Question 1

Find the absolute maximum and minimum values of f(x, y) = 4(x - x2) sin($\pi$y) on the rectangle 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 2 and the points where they are assumed.

Accepted Answer

A)  maximum 1 at   , minimum 0 at   
B)  maximum 2 at   , minimum -2 at   
C)  maximum 1 at   , minimum -1 at   
D)  maximum 1 at   , minimum 0 at (0, 0), (1, 0), (0, 2), and (1, 2) 
E)  maximum 1 at   , minimum 0 at   
A)  maximum 1 at   , minimum 0 at   
B)  maximum 2 at   , minimum -2 at   
C)  maximum 1 at   , minimum -1 at   
D)  maximum 1 at   , minimum 0 at (0, 0), (1, 0), (0, 2), and (1, 2) 
E)  maximum 1 at   , minimum 0 at

Question 2

Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy2z3 on the sphere x2 + y2 + z2 = 6.

Accepted Answer

A)  maximum 6, minimum -6 
B)  maximum 6   , minimum -6   
C)  maximum 6   , minimum -6   
D)  maximum 12, minimum -12 
E)  maximum 12   , minimum -12   
A)  maximum 6, minimum -6 
B)  maximum 6   , minimum -6   
C)  maximum 6   , minimum -6   
D)  maximum 12, minimum -12 
E)  maximum 12   , minimum -12

Question 3

Find the maximum and minimum distances from the origin to the ellipse 5x2 + 6xy + 5y2 - 8 = 0.

Accepted Answer

A)  maximum 2, minimum 1 
B)  maximum 4, minimum 1 
C)  maximum 2, minimum   
D)  maximum 4, minimum   
E)  maximum 4, minimum 2 
A)  maximum 2, minimum 1 
B)  maximum 4, minimum 1 
C)  maximum 2, minimum   
D)  maximum 4, minimum   
E)  maximum 4, minimum 2

Question 4

The extreme values of the function f(x , y, z) = 23 x + y2z subject to the constraintsx - z = 0 and y2 + z2 = 36 are given by:

Accepted Answer

A)  115, 128 
B)  - 115, 115 
C)  -128, 115 
D)  -128, 128 
E)  -128, -115 
A)  115, 128 
B)  - 115, 115 
C)  -128, 115 
D)  -128, 128 
E)  -128, -115

Question 5

Let  pi > 0, i = 1, 2, 3,..., n be real numbers such that    
Find the maximum value of   subject to the constraint

Accepted Answer

The answer of Let  pi > 0, i =...

Question 6

Find the point on the surface z = x2 + y2 closest to the point (1, 1, 0).

Accepted Answer

A)  (1, 1, 1) 
B)    
C)    
D)    
E)    
A)  (1, 1, 1) 
B)    
C)    
D)    
E)

Question 7

Suppose that a function f(x,y) has a critical point  (a, b) at an interior point in its domain and that f has continuous second order partials in a neighbourhood of (a, b).
If      , then f has no local extremum at (a, b).

Accepted Answer

A) True 
 B)False

Question 8

Find the point on the sphere x2 + y2 + z2 = 10 that is closest to the point (1, -8, 5).

Accepted Answer

A)    
B)    
C)  (0,   , 0) 
D)    
E)  (2, -16, 10) 
A)    
B)    
C)  (0,   , 0) 
D)    
E)  (2, -16, 10)

Question 9

Find the absolute maximum and minimum values of the linear function f(x, y) = -2x + y - 10 on the polygon 0 $\le$ x $\le$2, 0 $\le$ y $\le$ 2, y - x $\le$ 1.

Accepted Answer

A)  maximum -9, minimum -14 
B)  maximum -9, minimum -12 
C)  maximum -8, minimum -12 
D)  maximum -8, minimum -15 
E)  maximum -9, minimum -10 
A)  maximum -9, minimum -14 
B)  maximum -9, minimum -12 
C)  maximum -8, minimum -12 
D)  maximum -8, minimum -15 
E)  maximum -9, minimum -10

Question 10

Use Maple's fsolve routine to solve the non-linear system of equations   Quote the solution to 5 significant figures.

Accepted Answer

(x,y,z) = (0.120759,

Question 11

Let f(x, y, z) = x2 + 2y2 + 4z2. Find the point on the plane x + y + z = 14 at which f has its smallest value.

Accepted Answer

A)  (8, 4, 2) 
B)  (0, 0, 14) 
C)  (1, 2, 11) 
D)    
E)  (2, 4, 8) 
A)  (8, 4, 2) 
B)  (0, 0, 14) 
C)  (1, 2, 11) 
D)    
E)  (2, 4, 8)

Question 12

Find and classify the critical points of the Lagrange function L(   ,   ,...,   , λ) corresponding to the problem:extremize   subject to =   .

Accepted Answer

Critical point at ( @#IMG-DLM& , @#IMG-DLM& ,...,

Question 13

If a function f(x,y) has a local or absolute extreme value at the point (x0, y0) in its domain, then (x0, y0) must be either a critical point of f, a singular point of f, or a boundary point of the domain of f.

Accepted Answer

A) True 
 B)False

Question 14

Find and classify all critical points for the function f(x, y) = x3 - 12xy2 + y3 + 45y.

Accepted Answer

A)  (2, 1) and (-2, -1) are saddle points 
B)  (2, 1), (2, -1), (-2, 1), and (-2, -1) are saddle points 
C)  (2, 1) is a local maximum and (-2, -1) is a local minimum 
D)  (2, -1) and (-2, 1) are saddle points 
E)  There are no critical points. 
A)  (2, 1) and (-2, -1) are saddle points 
B)  (2, 1), (2, -1), (-2, 1), and (-2, -1) are saddle points 
C)  (2, 1) is a local maximum and (-2, -1) is a local minimum 
D)  (2, -1) and (-2, 1) are saddle points 
E)  There are no critical points.

Question 15

Find the absolute maximum and minimum of f(x, y) = 4x2 + 2xy - 3y2 on the unit square0 $\le$ x $\le$ 1, 0 $\le$y$\le$ 1.

Accepted Answer

A)  maximum   , minimum -3 
B)  maximum 4, minimum -3 
C)  maximum 4, minimum 0 
D)  maximum   , minimum -4 
E)  maximum   , minimum 0 
A)  maximum   , minimum -3 
B)  maximum 4, minimum -3 
C)  maximum 4, minimum 0 
D)  maximum   , minimum -4 
E)  maximum   , minimum 0

Question 16

Find and classify all critical points of f(x,y,z) = x3 + xz2 + 3x2 + y2 + 2z2 - 9x - 2y -10.

Accepted Answer

A)  local minimum at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) and saddle point at (1, 1, 0) 
B)  local minimum at (-2, 1, 3), (-2, 1, -3) and local maximum at (-3, 1, 0), (1, 1, 0) 
C)  local minimum at (1, 1, 0 ) and saddle point at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) 
D)  local maximum at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) and saddle point at (1, 1, 0) 
E)  local minimum at (-2, 1, 3) , (-3, 1, 0), local maximum at (1, 1, 0), and saddle point at (-2, 1, -3) 
A)  local minimum at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) and saddle point at (1, 1, 0) 
B)  local minimum at (-2, 1, 3), (-2, 1, -3) and local maximum at (-3, 1, 0), (1, 1, 0) 
C)  local minimum at (1, 1, 0 ) and saddle point at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) 
D)  local maximum at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) and saddle point at (1, 1, 0) 
E)  local minimum at (-2, 1, 3) , (-3, 1, 0), local maximum at (1, 1, 0), and saddle point at (-2, 1, -3)

Question 17

Find and classify the critical points of the following function: f(x, y) =   + 30x3 - 15y3.

Accepted Answer

A)  saddle points are (   , -   ) and (-   ,   ), maximum at (3, 3) and minimum at (-3, -3) 
B)  saddle points are (0, 0), (   , -   ) and (-   ,   ), minimum at (3, 3) and maximum at (-3, -3) 
C)  saddle points are (0, 0), (   , -   ), (-   ,   ), (3, 3), and (-3, -3) 
D)  saddle points are (0, 0), (   , -   ) and (-   ,   ), maximum at (3, 3) and minimum at (-3, -3) 
E)  saddle points are (0, 0), (   , -   ) and (-   ,   ), maximum at (3, -3) and minimum at (-3, 3) 
A)  saddle points are (   , -   ) and (-   ,   ), maximum at (3, 3) and minimum at (-3, -3) 
B)  saddle points are (0, 0), (   , -   ) and (-   ,   ), minimum at (3, 3) and maximum at (-3, -3) 
C)  saddle points are (0, 0), (   , -   ), (-   ,   ), (3, 3), and (-3, -3) 
D)  saddle points are (0, 0), (   , -   ) and (-   ,   ), maximum at (3, 3) and minimum at (-3, -3) 
E)  saddle points are (0, 0), (   , -   ) and (-   ,   ), maximum at (3, -3) and minimum at (-3, 3)

Question 18

Find the derivative of the function f(x) =   dt.

Accepted Answer

A)  6x   
B)  4x   
C)  2x   
D)  2x   
E)  2x   
A)  6x   
B)  4x   
C)  2x   
D)  2x   
E)  2x

Question 19

If the Lagrange function L corresponding to the problem of extremizing f(x, y, z) subject to the constraint g(x, y, z) = 0 has exactly two critical points, then f must attain its maximum value at one of the points and attain its minimum value at the other point.

Accepted Answer

A) True 
 B)False

Question 20

By first differentiating the integral, evaluate   dy for x > -1.

Accepted Answer

A)  ln (2x + 1) 
B)  ln (|x|) + 1 
C)  ln (x - 1) 
D)  ln (x + 1) 
E)  ln (x) + 1 
A)  ln (2x + 1) 
B)  ln (|x|) + 1 
C)  ln (x - 1) 
D)  ln (x + 1) 
E)  ln (x) + 1

Find the absolute maximum and minimum values of f(x, y) = 4(x - x²) sin( $\pi$ y) on the rectangle 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 2 and the points where they are assumed.

Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy²z³ on the sphere x² + y² + z² = 6.

Find the maximum and minimum distances from the origin to the ellipse 5x² + 6xy + 5y² - 8 = 0.

The extreme values of the function f(x , y, z) = 23 x + y²z subject to the constraintsx - z = 0 and y² + z² = 36 are given by:

Let pi > 0, i = 1, 2, 3,..., n be real numbers such that Find the maximum value of subject to the constraint

Find the point on the surface z = x² + y² closest to the point (1, 1, 0).

Suppose that a function f(x,y) has a critical point (a, b) at an interior point in its domain and that f has continuous second order partials in a neighbourhood of (a, b). If , then f has no local extremum at (a, b).

Find the point on the sphere x²+ y²+ z² = 10 that is closest to the point (1, -8, 5).

Find the absolute maximum and minimum values of the linear function f(x, y) = -2x + y - 10 on the polygon 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 2, y - x $\le$ 1.

Use Maple's fsolve routine to solve the non-linear system of equations Quote the solution to 5 significant figures.

Let f(x, y, z) = x² + 2y² + 4z². Find the point on the plane x + y + z = 14 at which f has its smallest value.

Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = .

If a function f(x,y) has a local or absolute extreme value at the point (x₀, y₀) in its domain, then (x₀, y₀) must be either a critical point of f, a singular point of f, or a boundary point of the domain of f.

Find and classify all critical points for the function f(x, y) = x³ - 12xy² + y³ + 45y.

Find the absolute maximum and minimum of f(x, y) = 4x² + 2xy - 3y² on the unit square0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1.

Find and classify all critical points of f(x,y,z) = x³ + xz² + 3x² + y² + 2z² - 9x - 2y -10.

Find and classify the critical points of the following function: f(x, y) = + 30x³ - 15y³.

Find the derivative of the function f(x) = dt.

If the Lagrange function L corresponding to the problem of extremizing f(x, y, z) subject to the constraint g(x, y, z) = 0 has exactly two critical points, then f must attain its maximum value at one of the points and attain its minimum value at the other point.

By first differentiating the integral, evaluate dy for x > -1.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 14: Applications of Partial Derivatives

Find the absolute maximum and minimum values of f(x, y) = 4(x - x2) sin( π\piπ y) on the rectangle 0 ≤\le≤ x ≤\le≤ 1, 0 ≤\le≤ y ≤\le≤ 2 and the points where they are assumed.

Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy2z3 on the sphere x2 + y2 + z2 = 6.

Find the maximum and minimum distances from the origin to the ellipse 5x2 + 6xy + 5y2 - 8 = 0.

The extreme values of the function f(x , y, z) = 23 x + y2z subject to the constraintsx - z = 0 and y2 + z2 = 36 are given by:

Let pi > 0, i = 1, 2, 3,..., n be real numbers such that Find the maximum value of subject to the constraint

Find the point on the surface z = x2 + y2 closest to the point (1, 1, 0).

Suppose that a function f(x,y) has a critical point (a, b) at an interior point in its domain and that f has continuous second order partials in a neighbourhood of (a, b). If , then f has no local extremum at (a, b).

Find the point on the sphere x2 + y2 + z2 = 10 that is closest to the point (1, -8, 5).

Find the absolute maximum and minimum values of the linear function f(x, y) = -2x + y - 10 on the polygon 0 ≤\le≤ x ≤\le≤ 2, 0 ≤\le≤ y ≤\le≤ 2, y - x ≤\le≤ 1.

Use Maple's fsolve routine to solve the non-linear system of equations Quote the solution to 5 significant figures.

Let f(x, y, z) = x2 + 2y2 + 4z2. Find the point on the plane x + y + z = 14 at which f has its smallest value.

Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = .

If a function f(x,y) has a local or absolute extreme value at the point (x0, y0) in its domain, then (x0, y0) must be either a critical point of f, a singular point of f, or a boundary point of the domain of f.

Find and classify all critical points for the function f(x, y) = x3 - 12xy2 + y3 + 45y.

Find the absolute maximum and minimum of f(x, y) = 4x2 + 2xy - 3y2 on the unit square0 ≤\le≤ x ≤\le≤ 1, 0 ≤\le≤ y ≤\le≤ 1.

Find and classify all critical points of f(x,y,z) = x3 + xz2 + 3x2 + y2 + 2z2 - 9x - 2y -10.

Find and classify the critical points of the following function: f(x, y) = + 30x3 - 15y3.

Find the derivative of the function f(x) = dt.

If the Lagrange function L corresponding to the problem of extremizing f(x, y, z) subject to the constraint g(x, y, z) = 0 has exactly two critical points, then f must attain its maximum value at one of the points and attain its minimum value at the other point.

By first differentiating the integral, evaluate dy for x > -1.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Find the absolute maximum and minimum values of f(x, y) = 4(x - x²) sin( $\pi$ y) on the rectangle 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 2 and the points where they are assumed.

Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy²z³ on the sphere x² + y² + z² = 6.

Find the maximum and minimum distances from the origin to the ellipse 5x² + 6xy + 5y² - 8 = 0.

The extreme values of the function f(x , y, z) = 23 x + y²z subject to the constraintsx - z = 0 and y² + z² = 36 are given by:

Find the point on the surface z = x² + y² closest to the point (1, 1, 0).

Find the point on the sphere x²+ y²+ z² = 10 that is closest to the point (1, -8, 5).

Find the absolute maximum and minimum values of the linear function f(x, y) = -2x + y - 10 on the polygon 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 2, y - x $\le$ 1.

Let f(x, y, z) = x² + 2y² + 4z². Find the point on the plane x + y + z = 14 at which f has its smallest value.

If a function f(x,y) has a local or absolute extreme value at the point (x₀, y₀) in its domain, then (x₀, y₀) must be either a critical point of f, a singular point of f, or a boundary point of the domain of f.

Find and classify all critical points for the function f(x, y) = x³ - 12xy² + y³ + 45y.

Find the absolute maximum and minimum of f(x, y) = 4x² + 2xy - 3y² on the unit square0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1.

Find and classify all critical points of f(x,y,z) = x³ + xz² + 3x² + y² + 2z² - 9x - 2y -10.

Find and classify the critical points of the following function: f(x, y) = + 30x³ - 15y³.