Question 1

Use the chain rule to find the values of   and   . at (u, v) = (0, 1), where z = x3y5,x = u - v, and y = u + v.

Accepted Answer

A)    = -2;   = -8 
B)    = 8;   = 2 
C)    = -2;   = 8 
D)    = -8;   = -2 
E)    = 2;   = 8 
A)    = -2;   = -8 
B)    = 8;   = 2 
C)    = -2;   = 8 
D)    = -8;   = -2 
E)    = 2;   = 8 A

Question 2

Evaluate the limit .

Accepted Answer

A)  1 
B)  0 
C)    
D)  -1 
E)  The limit does not exist. 
A)  1 
B)  0 
C)    
D)  -1 
E)  The limit does not exist. B

Question 3

Find the Maclaurin series of the function exy-3y by using Taylor series for functions of one variable.

Accepted Answer

A)          
B)          
C)          
D)          
E)          
A)          
B)          
C)          
D)          
E)          A

Question 4

Compute   and   if f(x, y) = xy + sin xy + x2 + 5xln y.

Accepted Answer

A)    =   sin xy + 2,   = -   sin xy -   
B)    = -   sin xy + 2,   = -   sin xy -   
C)    = -   sin xy + 2,   =   sin xy -   
D)    = -   sin xy + 2,   = -   sin xy +   
E)    = -   sin xy + 2,   = -   sin xy -   
A)    =   sin xy + 2,   = -   sin xy -   
B)    = -   sin xy + 2,   = -   sin xy -   
C)    = -   sin xy + 2,   =   sin xy -   
D)    = -   sin xy + 2,   = -   sin xy +   
E)    = -   sin xy + 2,   = -   sin xy -

Question 5

Which function f(x,y) has the level curves corresponding to c = -1, 0, and 1 shown in the figure below?

Accepted Answer

A)  f(x,y) =   
B)  f(x,y) =   
C)  f(x,y) =   
D)  f(x,y) =   
E)  f(x,y) =   
A)  f(x,y) =   
B)  f(x,y) =   
C)  f(x,y) =   
D)  f(x,y) =   
E)  f(x,y) =

Question 6

The equations x = u2 - v2 and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate   at that point.

Accepted Answer

A)  -   
B)    
C)    
D)  -   
E)    
A)  -   
B)    
C)    
D)  -   
E)

Question 7

The pressure P, the volume V, and the temperature T (in Kelvin) of a confined gas are related by the ideal gas law P V = kT , where k is a constant. If P = 0.5 pascal when V = 50 cm3 and T = 360 K, determine by approximately what percentage P changes if V and T change to52 cm3 and 351 K, respectively.

Accepted Answer

A)  6.5 % 
B)  1.5 % 
C)  -4.5 % 
D)  -6.5 % 
E)  -1.5 % 
A)  6.5 % 
B)  1.5 % 
C)  -4.5 % 
D)  -6.5 % 
E)  -1.5 %

Question 8

Evaluate the limit .

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  -1 
E)  The limit does not exist. 
A)  0 
B)  1 
C)  2 
D)  -1 
E)  The limit does not exist.

Question 9

Find   (x, y) and   (x, y) if f(x, y) =   .

Accepted Answer

A)    (x, y) =     ;   (x, y) =     
B)    (x, y) =     ;   (x, y) =     
C)    (x, y) =     ;   (x, y) =     
D)    (x, y) =     ;   (x, y) =     
E)    (x, y) =     ;   (x, y) =     
A)    (x, y) =     ;   (x, y) =     
B)    (x, y) =     ;   (x, y) =     
C)    (x, y) =     ;   (x, y) =     
D)    (x, y) =     ;   (x, y) =     
E)    (x, y) =     ;   (x, y) =

Question 10

Calculate and simplify ut - uxx - uyy for the function u =   .

Accepted Answer

A)  -   
B)    
C)    
D)  0 
E)  -   
A)  -   
B)    
C)    
D)  0 
E)  -

Question 11

The temperature in the xy-plane is a function T(x, y). At the point (a,b) the directional derivative of T in the direction of the vector 3 i + 4 j is 4 and the directional derivative in the direction of 5 i - 12 j is -2. Find  T(a,b).

Accepted Answer

A)    (68 i + 89 j) 
B)    (43 i - 17 j) 
C)    (-53 i + 117 j) 
D)    (28 i + 37 j) 
E)  -   (172 i + 89 j) 
A)    (68 i + 89 j) 
B)    (43 i - 17 j) 
C)    (-53 i + 117 j) 
D)    (28 i + 37 j) 
E)  -   (172 i + 89 j)

Question 12

Find f11(x, y) and f22(x, y) if f(x, y) = ex sin y + 2xy + y.

Accepted Answer

A)  f11(x, y) =   sin y + 2,   (x, y) =   sin y 
B)  f11(x, y) =   cos y, f22(x, y) =   sin y 
C)  f11(x, y) =   sin y, f22(x, y) =   cos y 
D)  f11(x, y) =   sin y, f22(x, y) =   sin y 
E)  f11(x, y) =   cos y, f22(x, y) =   cos y 
A)  f11(x, y) =   sin y + 2,   (x, y) =   sin y 
B)  f11(x, y) =   cos y, f22(x, y) =   sin y 
C)  f11(x, y) =   sin y, f22(x, y) =   cos y 
D)  f11(x, y) =   sin y, f22(x, y) =   sin y 
E)  f11(x, y) =   cos y, f22(x, y) =   cos y

Question 13

Use the linearization of f(x,y) = x3 e-y about the point (2, 0) to find an approximate value for f(2.1, 0.1).

Accepted Answer

A)  8.4 
B)  10.0 
C)  9.2 
D)  7.6 
E)  6.3 
A)  8.4 
B)  10.0 
C)  9.2 
D)  7.6 
E)  6.3

Question 14

Let h(t) = f(x, y), where f(x, y) =   +   , x = t, y = t2. Find   (2).

Accepted Answer

A)    - 2   
B)    + 4   
C)    - 4   
D)    + 2   
E)    - 4   
A)    - 2   
B)    + 4   
C)    - 4   
D)    + 2   
E)    - 4

Question 15

Find the Maclaurin polynomial of degree 5 for the function f(x, y) = cos(x - y2).

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 16

Find the domain of the function f(x, y) = ln(9 - x2 - 9y2).

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 17

The directional derivative of a function F(x,y) at a point P in the direction of the unit vector-   i +   j is equal to -4, while the directional derivative at P in the direction of the unit vector   i+   j is equal to0. The value  F(P) is equal to:

Accepted Answer

A)  20 
B)  ± 2   
C)    i -   j 
D)    i -   j 
E)  4 i - 2 j 
A)  20 
B)  ± 2   
C)    i -   j 
D)    i -   j 
E)  4 i - 2 j

Question 18

Find the equation of the plane tangent to the surface 5x2 - 2y2 + 2z = - 9 and parallel to the plane 5x -4y + z = 2.

Accepted Answer

A)    , t      
B)  5x - 4y + z = - 6 
C)  10x -4y + 2 = 0 
D)  5x - 4y + z = 9 
E)    , t      
A)    , t      
B)  5x - 4y + z = - 6 
C)  10x -4y + 2 = 0 
D)  5x - 4y + z = 9 
E)    , t

Question 19

z =   (y/x) + cos (x/y) satisfies the partial differential equationx   + y   = 0 provided (x, y) $\neq$(0, 0).

Accepted Answer

A) True 
 B)False

Question 20

Accepted Answer

A)    - ln (   ) 
B)    + ln (   ) 
C)  -   + ln (   ) 
D)  -   - ln (   ) 
E)    + ln (2) 
A)    - ln (   ) 
B)    + ln (   ) 
C)  -   + ln (   ) 
D)  -   - ln (   ) 
E)    + ln (2)

Use the chain rule to find the values of and . at (u, v) = (0, 1), where z = x³y⁵,x = u - v, and y = u + v.

Evaluate the limit .

Find the Maclaurin series of the function e^xy-3y by using Taylor series for functions of one variable.

Compute and if f(x, y) = xy + sin xy + x² + 5xln y.

Which function f(x,y) has the level curves corresponding to c = -1, 0, and 1 shown in the figure below?

The equations x = u² - v² and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate at that point.

Evaluate the limit .

Find (x, y) and (x, y) if f(x, y) = .

Calculate and simplify u_t - u_xx - u_yy for the function u = .

The temperature in the xy-plane is a function T(x, y). At the point (a,b) the directional derivative of T in the direction of the vector 3 i + 4 j is 4 and the directional derivative in the direction of 5 i - 12 j is -2. Find T(a,b).

Find f₁₁(x, y) and f₂₂(x, y) if f(x, y) = e^x sin y + 2xy + y.

Use the linearization of f(x,y) = x³ e^-y about the point (2, 0) to find an approximate value for f(2.1, 0.1).

Let h(t) = f(x, y), where f(x, y) = + , x = t, y = t². Find (2).

Find the Maclaurin polynomial of degree 5 for the function f(x, y) = cos(x - y²).

Find the domain of the function f(x, y) = ln(9 - x² - 9y²).

The directional derivative of a function F(x,y) at a point P in the direction of the unit vector- i + j is equal to -4, while the directional derivative at P in the direction of the unit vector i+ j is equal to0. The value ofF(P) is equal to:

Find the equation of the plane tangent to the surface 5x² - 2y² + 2z = - 9 and parallel to the plane 5x -4y + z = 2.

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

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 13: Partial Differentiation

Use the chain rule to find the values of and . at (u, v) = (0, 1), where z = x3y5,x = u - v, and y = u + v.

Evaluate the limit .

Find the Maclaurin series of the function exy-3y by using Taylor series for functions of one variable.

Compute and if f(x, y) = xy + sin xy + x2 + 5xln y.

Which function f(x,y) has the level curves corresponding to c = -1, 0, and 1 shown in the figure below?

The equations x = u2 - v2 and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate at that point.

Evaluate the limit .

Find (x, y) and (x, y) if f(x, y) = .

Calculate and simplify ut - uxx - uyy for the function u = .

The temperature in the xy-plane is a function T(x, y). At the point (a,b) the directional derivative of T in the direction of the vector 3 i + 4 j is 4 and the directional derivative in the direction of 5 i - 12 j is -2. Find T(a,b).

Find f11(x, y) and f22(x, y) if f(x, y) = ex sin y + 2xy + y.

Use the linearization of f(x,y) = x3 e-y about the point (2, 0) to find an approximate value for f(2.1, 0.1).

Let h(t) = f(x, y), where f(x, y) = + , x = t, y = t2. Find (2).

Find the Maclaurin polynomial of degree 5 for the function f(x, y) = cos(x - y2).

Find the domain of the function f(x, y) = ln(9 - x2 - 9y2).

The directional derivative of a function F(x,y) at a point P in the direction of the unit vector- i + j is equal to -4, while the directional derivative at P in the direction of the unit vector i+ j is equal to0. The value ofF(P) is equal to:

Find the equation of the plane tangent to the surface 5x2 - 2y2 + 2z = - 9 and parallel to the plane 5x -4y + z = 2.

z = (y/x) + cos (x/y) satisfies the partial differential equationx + y = 0 provided (x, y) ≠\neq= (0, 0).

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

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Use the chain rule to find the values of and . at (u, v) = (0, 1), where z = x³y⁵,x = u - v, and y = u + v.

Find the Maclaurin series of the function e^xy-3y by using Taylor series for functions of one variable.

Compute and if f(x, y) = xy + sin xy + x² + 5xln y.

The equations x = u² - v² and y = 2uv define u and v as functions of x and y in a neighbourhood of the point where u = 2 and v = 1. Evaluate at that point.

Calculate and simplify u_t - u_xx - u_yy for the function u = .

Find f₁₁(x, y) and f₂₂(x, y) if f(x, y) = e^x sin y + 2xy + y.

Use the linearization of f(x,y) = x³ e^-y about the point (2, 0) to find an approximate value for f(2.1, 0.1).

Let h(t) = f(x, y), where f(x, y) = + , x = t, y = t². Find (2).

Find the Maclaurin polynomial of degree 5 for the function f(x, y) = cos(x - y²).

Find the domain of the function f(x, y) = ln(9 - x² - 9y²).

Find the equation of the plane tangent to the surface 5x² - 2y² + 2z = - 9 and parallel to the plane 5x -4y + z = 2.