.

  • A. 17
  • B. 15
  • C. 19
  • D. 16
Show correct answer and explanation

Correct answer: D. 16

The Hessian is diag(2,14); its trace is 16.

Q117. For f(x,y)=x²+8y², find the trace of its Hessian.

  • A. 18
  • B. 19
  • C. 17
  • D. 21
Show correct answer and explanation

Correct answer: A. 18

The Hessian is diag(2,16); its trace is 18.

Q118. For f(x,y)=x²+9y², find the trace of its Hessian.

  • A. 23
  • B. 20
  • C. 21
  • D. 19
Show correct answer and explanation

Correct answer: B. 20

The Hessian is diag(2,18); its trace is 20.

Q119. For f(x,y)=x²+10y², find the trace of its Hessian.

  • A. 21
  • B. 25
  • C. 22
  • D. 23
Show correct answer and explanation

Correct answer: C. 22

The Hessian is diag(2,20); its trace is 22.

Q120. For f(x,y)=x²+11y², find the trace of its Hessian.

  • A. 25
  • B. 23
  • C. 27
  • D. 24
Show correct answer and explanation

Correct answer: D. 24

The Hessian is diag(2,22); its trace is 24.

Q121. For F(x,y)=(2y,3x), find the planar scalar curl.

  • A. 1
  • B. 2
  • C. 0
  • D. 4
Show correct answer and explanation

Correct answer: A. 1

Scalar curl is ∂Q/∂x−∂P/∂y=3−2=1.

Q122. For F(x,y)=(3y,4x), find the planar scalar curl.

  • A. 4
  • B. 1
  • C. 2
  • D. 0
Show correct answer and explanation

Correct answer: B. 1

Scalar curl is ∂Q/∂x−∂P/∂y=4−3=1.

Q123. For F(x,y)=(4y,5x), find the planar scalar curl.

  • A. 0
  • B. 4
  • C. 1
  • D. 2
Show correct answer and explanation

Correct answer: C. 1

Scalar curl is ∂Q/∂x−∂P/∂y=5−4=1.

Q124. For F(x,y)=(5y,6x), find the planar scalar curl.

  • A. 2
  • B. 0
  • C. 4
  • D. 1
Show correct answer and explanation

Correct answer: D. 1

Scalar curl is ∂Q/∂x−∂P/∂y=6−5=1.

Q125. For F(x,y)=(6y,7x), find the planar scalar curl.

  • A. 1
  • B. 2
  • C. 0
  • D. 4
Show correct answer and explanation

Correct answer: A. 1

Scalar curl is ∂Q/∂x−∂P/∂y=7−6=1.

Q126. For F(x,y)=(7y,8x), find the planar scalar curl.

  • A. 4
  • B. 1
  • C. 2
  • D. 0
Show correct answer and explanation

Correct answer: B. 1

Scalar curl is ∂Q/∂x−∂P/∂y=8−7=1.

Q127. For F(x,y)=(8y,9x), find the planar scalar curl.

  • A. 0
  • B. 4
  • C. 1
  • D. 2
Show correct answer and explanation

Correct answer: C. 1

Scalar curl is ∂Q/∂x−∂P/∂y=9−8=1.

Q128. For F(x,y)=(9y,10x), find the planar scalar curl.

  • A. 2
  • B. 0
  • C. 4
  • D. 1
Show correct answer and explanation

Correct answer: D. 1

Scalar curl is ∂Q/∂x−∂P/∂y=10−9=1.

Q129. For F(x,y)=(10y,11x), find the planar scalar curl.

  • A. 1
  • B. 2
  • C. 0
  • D. 4
Show correct answer and explanation

Correct answer: A. 1

Scalar curl is ∂Q/∂x−∂P/∂y=11−10=1.

Q130. For F(x,y)=(11y,12x), find the planar scalar curl.

  • A. 4
  • B. 1
  • C. 2
  • D. 0
Show correct answer and explanation

Correct answer: B. 1

Scalar curl is ∂Q/∂x−∂P/∂y=12−11=1.

Q131. Integrate the constant scalar field 2 along a curve of arc length 3.

  • A. 5
  • B. 9
  • C. 6
  • D. 7
Show correct answer and explanation

Correct answer: C. 6

The scalar line integral equals 2×length=6.

Q132. Integrate the constant scalar field 3 along a curve of arc length 4.

  • A. 13
  • B. 11
  • C. 15
  • D. 12
Show correct answer and explanation

Correct answer: D. 12

The scalar line integral equals 3×length=12.

Q133. Integrate the constant scalar field 4 along a curve of arc length 5.

  • A. 20
  • B. 21
  • C. 19
  • D. 23
Show correct answer and explanation

Correct answer: A. 20

The scalar line integral equals 4×length=20.

Q134. Integrate the constant scalar field 5 along a curve of arc length 6.

  • A. 33
  • B. 30
  • C. 31
  • D. 29
Show correct answer and explanation

Correct answer: B. 30

The scalar line integral equals 5×length=30.

Q135. Integrate the constant scalar field 6 along a curve of arc length 7.

  • A. 41
  • B. 45
  • C. 42
  • D. 43
Show correct answer and explanation

Correct answer: C. 42

The scalar line integral equals 6×length=42.

Q136. Integrate the constant scalar field 7 along a curve of arc length 8.

  • A. 57
  • B. 55
  • C. 59
  • D. 56
Show correct answer and explanation

Correct answer: D. 56

The scalar line integral equals 7×length=56.

Q137. Integrate the constant scalar field 8 along a curve of arc length 9.

  • A. 72
  • B. 73
  • C. 71
  • D. 75
Show correct answer and explanation

Correct answer: A. 72

The scalar line integral equals 8×length=72.

Q138. Integrate the constant scalar field 9 along a curve of arc length 10.

  • A. 93
  • B. 90
  • C. 91
  • D. 89
Show correct answer and explanation

Correct answer: B. 90

The scalar line integral equals 9×length=90.

Q139. Integrate the constant scalar field 10 along a curve of arc length 11.

  • A. 109
  • B. 113
  • C. 110
  • D. 111
Show correct answer and explanation

Correct answer: C. 110

The scalar line integral equals 10×length=110.

Q140. Integrate the constant scalar field 11 along a curve of arc length 12.

  • A. 133
  • B. 131
  • C. 135
  • D. 132
Show correct answer and explanation

Correct answer: D. 132

The scalar line integral equals 11×length=132.

Q141. For F=(2,0), find ∫ F·dr along the straight path from (0,0) to (3,0).

  • A. 6
  • B. 7
  • C. 5
  • D. 9
Show correct answer and explanation

Correct answer: A. 6

The constant vector dotted with displacement (3,0) gives 6.

Q142. For F=(3,0), find ∫ F·dr along the straight path from (0,0) to (4,0).

  • A. 15
  • B. 12
  • C. 13
  • D. 11
Show correct answer and explanation

Correct answer: B. 12

The constant vector dotted with displacement (4,0) gives 12.

Q143. For F=(4,0), find ∫ F·dr along the straight path from (0,0) to (5,0).

  • A. 19
  • B. 23
  • C. 20
  • D. 21
Show correct answer and explanation

Correct answer: C. 20

The constant vector dotted with displacement (5,0) gives 20.

Q144. For F=(5,0), find ∫ F·dr along the straight path from (0,0) to (6,0).

  • A. 31
  • B. 29
  • C. 33
  • D. 30
Show correct answer and explanation

Correct answer: D. 30

The constant vector dotted with displacement (6,0) gives 30.

Q145. For F=(6,0), find ∫ F·dr along the straight path from (0,0) to (7,0).

  • A. 42
  • B. 43
  • C. 41
  • D. 45
Show correct answer and explanation

Correct answer: A. 42

The constant vector dotted with displacement (7,0) gives 42.

Q146. For F=(7,0), find ∫ F·dr along the straight path from (0,0) to (8,0).

  • A. 59
  • B. 56
  • C. 57
  • D. 55
Show correct answer and explanation

Correct answer: B. 56

The constant vector dotted with displacement (8,0) gives 56.

Q147. For F=(8,0), find ∫ F·dr along the straight path from (0,0) to (9,0).

  • A. 71
  • B. 75
  • C. 72
  • D. 73
Show correct answer and explanation

Correct answer: C. 72

The constant vector dotted with displacement (9,0) gives 72.

Q148. For F=(9,0), find ∫ F·dr along the straight path from (0,0) to (10,0).

  • A. 91
  • B. 89
  • C. 93
  • D. 90
Show correct answer and explanation

Correct answer: D. 90

The constant vector dotted with displacement (10,0) gives 90.

Q149. For F=(10,0), find ∫ F·dr along the straight path from (0,0) to (11,0).

  • A. 110
  • B. 111
  • C. 109
  • D. 113
Show correct answer and explanation

Correct answer: A. 110

The constant vector dotted with displacement (11,0) gives 110.

Q150. For F=(11,0), find ∫ F·dr along the straight path from (0,0) to (12,0).

  • A. 135
  • B. 132
  • C. 133
  • D. 131
Show correct answer and explanation

Correct answer: B. 132

The constant vector dotted with displacement (12,0) gives 132.

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