Calculus 3 Practice Test

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34 questions. Calculator allowed. Show your working. steplyclass.com/calculus-3/practice-test/

1. 1

   Find the distance (1, 2, 3) (4, 6, 15).
   1. A) d = 4
   2. B) d = 169
   3. C) d = 3
   4. D) d = 13

2. 2

   Find the distance (2, 0, −3) (4, 3, 1).

   Answer: ____________________

3. 3

   Vector from (1, 2, 3) to (4, 6, 3)

   Answer: ____________________

4. 4

   Vector from (1, 1, 1) to (2, 3, 4)

   Answer: ____________________

5. 5

   Calculate <1, 2, 3> • <4, −5, 6>

   Answer: ____________________

6. 6

   Angle between <1, 1, 1> and <1, −1, 2>

   Answer: ____________________

7. 7

   Calculate <1, 2, 3> × <4, 5, 6>
   1. A) (3, −6, 3)
   2. B) (−3, 6, −3)
   3. C) (−3, −6, −3)
   4. D) (3, 6, −3)

8. 8

   Calculate <1, 2, −3> × <2, 4, 1>

   Answer: ____________________

9. 9

   Line through (1, 2, 3) and (4, 6, 8)

   Answer: ____________________

10. 10

    Symmetric equations of the line through (−2, 0, 1) parallel to <1, 2, −3>

    Answer: ____________________

11. 11

    Plane through (1, 2, 3) with normal <2, −1, 4>
    1. A) 0
    2. B) 12
    3. C) −12

12. 12

    Plane through (1, 0, 2), (3, 1, 0) and (0, 2, 1)

    Answer: ____________________

13. 13

    Partial derivatives of f(x, y) = x² y + 3y

    Answer: ____________________

14. 14

    Second partial derivatives of f(x, y) = 4x³ − 2xy + y⁴

    Answer: ____________________

15. 15

    Tangent plane to z = x² + y² at (1, 2). Find z.

    Answer: ____________________

16. 16

    Linear approximation of f(x, y) = x³ − y² at (2, 1) to estimate f(1.9, 1.1)

    Answer: ____________________

17. 17

    z = x² y, x = t², y = 3t, find dz/dt

    Answer: ____________________

18. 18

    z = x² + y³, x = 3t², y = 2t, find dz/dt

    Answer: ____________________

19. 19

    Find the gradient of f(x, y) = x² y at (1, 2).

    Answer: ____________________

20. 20

    Maximum rate of change of f(x, y) = 3x + 4y at (1, 1)

    Answer: ____________________

21. 21

    Local max and min of f(x, y) = x² + y² − 4x + 6y

    Answer: ____________________

22. 22

    Classify the critical points of f(x, y) = x² − 2xy + 3y² − 4x

    Answer: ____________________

23. 23

    Double integral of xy over 0 ≤ x ≤ 2, 0 ≤ y ≤ 1

    Answer: ____________________

24. 24

    Double integral of 3x² y² over −1 ≤ x ≤ 1, 0 ≤ y ≤ 2

    Answer: ____________________

25. 25

    Double integral of x over the region between y = x² and y = x
    1. A) 0
    2. B) 1
    3. C) 112

26. 26

    Double integral of 4xy over 0 ≤ x ≤ 1, x³ ≤ y ≤ x

    Answer: ____________________

27. 27

    Double integral of x² + y² over the disk x² + y² ≤ 4

    Answer: ____________________

28. 28

    Double integral of x² + y² over the upper half of the ring 1 ≤ x² + y² ≤ 9

    Answer: ____________________

29. 29

    Triple integral of xyz over 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3

    Answer: ____________________

30. 30

    Find the volume of the solid under the plane 3x + 2y + z = 6 in the first octant.

    Answer: ____________________

31. 31

    Line integral of F = <y, x> along the segment from (0, 0) to (1, 2)

    Answer: ____________________

32. 32

    Line integral of F = <x − y, x + y> along the segment from (1, 1) to (3, 2)

    Answer: ____________________

33. 33

    Use Green's theorem on ∮ (x² − y) dx + (x + y²) dy around the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
    1. A) −2
    2. B) 1
    3. C) 2
    4. D) −1

34. 34

    Use Green's theorem on ∮ (y² − x) dx + (2xy + x²) dy around the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2

    Answer: ____________________