Question 1

Describe in words the region of $\mathbb { R } ^ { 3 }$ represented by the inequality $1 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 25$ .

Accepted Answer

All points on and be

Question 2

Find the coordinates of the point(s) of intersection of the line x = 1 - t, y = 1 - t, z = 4t and the surface z = $x ^ { 2 } + 2 y ^ { 2 }$ .

Accepted Answer

The answer of Find the coordinates of the point(s) of...

Question 3

Find the range of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

Accepted Answer

A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , 0 )$ 
D)  $( - \infty , \infty )$ 
E)  $( 1 , \infty )$ 
F) $[ 1 , \infty )$ 
G) $( \sqrt { 2 } , \infty )$ 
H) $[ \sqrt { 2 } , \infty )$ 
A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , 0 )$ 
D)  $( - \infty , \infty )$ 
E)  $( 1 , \infty )$ 
F) $[ 1 , \infty )$ 
G) $( \sqrt { 2 } , \infty )$ 
H) $[ \sqrt { 2 } , \infty )$

Question 4

Find the distance from the point P(1, 1, 1) to the line passing through the points Q(2, 0, 1) and R(0, 3, 2).

Accepted Answer

The answer of Find the distance from the point P(1,...

Question 5

Identify the surface $x = y ^ { 2 } - z ^ { 2 }$ .

Accepted Answer

A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)hyperbolic paraboloid 
A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)hyperbolic paraboloid

Question 6

Find the distance between the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ and the sphere $( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4$

Accepted Answer

The closest distance between t

Question 7

Consider the points P = (1, 2, 3), Q = (2, -1, 0) , and R = (-1, 4, 1). Find the area of the triangle PQR.

Accepted Answer

The answer of Consider the points P = (1, 2,...

Question 8

Determine whether the lines L1: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L2: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Accepted Answer

Question 9

Find the range of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

Accepted Answer

A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , \infty )$ 
D)  $( - \infty , 0 )$ 
E)  $( 1 , \infty )$ 
F) $[ 1 , \infty )$ 
G) $( \sqrt { 2 } , \infty )$ 
H) $[ \sqrt { 2 } , \infty )$ 
A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , \infty )$ 
D)  $( - \infty , 0 )$ 
E)  $( 1 , \infty )$ 
F) $[ 1 , \infty )$ 
G) $( \sqrt { 2 } , \infty )$ 
H) $[ \sqrt { 2 } , \infty )$

Question 10

New Orleans is situated at latitude 30° N and longitude 90° W, and New York is situated at latitude 41° N and longitude 74° W. Find the distance from New Orleans to New York, assuming that the radius of the earth is 3960 miles.

Accepted Answer

The answer of New Orleans is situated at latitude 30°...

Question 11

Describe the surface whose equation in cylindrical coordinates is $\beta = 3$ .

Accepted Answer

A) Cylinder with vertical axis 
B) Cylinder with horizontal axis 
C) Sphere 
D) Vertical plane or half-plane 
E) Horizontal plane or half-plane
F)Paraboloid
G)Cone or half-cone with vertical axis
H)Cone or half-cone with horizontal axis 
A) Cylinder with vertical axis 
B) Cylinder with horizontal axis 
C) Sphere 
D) Vertical plane or half-plane 
E) Horizontal plane or half-plane
F)Paraboloid
G)Cone or half-cone with vertical axis
H)Cone or half-cone with horizontal axis

Question 12

Find $\mathbf { a } \cdot \mathbf { b }$ , given that $| \mathbf { a } |$ = 2, $| \mathbf { b } |$ = 3, and the angle between a and b is $\frac { 5 \pi } { 6 }$ .

Accepted Answer

A)  $3 \sqrt { 2 }$ 
B) 3 
C)  $3 \sqrt { 3 }$ 
D)  $2 \sqrt { 3 }$ 
E)  $- 3 \sqrt { 2 }$ 
F)-3

G) $- 3 \sqrt { 3 }$ 
H) $- 2 \sqrt { 3 }$ 
A)  $3 \sqrt { 2 }$ 
B) 3 
C)  $3 \sqrt { 3 }$ 
D)  $2 \sqrt { 3 }$ 
E)  $- 3 \sqrt { 2 }$ 
F)-3

G) $- 3 \sqrt { 3 }$ 
H) $- 2 \sqrt { 3 }$

Question 13

Identify the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ .

Accepted Answer

A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution) 
A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 14

Let $| \mathbf { a } | = 1$ and $| \mathbf { b } | = 2$ and the angle between a and b be 150°. Find
(i) $\mathbf { a } \cdot \mathbf { b }$ (ii) $\mathbf { b } \cdot ( 3 \mathbf { a } + \mathbf { b } )$ (iii) $( \mathbf { a } + \mathbf { b } ) \cdot ( \mathbf { a } - \mathbf { b } )$ (iv) $| 2 a - b | ^ { 2 }$

Accepted Answer

(i) @#IMG-DLM& , (ii

Question 15

Find the equation of the plane containing the points (5, 3, 1), (1, 8, 4), and (-1, 3, -2).

Accepted Answer

The answer of Find the equation of the plane containing...

Question 16

Let A, B, and C be three points on the unit sphere centered at the origin whose spherical coordinates are A = (1, 0, 0), B = $\left( 1,0 , \frac { \pi } { 4 } \right)$ , and C = $\left( 1 , \frac { \pi } { 2 } , \frac { \pi } { 3 } \right)$ .(a) Find the angle between $\overrightarrow { O A }$ and $\overrightarrow { O B }$ .(b) Find the angle between $\overrightarrow { O A }$ and $\overrightarrow { O C }$ .(c) Find the angle between $\overrightarrow { O B }$ and $\overrightarrow { O C }$ .

Accepted Answer

(a) 45°
(b

Question 17

Find the intersections of the line passing through the points (-1, 3, 4) and (3, 5, 2) with the yz-plane, the xz-plane, and the xy-plane.

Accepted Answer

(0, 3.5, 3

Question 18

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find parametric equations for the line through P which lies in the plane and is perpendicular to L.

Accepted Answer

x = 1 + 3t

Question 19

Given points A = $\left( \sqrt { 5 } , \frac { \pi } { 6 } , \pi \right)$ and B = $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$ in spherical coordinates, find the distance between the two points.

Accepted Answer

A = @#IMG-DLM& and B = @#IMG-DLM& in rec

Question 20

Find a parametric equation of the line which is the intersection of the planes - x + 3y + z = 7 and x + y = 1.

Accepted Answer

@#IMG-DLM& . (Other

Describe in words the region of $\mathbb { R } ^ { 3 }$ represented by the inequality $1 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 25$ .

Find the coordinates of the point(s) of intersection of the line x = 1 - t, y = 1 - t, z = 4t and the surface z = $x ^ { 2 } + 2 y ^ { 2 }$ .

Find the range of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

Find the distance from the point P(1, 1, 1) to the line passing through the points Q(2, 0, 1) and R(0, 3, 2).

Identify the surface $x = y ^ { 2 } - z ^ { 2 }$ .

Find the distance between the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ and the sphere $( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4$

Consider the points P = (1, 2, 3), Q = (2, -1, 0) , and R = (-1, 4, 1). Find the area of the triangle PQR.

Determine whether the lines L₁: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L₂: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Find the range of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

New Orleans is situated at latitude 30° N and longitude 90° W, and New York is situated at latitude 41° N and longitude 74° W. Find the distance from New Orleans to New York, assuming that the radius of the earth is 3960 miles.

Describe the surface whose equation in cylindrical coordinates is $\beta = 3$ .

Find $\mathbf { a } \cdot \mathbf { b }$ , given that $| \mathbf { a } |$ = 2, $| \mathbf { b } |$ = 3, and the angle between a and b is $\frac { 5 \pi } { 6 }$ .

Identify the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ .

Find the equation of the plane containing the points (5, 3, 1), (1, 8, 4), and (-1, 3, -2).

Find the intersections of the line passing through the points (-1, 3, 4) and (3, 5, 2) with the yz-plane, the xz-plane, and the xy-plane.

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find parametric equations for the line through P which lies in the plane and is perpendicular to L.

Given points A = $\left( \sqrt { 5 } , \frac { \pi } { 6 } , \pi \right)$ and B = $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$ in spherical coordinates, find the distance between the two points.

Find a parametric equation of the line which is the intersection of the planes - x + 3y + z = 7 and x + y = 1.

Functions and Models

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Exam 9: Vectors and the Geometry of Space

Describe in words the region of R3\mathbb { R } ^ { 3 }R3 represented by the inequality 1≤x2+y2+z2≤251 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 251≤x2+y2+z2≤25 .

Find the coordinates of the point(s) of intersection of the line x = 1 - t, y = 1 - t, z = 4t and the surface z = x2+2y2x ^ { 2 } + 2 y ^ { 2 }x2+2y2 .

Find the range of the function f(x, y) = x−y2\sqrt { x - y ^ { 2 } }x−y2​ .

Find the distance from the point P(1, 1, 1) to the line passing through the points Q(2, 0, 1) and R(0, 3, 2).

Identify the surface x=y2−z2x = y ^ { 2 } - z ^ { 2 }x=y2−z2 .

Find the distance between the sphere x2+y2+z2=36x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36x2+y2+z2=36 and the sphere (x−2)2+(y+1)2+(z−2)2=4( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4(x−2)2+(y+1)2+(z−2)2=4

Consider the points P = (1, 2, 3), Q = (2, -1, 0) , and R = (-1, 4, 1). Find the area of the triangle PQR.

Determine whether the lines L1: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L2: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Find the range of the function f(x, y) = ln⁡(x−y2)\ln \left( x - y ^ { 2 } \right)ln(x−y2) .

New Orleans is situated at latitude 30° N and longitude 90° W, and New York is situated at latitude 41° N and longitude 74° W. Find the distance from New Orleans to New York, assuming that the radius of the earth is 3960 miles.

Describe the surface whose equation in cylindrical coordinates is β=3\beta = 3β=3 .

Find a⋅b\mathbf { a } \cdot \mathbf { b }a⋅b , given that ∣a∣| \mathbf { a } |∣a∣ = 2, ∣b∣| \mathbf { b } |∣b∣ = 3, and the angle between a and b is 5π6\frac { 5 \pi } { 6 }65π​ .

Identify the surface x2=y2+z2x ^ { 2 } = y ^ { 2 } + z ^ { 2 }x2=y2+z2 .

Find the equation of the plane containing the points (5, 3, 1), (1, 8, 4), and (-1, 3, -2).

Find the intersections of the line passing through the points (-1, 3, 4) and (3, 5, 2) with the yz-plane, the xz-plane, and the xy-plane.

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find parametric equations for the line through P which lies in the plane and is perpendicular to L.

Given points A = (5,π6,π)\left( \sqrt { 5 } , \frac { \pi } { 6 } , \pi \right)(5​,6π​,π) and B = (2,π4,π2)\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)(2,4π​,2π​) in spherical coordinates, find the distance between the two points.

Find a parametric equation of the line which is the intersection of the planes - x + 3y + z = 7 and x + y = 1.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Describe in words the region of $\mathbb { R } ^ { 3 }$ represented by the inequality $1 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 25$ .

Find the coordinates of the point(s) of intersection of the line x = 1 - t, y = 1 - t, z = 4t and the surface z = $x ^ { 2 } + 2 y ^ { 2 }$ .

Find the range of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

Identify the surface $x = y ^ { 2 } - z ^ { 2 }$ .

Find the distance between the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ and the sphere $( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4$

Determine whether the lines L₁: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L₂: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Find the range of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

Describe the surface whose equation in cylindrical coordinates is $\beta = 3$ .

Find $\mathbf { a } \cdot \mathbf { b }$ , given that $| \mathbf { a } |$ = 2, $| \mathbf { b } |$ = 3, and the angle between a and b is $\frac { 5 \pi } { 6 }$ .

Identify the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ .

Given points A = $\left( \sqrt { 5 } , \frac { \pi } { 6 } , \pi \right)$ and B = $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$ in spherical coordinates, find the distance between the two points.