Calculus 3 Practice Test
34 questions from 17 lessons · 105 minutes · percent score and grade · step-by-step answers
How it works
- 34 questions, about 105 minutes (a timer counts down; you can hide it). Solve it right here: no sign-up.
- 5 questions have choices. For the others you type the answer: a number like 4, −2, 7/2 or 3.5, or for example x = 4, x = 2 or x = −3, (3, 5), 2x + 1. Use ^ for powers (x^2) and sqrt for √. Any form with the same value counts (7/2 = 3.5).
- One question at a time: Back, Next, Mark for review and the question list at the bottom. A calculator is on the screen.
- At the end: your percent and grade, every question right or wrong, why a wrong choice is wrong, and its step-by-step solution. Then practise these lessons next: the lessons of the questions you got wrong, with new questions.
Your answers and results are kept in this browser only, so you can close the page and come back.
Formulas: the Calculus 3 formula sheet.
1 Question 1
Find the distance (1, 2, 3) (4, 6, 15).- A) d = 4
- B) d = 169
- C) d = 3
- D) d = 13
2 Question 2
Find the distance (2, 0, −3) (4, 3, 1).Type the answer.
3 Question 3
Vector from (1, 2, 3) to (4, 6, 3)Type the answer.
4 Question 4
Vector from (1, 1, 1) to (2, 3, 4)Type the answer.
5 Question 5
Calculate <1, 2, 3> • <4, −5, 6>Type the answer.
6 Question 6
Angle between <1, 1, 1> and <1, −1, 2>Type the answer.
7 Question 7
Calculate <1, 2, 3> × <4, 5, 6>- A) (3, −6, 3)
- B) (−3, 6, −3)
- C) (−3, −6, −3)
- D) (3, 6, −3)
8 Question 8
Calculate <1, 2, −3> × <2, 4, 1>Type the answer.
9 Question 9
Line through (1, 2, 3) and (4, 6, 8)Type the answer.
10 Question 10
Symmetric equations of the line through (−2, 0, 1) parallel to <1, 2, −3>Type the answer.
11 Question 11
Plane through (1, 2, 3) with normal <2, −1, 4>- A) 0
- B) 12
- C) −12
12 Question 12
Plane through (1, 0, 2), (3, 1, 0) and (0, 2, 1)Type the answer.
13 Question 13
Partial derivatives of f(x, y) = x2 y + 3yType the answer.
14 Question 14
Second partial derivatives of f(x, y) = 4x3 − 2xy + y4Type the answer.
15 Question 15
Tangent plane to z = x2 + y2 at (1, 2). Find z.Type the answer.
16 Question 16
Linear approximation of f(x, y) = x3 − y2 at (2, 1) to estimate f(1.9, 1.1)Type the answer.
17 Question 17
z = x2 y, x = t2, y = 3t, find dz/dtType the answer.
18 Question 18
z = x2 + y3, x = 3t2, y = 2t, find dz/dtType the answer.
19 Question 19
Find the gradient of f(x, y) = x2 y at (1, 2).Type the answer.
20 Question 20
Maximum rate of change of f(x, y) = 3x + 4y at (1, 1)Type the answer.
21 Question 21
Local max and min of f(x, y) = x2 + y2 − 4x + 6yType the answer.
22 Question 22
Classify the critical points of f(x, y) = x2 − 2xy + 3y2 − 4xType the answer.
23 Question 23
Double integral of xy over 0 ≤ x ≤ 2, 0 ≤ y ≤ 1Type the answer.
24 Question 24
Double integral of 3x2 y2 over −1 ≤ x ≤ 1, 0 ≤ y ≤ 2Type the answer.
25 Question 25
Double integral of x over the region between y = x2 and y = x- A) 0
- B) 1
- C) 112
26 Question 26
Double integral of 4xy over 0 ≤ x ≤ 1, x3 ≤ y ≤ xType the answer.
27 Question 27
Double integral of x2 + y2 over the disk x2 + y2 ≤ 4Type the answer.
28 Question 28
Double integral of x2 + y2 over the upper half of the ring 1 ≤ x2 + y2 ≤ 9Type the answer.
29 Question 29
Triple integral of xyz over 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3Type the answer.
30 Question 30
Find the volume of the solid under the plane 3x + 2y + z = 6 in the first octant.Type the answer.
31 Question 31
Line integral of F = <y, x> along the segment from (0, 0) to (1, 2)Type the answer.
32 Question 32
Line integral of F = <x − y, x + y> along the segment from (1, 1) to (3, 2)Type the answer.
33 Question 33
Use Green's theorem on ∮ (x2 − y) dx + (x + y2) dy around the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1- A) −2
- B) 1
- C) 2
- D) −1
34 Question 34
Use Green's theorem on ∮ (y2 − x) dx + (2xy + x2) dy around the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2Type the answer.