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Multivariable Calculus — One Question at a Time

Multivariable Calculus — One Question at a Time

Practise 150 Multivariable Calculus multiple-choice questions covering partial derivatives, gradients, double integrals, vector calculus, Fourier series and Laplace transforms. Work through one question at a time, check the correct answer with an explanation, and select Next Question to continue. This is self-paced practice without automatic scoring.

Before you practise: Read Multivariable Calculus study material and examples. You can also browse all 1,576 interactive MCQs.

Practice Set 1 · Question 1 of 150

Question 1. For f(x,y)=x²+2y, find f(2,3).

  1. 7
  2. 10
  3. 11
  4. 12
Show Answer and Explanation

Correct answer: B. 10

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

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Practice Set 1 · Question 2 of 150

Question 2. For f=x²y+3y, find ∂f/∂x.

  1. 2xy
  2. x²+3
  3. 2x+3
  4. 2y
Show Answer and Explanation

Correct answer: A. 2xy

Treat y as constant, so derivative is 2xy.

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Practice Set 1 · Question 3 of 150

Question 3. For f=x²y+3y, find ∂f/∂y.

  1. 2xy
  2. x²y
  3. x²+3
  4. 3y
Show Answer and Explanation

Correct answer: C. x²+3

Treat x as constant: derivative is x²+3.

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Practice Set 1 · Question 4 of 150

Question 4. Find ∇f for f=x²+y².

  1. (x,y)
  2. (2x,2y)
  3. (2,2)
  4. (x²,y²)
Show Answer and Explanation

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

Gradient collects first partial derivatives.

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Practice Set 1 · Question 5 of 150

Question 5. For a unit vector u, what equals the directional derivative of differentiable f?

  1. f·u
  2. ∇f+u
  3. ∇f×u
  4. ∇f·u
Show Answer and Explanation

Correct answer: D. ∇f·u

A directional derivative equals gradient dot unit direction.

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Practice Set 1 · Question 6 of 150

Question 6. Evaluate ∫₀¹ ∫₀¹ 1 dy dx.

  1. 1
  2. 2
  3. 0
  4. 1/2
Show Answer and Explanation

Correct answer: A. 1

Integrating 1 over the unit square gives area 1.

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Practice Set 1 · Question 7 of 150

Question 7. What is the integral of 1 over the unit cube?

  1. 0
  2. 3
  3. 1
  4. 6
Show Answer and Explanation

Correct answer: C. 1

Its value equals the cube’s volume, 1.

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Practice Set 1 · Question 8 of 150

Question 8. For a scalar field equal to 2 along a curve of length 3, find ∫C 2 ds.

  1. 3
  2. 6
  3. 2
  4. 9
Show Answer and Explanation

Correct answer: B. 6

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

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Practice Set 1 · Question 9 of 150

Question 9. A 2π-periodic odd integrable function has which Fourier terms?

  1. Only cosine terms
  2. Constant and cosine terms
  3. Only sine terms
  4. No terms at all
Show Answer and Explanation

Correct answer: C. Only sine terms

Oddness forces the constant and cosine coefficients to vanish.

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Practice Set 1 · Question 10 of 150

Question 10. What is the Laplace transform of the constant function 1 for Re(s)>0?

  1. s
  2. 1/s
  3. s²
  4. 1/s²
Show Answer and Explanation

Correct answer: B. 1/s

∫₀∞ e^(−st)dt = 1/s for Re(s)>0.

Continue to Practice Set 2

Practice Set 2 · Question 11 of 150

Question 11. What is lim((x,y)→(0,0)) (x²+y²)?

  1. 0
  2. 1
  3. 2
  4. Does not exist
Show Answer and Explanation

Correct answer: A. 0

Both squared terms approach 0.

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Practice Set 2 · Question 12 of 150

Question 12. Find ∂²f/∂x∂y for f=x²y.

  1. x²
  2. 2x
  3. 2y
  4. 0
Show Answer and Explanation

Correct answer: B. 2x

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

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Practice Set 2 · Question 13 of 150

Question 13. If z=x²+y² with x=t and y=2t, find dz/dt.

  1. 2t
  2. 4t
  3. 5t
  4. 10t
Show Answer and Explanation

Correct answer: D. 10t

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

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Practice Set 2 · Question 14 of 150

Question 14. At a differentiable interior local extremum of f(x,y), what is necessarily true?

  1. ∇f=0
  2. f=0
  3. ∇f=(1,1)
  4. f is linear
Show Answer and Explanation

Correct answer: A. ∇f=0

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

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Practice Set 2 · Question 15 of 150

Question 15. Evaluate ∫₀² ∫₀³ 1 dy dx.

  1. 3
  2. 5
  3. 6
  4. 12
Show Answer and Explanation

Correct answer: C. 6

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

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Practice Set 2 · Question 16 of 150

Question 16. What is x²+y² in polar coordinates?

  1. r
  2. r²
  3. r sinθ
  4. r cosθ
Show Answer and Explanation

Correct answer: B. r²

With x=r cosθ and y=r sinθ, x²+y²=r².

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Practice Set 2 · Question 17 of 150

Question 17. For F=(P,Q) in the plane, what is the scalar curl expression?

  1. ∂P/∂x+∂Q/∂y
  2. ∂P/∂y+∂Q/∂x
  3. P+Q
  4. ∂Q/∂x−∂P/∂y
Show Answer and Explanation

Correct answer: D. ∂Q/∂x−∂P/∂y

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

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Practice Set 2 · Question 18 of 150

Question 18. 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?

  1. ∬(∂Q/∂x−∂P/∂y)dA
  2. ∬(P+Q)dA
  3. ∫P dx only
  4. ∫Q dy only
Show Answer and Explanation

Correct answer: A. ∬(∂Q/∂x−∂P/∂y)dA

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

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Practice Set 2 · Question 19 of 150

Question 19. For an even integrable function on [−π,π], which Fourier coefficients vanish?

  1. Cosine coefficients
  2. Constant term
  3. Sine coefficients
  4. All coefficients
Show 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.

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Practice Set 2 · Question 20 of 150

Question 20. For a real constant a, what is L{e^(at)}(s) for Re(s)>a?

  1. s−a
  2. 1/(s−a)
  3. 1/(s+a)
  4. a/s
Show Answer and Explanation

Correct answer: B. 1/(s−a)

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

Continue to Practice Set 3

Practice Set 3 · Question 21 of 150

Question 21. What is the domain of f(x,y)=√(9−x²−y²)?

  1. All R²
  2. x²+y²<9 only
  3. x²+y²≤9
  4. x²+y²≥9
Show Answer and Explanation

Correct answer: C. x²+y²≤9

The expression under the square root must be nonnegative.

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Practice Set 3 · Question 22 of 150

Question 22. Does lim((x,y)→(0,0)) xy/(x²+y²) exist?

  1. Yes, 0
  2. Yes, 1/2
  3. Yes, −1/2
  4. No
Show Answer and Explanation

Correct answer: D. No

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

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Practice Set 3 · Question 23 of 150

Question 23. Find ∇f for f=3x+4y.

  1. (3,4)
  2. (x,y)
  3. (4,3)
  4. (7,0)
Show Answer and Explanation

Correct answer: A. (3,4)

The two partial derivatives are 3 and 4.

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Practice Set 3 · Question 24 of 150

Question 24. Find the Hessian of f=x²+3y².

  1. diag(1,3)
  2. diag(2,6)
  3. diag(2,3)
  4. diag(x,y)
Show Answer and Explanation

Correct answer: B. diag(2,6)

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

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Practice Set 3 · Question 25 of 150

Question 25. Find ∫₀¹∫₀² x dy dx.

  1. 1/2
  2. 1
  3. 2
  4. 4
Show Answer and Explanation

Correct answer: B. 1

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

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Practice Set 3 · Question 26 of 150

Question 26. Which factor appears in the polar area element dA?

  1. 1
  2. r²
  3. sin θ
  4. r
Show Answer and Explanation

Correct answer: D. r

The Jacobian gives dA=r dr dθ.

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Practice Set 3 · Question 27 of 150

Question 27. Find div F for F=(x,y,z).

  1. 3
  2. x+y+z
  3. 1
  4. 0
Show Answer and Explanation

Correct answer: A. 3

Each of the three diagonal partial derivatives is 1.

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Practice Set 3 · Question 28 of 150

Question 28. For F=(1,0), find ∫C F·dr along a straight segment from (0,0) to (2,0).

  1. 0
  2. 1
  3. 2
  4. 4
Show Answer and Explanation

Correct answer: C. 2

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

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Practice Set 3 · Question 29 of 150

Question 29. Stokes’ theorem equates boundary circulation with which quantity?

  1. Time derivative
  2. Flux of curl through an oriented surface
  3. Surface area alone
  4. Volume alone
Show Answer and Explanation

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

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

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Practice Set 3 · Question 30 of 150

Question 30. Find the unilateral Z-transform of a_n=1 for n≥0 and |z|>1.

  1. z/(z−1)
  2. 1/z
  3. z−1
  4. 1/(1−z)
Show Answer and Explanation

Correct answer: A. z/(z−1)

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

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Practice Set 4 · Question 31 of 150

Question 31. For f(x,y)=x²+3y, evaluate f(1,2).

  1. 6
  2. 10
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

Substitution gives 1²+3×2=7.

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Practice Set 4 · Question 32 of 150

Question 32. For f(x,y)=x²+4y, evaluate f(2,3).

  1. 17
  2. 15
  3. 19
  4. 16
Show Answer and Explanation

Correct answer: D. 16

Substitution gives 2²+4×3=16.

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Practice Set 4 · Question 33 of 150

Question 33. For f(x,y)=x²+5y, evaluate f(3,4).

  1. 29
  2. 30
  3. 28
  4. 32
Show Answer and Explanation

Correct answer: A. 29

Substitution gives 3²+5×4=29.

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Practice Set 4 · Question 34 of 150

Question 34. For f(x,y)=x²+6y, evaluate f(4,5).

  1. 49
  2. 46
  3. 47
  4. 45
Show Answer and Explanation

Correct answer: B. 46

Substitution gives 4²+6×5=46.

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Practice Set 4 · Question 35 of 150

Question 35. For f(x,y)=x²+7y, evaluate f(5,6).

  1. 66
  2. 70
  3. 67
  4. 68
Show Answer and Explanation

Correct answer: C. 67

Substitution gives 5²+7×6=67.

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Practice Set 4 · Question 36 of 150

Question 36. For f(x,y)=x²+8y, evaluate f(6,7).

  1. 93
  2. 91
  3. 95
  4. 92
Show Answer and Explanation

Correct answer: D. 92

Substitution gives 6²+8×7=92.

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Practice Set 4 · Question 37 of 150

Question 37. For f(x,y)=x²+9y, evaluate f(7,8).

  1. 121
  2. 122
  3. 120
  4. 124
Show Answer and Explanation

Correct answer: A. 121

Substitution gives 7²+9×8=121.

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Practice Set 4 · Question 38 of 150

Question 38. For f(x,y)=x²+10y, evaluate f(8,9).

  1. 157
  2. 154
  3. 155
  4. 153
Show Answer and Explanation

Correct answer: B. 154

Substitution gives 8²+10×9=154.

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Practice Set 4 · Question 39 of 150

Question 39. For f(x,y)=x²+11y, evaluate f(9,10).

  1. 190
  2. 194
  3. 191
  4. 192
Show Answer and Explanation

Correct answer: C. 191

Substitution gives 9²+11×10=191.

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Practice Set 4 · Question 40 of 150

Question 40. For f(x,y)=x²+12y, evaluate f(10,11).

  1. 233
  2. 231
  3. 235
  4. 232
Show Answer and Explanation

Correct answer: D. 232

Substitution gives 10²+12×11=232.

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Practice Set 5 · Question 41 of 150

Question 41. For f(x,y)=2x²y, evaluate ∂f/∂x at (1,2).

  1. 8
  2. 9
  3. 7
  4. 11
Show Answer and Explanation

Correct answer: A. 8

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

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Practice Set 5 · Question 42 of 150

Question 42. For f(x,y)=3x²y, evaluate ∂f/∂x at (2,2).

  1. 27
  2. 24
  3. 25
  4. 23
Show Answer and Explanation

Correct answer: B. 24

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

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Practice Set 5 · Question 43 of 150

Question 43. For f(x,y)=4x²y, evaluate ∂f/∂x at (3,2).

  1. 47
  2. 51
  3. 48
  4. 49
Show Answer and Explanation

Correct answer: C. 48

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

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Practice Set 5 · Question 44 of 150

Question 44. For f(x,y)=5x²y, evaluate ∂f/∂x at (4,2).

  1. 81
  2. 79
  3. 83
  4. 80
Show Answer and Explanation

Correct answer: D. 80

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

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Practice Set 5 · Question 45 of 150

Question 45. For f(x,y)=6x²y, evaluate ∂f/∂x at (5,2).

  1. 120
  2. 121
  3. 119
  4. 123
Show Answer and Explanation

Correct answer: A. 120

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

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Practice Set 5 · Question 46 of 150

Question 46. For f(x,y)=7x²y, evaluate ∂f/∂x at (6,2).

  1. 171
  2. 168
  3. 169
  4. 167
Show Answer and Explanation

Correct answer: B. 168

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

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Practice Set 5 · Question 47 of 150

Question 47. For f(x,y)=8x²y, evaluate ∂f/∂x at (7,2).

  1. 223
  2. 227
  3. 224
  4. 225
Show Answer and Explanation

Correct answer: C. 224

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

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Practice Set 5 · Question 48 of 150

Question 48. For f(x,y)=9x²y, evaluate ∂f/∂x at (8,2).

  1. 289
  2. 287
  3. 291
  4. 288
Show Answer and Explanation

Correct answer: D. 288

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

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Practice Set 5 · Question 49 of 150

Question 49. For f(x,y)=10x²y, evaluate ∂f/∂x at (9,2).

  1. 360
  2. 361
  3. 359
  4. 363
Show Answer and Explanation

Correct answer: A. 360

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

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Practice Set 5 · Question 50 of 150

Question 50. For f(x,y)=11x²y, evaluate ∂f/∂x at (10,2).

  1. 443
  2. 440
  3. 441
  4. 439
Show Answer and Explanation

Correct answer: B. 440

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

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Practice Set 6 · Question 51 of 150

Question 51. For f(x,y)=2xy², evaluate ∂f/∂y at (1,3).

  1. 11
  2. 15
  3. 12
  4. 13
Show Answer and Explanation

Correct answer: C. 12

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

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Practice Set 6 · Question 52 of 150

Question 52. For f(x,y)=3xy², evaluate ∂f/∂y at (2,3).

  1. 37
  2. 35
  3. 39
  4. 36
Show Answer and Explanation

Correct answer: D. 36

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

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Practice Set 6 · Question 53 of 150

Question 53. For f(x,y)=4xy², evaluate ∂f/∂y at (3,3).

  1. 72
  2. 73
  3. 71
  4. 75
Show Answer and Explanation

Correct answer: A. 72

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

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Practice Set 6 · Question 54 of 150

Question 54. For f(x,y)=5xy², evaluate ∂f/∂y at (4,3).

  1. 123
  2. 120
  3. 121
  4. 119
Show Answer and Explanation

Correct answer: B. 120

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

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Practice Set 6 · Question 55 of 150

Question 55. For f(x,y)=6xy², evaluate ∂f/∂y at (5,3).

  1. 179
  2. 183
  3. 180
  4. 181
Show Answer and Explanation

Correct answer: C. 180

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

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Practice Set 6 · Question 56 of 150

Question 56. For f(x,y)=7xy², evaluate ∂f/∂y at (6,3).

  1. 253
  2. 251
  3. 255
  4. 252
Show Answer and Explanation

Correct answer: D. 252

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

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Practice Set 6 · Question 57 of 150

Question 57. For f(x,y)=8xy², evaluate ∂f/∂y at (7,3).

  1. 336
  2. 337
  3. 335
  4. 339
Show Answer and Explanation

Correct answer: A. 336

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

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Practice Set 6 · Question 58 of 150

Question 58. For f(x,y)=9xy², evaluate ∂f/∂y at (8,3).

  1. 435
  2. 432
  3. 433
  4. 431
Show Answer and Explanation

Correct answer: B. 432

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

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Practice Set 6 · Question 59 of 150

Question 59. For f(x,y)=10xy², evaluate ∂f/∂y at (9,3).

  1. 539
  2. 543
  3. 540
  4. 541
Show Answer and Explanation

Correct answer: C. 540

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

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Practice Set 6 · Question 60 of 150

Question 60. For f(x,y)=11xy², evaluate ∂f/∂y at (10,3).

  1. 661
  2. 659
  3. 663
  4. 660
Show Answer and Explanation

Correct answer: D. 660

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

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Practice Set 7 · Question 61 of 150

Question 61. For f(x,y)=2x+3y, find the x-component of ∇f.

  1. 2
  2. 3
  3. 1
  4. 5
Show Answer and Explanation

Correct answer: A. 2

The x-component is ∂f/∂x=2.

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Practice Set 7 · Question 62 of 150

Question 62. For f(x,y)=3x+4y, find the x-component of ∇f.

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

The x-component is ∂f/∂x=3.

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Practice Set 7 · Question 63 of 150

Question 63. For f(x,y)=4x+5y, find the x-component of ∇f.

  1. 3
  2. 7
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

The x-component is ∂f/∂x=4.

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Practice Set 7 · Question 64 of 150

Question 64. For f(x,y)=5x+6y, find the x-component of ∇f.

  1. 6
  2. 4
  3. 8
  4. 5
Show Answer and Explanation

Correct answer: D. 5

The x-component is ∂f/∂x=5.

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Practice Set 7 · Question 65 of 150

Question 65. For f(x,y)=6x+7y, find the x-component of ∇f.

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

The x-component is ∂f/∂x=6.

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Practice Set 7 · Question 66 of 150

Question 66. For f(x,y)=7x+8y, find the x-component of ∇f.

  1. 10
  2. 7
  3. 8
  4. 6
Show Answer and Explanation

Correct answer: B. 7

The x-component is ∂f/∂x=7.

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Practice Set 7 · Question 67 of 150

Question 67. For f(x,y)=8x+9y, find the x-component of ∇f.

  1. 7
  2. 11
  3. 8
  4. 9
Show Answer and Explanation

Correct answer: C. 8

The x-component is ∂f/∂x=8.

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Practice Set 7 · Question 68 of 150

Question 68. For f(x,y)=9x+10y, find the x-component of ∇f.

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

The x-component is ∂f/∂x=9.

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Practice Set 7 · Question 69 of 150

Question 69. For f(x,y)=10x+11y, find the x-component of ∇f.

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

The x-component is ∂f/∂x=10.

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Practice Set 7 · Question 70 of 150

Question 70. For f(x,y)=11x+12y, find the x-component of ∇f.

  1. 14
  2. 11
  3. 12
  4. 10
Show Answer and Explanation

Correct answer: B. 11

The x-component is ∂f/∂x=11.

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Practice Set 8 · Question 71 of 150

Question 71. Find the divergence of F=(2x,3y,z).

  1. 5
  2. 9
  3. 6
  4. 7
Show Answer and Explanation

Correct answer: C. 6

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

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Practice Set 8 · Question 72 of 150

Question 72. Find the divergence of F=(3x,4y,z).

  1. 9
  2. 7
  3. 11
  4. 8
Show Answer and Explanation

Correct answer: D. 8

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

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Practice Set 8 · Question 73 of 150

Question 73. Find the divergence of F=(4x,5y,z).

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

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

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Practice Set 8 · Question 74 of 150

Question 74. Find the divergence of F=(5x,6y,z).

  1. 15
  2. 12
  3. 13
  4. 11
Show Answer and Explanation

Correct answer: B. 12

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

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Practice Set 8 · Question 75 of 150

Question 75. Find the divergence of F=(6x,7y,z).

  1. 13
  2. 17
  3. 14
  4. 15
Show Answer and Explanation

Correct answer: C. 14

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

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Practice Set 8 · Question 76 of 150

Question 76. Find the divergence of F=(7x,8y,z).

  1. 17
  2. 15
  3. 19
  4. 16
Show Answer and Explanation

Correct answer: D. 16

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

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Practice Set 8 · Question 77 of 150

Question 77. Find the divergence of F=(8x,9y,z).

  1. 18
  2. 19
  3. 17
  4. 21
Show Answer and Explanation

Correct answer: A. 18

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

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Practice Set 8 · Question 78 of 150

Question 78. Find the divergence of F=(9x,10y,z).

  1. 23
  2. 20
  3. 21
  4. 19
Show Answer and Explanation

Correct answer: B. 20

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

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Practice Set 8 · Question 79 of 150

Question 79. Find the divergence of F=(10x,11y,z).

  1. 21
  2. 25
  3. 22
  4. 23
Show Answer and Explanation

Correct answer: C. 22

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

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Practice Set 8 · Question 80 of 150

Question 80. Find the divergence of F=(11x,12y,z).

  1. 25
  2. 23
  3. 27
  4. 24
Show Answer and Explanation

Correct answer: D. 24

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

Next Question →

Practice Set 9 · Question 81 of 150

Question 81. Evaluate ∫₀^2∫₀^3 1 dy dx.

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

The integral equals rectangle area 2×3=6.

Next Question →

Practice Set 9 · Question 82 of 150

Question 82. Evaluate ∫₀^3∫₀^4 1 dy dx.

  1. 15
  2. 12
  3. 13
  4. 11
Show Answer and Explanation

Correct answer: B. 12

The integral equals rectangle area 3×4=12.

Next Question →

Practice Set 9 · Question 83 of 150

Question 83. Evaluate ∫₀^4∫₀^5 1 dy dx.

  1. 19
  2. 23
  3. 20
  4. 21
Show Answer and Explanation

Correct answer: C. 20

The integral equals rectangle area 4×5=20.

Next Question →

Practice Set 9 · Question 84 of 150

Question 84. Evaluate ∫₀^5∫₀^6 1 dy dx.

  1. 31
  2. 29
  3. 33
  4. 30
Show Answer and Explanation

Correct answer: D. 30

The integral equals rectangle area 5×6=30.

Next Question →

Practice Set 9 · Question 85 of 150

Question 85. Evaluate ∫₀^6∫₀^7 1 dy dx.

  1. 42
  2. 43
  3. 41
  4. 45
Show Answer and Explanation

Correct answer: A. 42

The integral equals rectangle area 6×7=42.

Next Question →

Practice Set 9 · Question 86 of 150

Question 86. Evaluate ∫₀^7∫₀^8 1 dy dx.

  1. 59
  2. 56
  3. 57
  4. 55
Show Answer and Explanation

Correct answer: B. 56

The integral equals rectangle area 7×8=56.

Next Question →

Practice Set 9 · Question 87 of 150

Question 87. Evaluate ∫₀^8∫₀^9 1 dy dx.

  1. 71
  2. 75
  3. 72
  4. 73
Show Answer and Explanation

Correct answer: C. 72

The integral equals rectangle area 8×9=72.

Next Question →

Practice Set 9 · Question 88 of 150

Question 88. Evaluate ∫₀^9∫₀^10 1 dy dx.

  1. 91
  2. 89
  3. 93
  4. 90
Show Answer and Explanation

Correct answer: D. 90

The integral equals rectangle area 9×10=90.

Next Question →

Practice Set 9 · Question 89 of 150

Question 89. Evaluate ∫₀^10∫₀^11 1 dy dx.

  1. 110
  2. 111
  3. 109
  4. 113
Show Answer and Explanation

Correct answer: A. 110

The integral equals rectangle area 10×11=110.

Next Question →

Practice Set 9 · Question 90 of 150

Question 90. Evaluate ∫₀^11∫₀^12 1 dy dx.

  1. 135
  2. 132
  3. 133
  4. 131
Show Answer and Explanation

Correct answer: B. 132

The integral equals rectangle area 11×12=132.

Next Question →

Practice Set 10 · Question 91 of 150

Question 91. Evaluate ∫₀¹∫₀^3 2x dy dx.

  1. 2
  2. 6
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: C. 3

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

Next Question →

Practice Set 10 · Question 92 of 150

Question 92. Evaluate ∫₀¹∫₀^4 3x dy dx.

  1. 7
  2. 5
  3. 9
  4. 6
Show Answer and Explanation

Correct answer: D. 6

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

Next Question →

Practice Set 10 · Question 93 of 150

Question 93. Evaluate ∫₀¹∫₀^5 4x dy dx.

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

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

Next Question →

Practice Set 10 · Question 94 of 150

Question 94. Evaluate ∫₀¹∫₀^6 5x dy dx.

  1. 18
  2. 15
  3. 16
  4. 14
Show Answer and Explanation

Correct answer: B. 15

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

Next Question →

Practice Set 10 · Question 95 of 150

Question 95. Evaluate ∫₀¹∫₀^7 6x dy dx.

  1. 20
  2. 24
  3. 21
  4. 22
Show Answer and Explanation

Correct answer: C. 21

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

Next Question →

Practice Set 10 · Question 96 of 150

Question 96. Evaluate ∫₀¹∫₀^8 7x dy dx.

  1. 29
  2. 27
  3. 31
  4. 28
Show Answer and Explanation

Correct answer: D. 28

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

Next Question →

Practice Set 10 · Question 97 of 150

Question 97. Evaluate ∫₀¹∫₀^9 8x dy dx.

  1. 36
  2. 37
  3. 35
  4. 39
Show Answer and Explanation

Correct answer: A. 36

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

Next Question →

Practice Set 10 · Question 98 of 150

Question 98. Evaluate ∫₀¹∫₀^10 9x dy dx.

  1. 48
  2. 45
  3. 46
  4. 44
Show Answer and Explanation

Correct answer: B. 45

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

Next Question →

Practice Set 10 · Question 99 of 150

Question 99. Evaluate ∫₀¹∫₀^11 10x dy dx.

  1. 54
  2. 58
  3. 55
  4. 56
Show Answer and Explanation

Correct answer: C. 55

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

Next Question →

Practice Set 10 · Question 100 of 150

Question 100. Evaluate ∫₀¹∫₀^12 11x dy dx.

  1. 67
  2. 65
  3. 69
  4. 66
Show Answer and Explanation

Correct answer: D. 66

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

Next Question →

Practice Set 11 · Question 101 of 150

Question 101. If z=x²+y², x=2t and y=3t, find dz/dt at t=1.

  1. 26
  2. 27
  3. 25
  4. 29
Show Answer and Explanation

Correct answer: A. 26

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

Next Question →

Practice Set 11 · Question 102 of 150

Question 102. If z=x²+y², x=3t and y=4t, find dz/dt at t=1.

  1. 53
  2. 50
  3. 51
  4. 49
Show Answer and Explanation

Correct answer: B. 50

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

Next Question →

Practice Set 11 · Question 103 of 150

Question 103. If z=x²+y², x=4t and y=5t, find dz/dt at t=1.

  1. 81
  2. 85
  3. 82
  4. 83
Show Answer and Explanation

Correct answer: C. 82

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

Next Question →

Practice Set 11 · Question 104 of 150

Question 104. If z=x²+y², x=5t and y=6t, find dz/dt at t=1.

  1. 123
  2. 121
  3. 125
  4. 122
Show Answer and Explanation

Correct answer: D. 122

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

Next Question →

Practice Set 11 · Question 105 of 150

Question 105. If z=x²+y², x=6t and y=7t, find dz/dt at t=1.

  1. 170
  2. 171
  3. 169
  4. 173
Show Answer and Explanation

Correct answer: A. 170

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

Next Question →

Practice Set 11 · Question 106 of 150

Question 106. If z=x²+y², x=7t and y=8t, find dz/dt at t=1.

  1. 229
  2. 226
  3. 227
  4. 225
Show Answer and Explanation

Correct answer: B. 226

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

Next Question →

Practice Set 11 · Question 107 of 150

Question 107. If z=x²+y², x=8t and y=9t, find dz/dt at t=1.

  1. 289
  2. 293
  3. 290
  4. 291
Show Answer and Explanation

Correct answer: C. 290

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

Next Question →

Practice Set 11 · Question 108 of 150

Question 108. If z=x²+y², x=9t and y=10t, find dz/dt at t=1.

  1. 363
  2. 361
  3. 365
  4. 362
Show Answer and Explanation

Correct answer: D. 362

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

Next Question →

Practice Set 11 · Question 109 of 150

Question 109. If z=x²+y², x=10t and y=11t, find dz/dt at t=1.

  1. 442
  2. 443
  3. 441
  4. 445
Show Answer and Explanation

Correct answer: A. 442

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

Next Question →

Practice Set 11 · Question 110 of 150

Question 110. If z=x²+y², x=11t and y=12t, find dz/dt at t=1.

  1. 533
  2. 530
  3. 531
  4. 529
Show Answer and Explanation

Correct answer: B. 530

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

Next Question →

Practice Set 12 · Question 111 of 150

Question 111. For f(x,y)=x²+2y², find the trace of its Hessian.

  1. 5
  2. 9
  3. 6
  4. 7
Show Answer and Explanation

Correct answer: C. 6

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

Next Question →

Practice Set 12 · Question 112 of 150

Question 112. For f(x,y)=x²+3y², find the trace of its Hessian.

  1. 9
  2. 7
  3. 11
  4. 8
Show Answer and Explanation

Correct answer: D. 8

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

Next Question →

Practice Set 12 · Question 113 of 150

Question 113. For f(x,y)=x²+4y², find the trace of its Hessian.

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

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

Next Question →

Practice Set 12 · Question 114 of 150

Question 114. For f(x,y)=x²+5y², find the trace of its Hessian.

  1. 15
  2. 12
  3. 13
  4. 11
Show Answer and Explanation

Correct answer: B. 12

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

Next Question →

Practice Set 12 · Question 115 of 150

Question 115. For f(x,y)=x²+6y², find the trace of its Hessian.

  1. 13
  2. 17
  3. 14
  4. 15
Show Answer and Explanation

Correct answer: C. 14

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

Next Question →

Practice Set 12 · Question 116 of 150

Question 116. For f(x,y)=x²+7y², find the trace of its Hessian.

  1. 17
  2. 15
  3. 19
  4. 16
Show Answer and Explanation

Correct answer: D. 16

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

Next Question →

Practice Set 12 · Question 117 of 150

Question 117. For f(x,y)=x²+8y², find the trace of its Hessian.

  1. 18
  2. 19
  3. 17
  4. 21
Show Answer and Explanation

Correct answer: A. 18

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

Next Question →

Practice Set 12 · Question 118 of 150

Question 118. For f(x,y)=x²+9y², find the trace of its Hessian.

  1. 23
  2. 20
  3. 21
  4. 19
Show Answer and Explanation

Correct answer: B. 20

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

Next Question →

Practice Set 12 · Question 119 of 150

Question 119. For f(x,y)=x²+10y², find the trace of its Hessian.

  1. 21
  2. 25
  3. 22
  4. 23
Show Answer and Explanation

Correct answer: C. 22

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

Next Question →

Practice Set 12 · Question 120 of 150

Question 120. For f(x,y)=x²+11y², find the trace of its Hessian.

  1. 25
  2. 23
  3. 27
  4. 24
Show Answer and Explanation

Correct answer: D. 24

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

Next Question →

Practice Set 13 · Question 121 of 150

Question 121. For F(x,y)=(2y,3x), find the planar scalar curl.

  1. 1
  2. 2
  3. 0
  4. 4
Show Answer and Explanation

Correct answer: A. 1

Scalar curl is ∂Q/∂x−∂P/∂y=3−2=1.

Next Question →

Practice Set 13 · Question 122 of 150

Question 122. For F(x,y)=(3y,4x), find the planar scalar curl.

  1. 4
  2. 1
  3. 2
  4. 0
Show Answer and Explanation

Correct answer: B. 1

Scalar curl is ∂Q/∂x−∂P/∂y=4−3=1.

Next Question →

Practice Set 13 · Question 123 of 150

Question 123. For F(x,y)=(4y,5x), find the planar scalar curl.

  1. 0
  2. 4
  3. 1
  4. 2
Show Answer and Explanation

Correct answer: C. 1

Scalar curl is ∂Q/∂x−∂P/∂y=5−4=1.

Next Question →

Practice Set 13 · Question 124 of 150

Question 124. For F(x,y)=(5y,6x), find the planar scalar curl.

  1. 2
  2. 0
  3. 4
  4. 1
Show Answer and Explanation

Correct answer: D. 1

Scalar curl is ∂Q/∂x−∂P/∂y=6−5=1.

Next Question →

Practice Set 13 · Question 125 of 150

Question 125. For F(x,y)=(6y,7x), find the planar scalar curl.

  1. 1
  2. 2
  3. 0
  4. 4
Show Answer and Explanation

Correct answer: A. 1

Scalar curl is ∂Q/∂x−∂P/∂y=7−6=1.

Next Question →

Practice Set 13 · Question 126 of 150

Question 126. For F(x,y)=(7y,8x), find the planar scalar curl.

  1. 4
  2. 1
  3. 2
  4. 0
Show Answer and Explanation

Correct answer: B. 1

Scalar curl is ∂Q/∂x−∂P/∂y=8−7=1.

Next Question →

Practice Set 13 · Question 127 of 150

Question 127. For F(x,y)=(8y,9x), find the planar scalar curl.

  1. 0
  2. 4
  3. 1
  4. 2
Show Answer and Explanation

Correct answer: C. 1

Scalar curl is ∂Q/∂x−∂P/∂y=9−8=1.

Next Question →

Practice Set 13 · Question 128 of 150

Question 128. For F(x,y)=(9y,10x), find the planar scalar curl.

  1. 2
  2. 0
  3. 4
  4. 1
Show Answer and Explanation

Correct answer: D. 1

Scalar curl is ∂Q/∂x−∂P/∂y=10−9=1.

Next Question →

Practice Set 13 · Question 129 of 150

Question 129. For F(x,y)=(10y,11x), find the planar scalar curl.

  1. 1
  2. 2
  3. 0
  4. 4
Show Answer and Explanation

Correct answer: A. 1

Scalar curl is ∂Q/∂x−∂P/∂y=11−10=1.

Next Question →

Practice Set 13 · Question 130 of 150

Question 130. For F(x,y)=(11y,12x), find the planar scalar curl.

  1. 4
  2. 1
  3. 2
  4. 0
Show Answer and Explanation

Correct answer: B. 1

Scalar curl is ∂Q/∂x−∂P/∂y=12−11=1.

Next Question →

Practice Set 14 · Question 131 of 150

Question 131. Integrate the constant scalar field 2 along a curve of arc length 3.

  1. 5
  2. 9
  3. 6
  4. 7
Show Answer and Explanation

Correct answer: C. 6

The scalar line integral equals 2×length=6.

Next Question →

Practice Set 14 · Question 132 of 150

Question 132. Integrate the constant scalar field 3 along a curve of arc length 4.

  1. 13
  2. 11
  3. 15
  4. 12
Show Answer and Explanation

Correct answer: D. 12

The scalar line integral equals 3×length=12.

Next Question →

Practice Set 14 · Question 133 of 150

Question 133. Integrate the constant scalar field 4 along a curve of arc length 5.

  1. 20
  2. 21
  3. 19
  4. 23
Show Answer and Explanation

Correct answer: A. 20

The scalar line integral equals 4×length=20.

Next Question →

Practice Set 14 · Question 134 of 150

Question 134. Integrate the constant scalar field 5 along a curve of arc length 6.

  1. 33
  2. 30
  3. 31
  4. 29
Show Answer and Explanation

Correct answer: B. 30

The scalar line integral equals 5×length=30.

Next Question →

Practice Set 14 · Question 135 of 150

Question 135. Integrate the constant scalar field 6 along a curve of arc length 7.

  1. 41
  2. 45
  3. 42
  4. 43
Show Answer and Explanation

Correct answer: C. 42

The scalar line integral equals 6×length=42.

Next Question →

Practice Set 14 · Question 136 of 150

Question 136. Integrate the constant scalar field 7 along a curve of arc length 8.

  1. 57
  2. 55
  3. 59
  4. 56
Show Answer and Explanation

Correct answer: D. 56

The scalar line integral equals 7×length=56.

Next Question →

Practice Set 14 · Question 137 of 150

Question 137. Integrate the constant scalar field 8 along a curve of arc length 9.

  1. 72
  2. 73
  3. 71
  4. 75
Show Answer and Explanation

Correct answer: A. 72

The scalar line integral equals 8×length=72.

Next Question →

Practice Set 14 · Question 138 of 150

Question 138. Integrate the constant scalar field 9 along a curve of arc length 10.

  1. 93
  2. 90
  3. 91
  4. 89
Show Answer and Explanation

Correct answer: B. 90

The scalar line integral equals 9×length=90.

Next Question →

Practice Set 14 · Question 139 of 150

Question 139. Integrate the constant scalar field 10 along a curve of arc length 11.

  1. 109
  2. 113
  3. 110
  4. 111
Show Answer and Explanation

Correct answer: C. 110

The scalar line integral equals 10×length=110.

Next Question →

Practice Set 14 · Question 140 of 150

Question 140. Integrate the constant scalar field 11 along a curve of arc length 12.

  1. 133
  2. 131
  3. 135
  4. 132
Show Answer and Explanation

Correct answer: D. 132

The scalar line integral equals 11×length=132.

Next Question →

Practice Set 15 · Question 141 of 150

Question 141. For F=(2,0), find ∫ F·dr along the straight path from (0,0) to (3,0).

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

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

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Practice Set 15 · Question 142 of 150

Question 142. For F=(3,0), find ∫ F·dr along the straight path from (0,0) to (4,0).

  1. 15
  2. 12
  3. 13
  4. 11
Show Answer and Explanation

Correct answer: B. 12

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

Next Question →

Practice Set 15 · Question 143 of 150

Question 143. For F=(4,0), find ∫ F·dr along the straight path from (0,0) to (5,0).

  1. 19
  2. 23
  3. 20
  4. 21
Show Answer and Explanation

Correct answer: C. 20

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

Next Question →

Practice Set 15 · Question 144 of 150

Question 144. For F=(5,0), find ∫ F·dr along the straight path from (0,0) to (6,0).

  1. 31
  2. 29
  3. 33
  4. 30
Show Answer and Explanation

Correct answer: D. 30

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

Next Question →

Practice Set 15 · Question 145 of 150

Question 145. For F=(6,0), find ∫ F·dr along the straight path from (0,0) to (7,0).

  1. 42
  2. 43
  3. 41
  4. 45
Show Answer and Explanation

Correct answer: A. 42

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

Next Question →

Practice Set 15 · Question 146 of 150

Question 146. For F=(7,0), find ∫ F·dr along the straight path from (0,0) to (8,0).

  1. 59
  2. 56
  3. 57
  4. 55
Show Answer and Explanation

Correct answer: B. 56

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

Next Question →

Practice Set 15 · Question 147 of 150

Question 147. For F=(8,0), find ∫ F·dr along the straight path from (0,0) to (9,0).

  1. 71
  2. 75
  3. 72
  4. 73
Show Answer and Explanation

Correct answer: C. 72

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

Next Question →

Practice Set 15 · Question 148 of 150

Question 148. For F=(9,0), find ∫ F·dr along the straight path from (0,0) to (10,0).

  1. 91
  2. 89
  3. 93
  4. 90
Show Answer and Explanation

Correct answer: D. 90

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

Next Question →

Practice Set 15 · Question 149 of 150

Question 149. For F=(10,0), find ∫ F·dr along the straight path from (0,0) to (11,0).

  1. 110
  2. 111
  3. 109
  4. 113
Show Answer and Explanation

Correct answer: A. 110

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

Next Question →

Practice Set 15 · Question 150 of 150

Question 150. For F=(11,0), find ∫ F·dr along the straight path from (0,0) to (12,0).

  1. 135
  2. 132
  3. 133
  4. 131
Show Answer and Explanation

Correct answer: B. 132

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

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