Question 1

Evaluate the limit.

Accepted Answer

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

Question 2

Find   and   if z = f(x, y) =     sin(x - 2y).

Accepted Answer

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

Question 3

Find   if z = sin(uv)cos(uv), where u = et and y = e2t.

Accepted Answer

A)  3     
B)  3     
C)  3     
D)      
E)      
A)  3     
B)  3     
C)  3     
D)      
E)

Question 4

Let g be a function of a single variable having continuous second order derivatives, and letu(x, y) = g(y + mx) , for some constant real number m. Determine all values of m such that u(x, y) satisfies the partial differential equation   - 10   + 24   = 0.

Accepted Answer

A)  - 2 and 12 
B)  - 4 and - 6 
C)  2 and - 12 
D)  4 and 6 
E)  3 and 8 
A)  - 2 and 12 
B)  - 4 and - 6 
C)  2 and - 12 
D)  4 and 6 
E)  3 and 8

Question 5

Describe the level curves of the function f(x,y) =   , where a > 0.

Accepted Answer

A)  all circles in the xy-plane 
B)  all circles in the xy-plane having centres at the origin. 
C)  all circles in the xy-plane having centres at the origin and radii less than a 
D)  the origin together with all circles in the xy-plane having centres at the origin and radii in the interval (0, a] 
E)  a circle in the xy -plane having its centre at the origin and radius = a 
A)  all circles in the xy-plane 
B)  all circles in the xy-plane having centres at the origin. 
C)  all circles in the xy-plane having centres at the origin and radii less than a 
D)  the origin together with all circles in the xy-plane having centres at the origin and radii in the interval (0, a] 
E)  a circle in the xy -plane having its centre at the origin and radius = a

Question 6

Find the equation of the normal line to the surface z =   at the point(2, -2, 1).

Accepted Answer

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

Question 7

Assume that the relation   - 65 +   = 0 defines z as a differentiable functionof x and y near the point (x , y) = (4 , 0).
(a) If z = f(x , y) , find   and   at (x,y) = (4 , 0).
(b) If x =   +   , y =   , find   at (u , v) =(0 , 2)\.
Hints :Part (a): First , find the value of z at (x , y) =(4 , 0).
Part (b): Use the chain rule!

Accepted Answer

(a) @#IMG-DLM& = @#IMG-DLM& ,

Question 8

The graph below shows level curves f(x,y) = C for a function f and equally spaced values of C. Which of the following functions f is consistent with the graph?
(a) f(x,y) =   +   , (b) f(x,y) =   (c) f(x,y) =

Accepted Answer

A)  function (a) 
B)  function (b) 
C)  function (c) 
D)  all three functions 
E)  none of the functions 
A)  function (a) 
B)  function (b) 
C)  function (c) 
D)  all three functions 
E)  none of the functions

Question 9

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy2 + x2y + z3 = 3 and x3 - y3 - xyz = -1.

Accepted Answer

A)  ±   ( i + j - 2 k ) 
B)  ±   ( i - j - 2 k ) 
C)  ±   ( i - 2 j - 2 k ) 
D)  ±   ( i + j - k ) 
E)  ±   ( i + j - 2 k ) 
A)  ±   ( i + j - 2 k ) 
B)  ±   ( i - j - 2 k ) 
C)  ±   ( i - 2 j - 2 k ) 
D)  ±   ( i + j - k ) 
E)  ±   ( i + j - 2 k )

Question 10

Identify the function f(x,y) whose domain is the shaded region 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 11

Find an equation of the plane tangent to the surface z = x2 - y2 at the point (2, 1, 3).

Accepted Answer

A)  4x - 2y + z = 9 
B)  4x - 2y - z = 3 
C)  4x + 2y + z = 13 
D)  4x + 2y - z = 7 
E)  4x - 2y + z = -9 
A)  4x - 2y + z = 9 
B)  4x - 2y - z = 3 
C)  4x + 2y + z = 13 
D)  4x + 2y - z = 7 
E)  4x - 2y + z = -9

Question 12

Given z = f(x, y) =   y -   , find   .

Accepted Answer

A)  3   y -   
B)  3   y - y   
C)    y - x   
D)    
E)    
A)  3   y -   
B)  3   y - y   
C)    y - x   
D)    
E)

Question 13

Assuming that the function f has continuous partial derivatives of orders 1 and 2, calculate and simplify   f(   y, x   ).

Accepted Answer

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

Question 14

Accepted Answer

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

Question 15

Evaluate   (3, 2, 1) if f(x, y, z) = x   (yz).

Accepted Answer

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

Question 16

Given that the relation y2 + y   = 14 - sin(xz2) +   implicitly defines x as a differentiable function of y and z, find   at the point (0, 3, 4).

Accepted Answer

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

Question 17

Show that the function g(x,y) =   is continuous at (x, y) = (0, 0).

Accepted Answer

First you need to verify that the three

Question 18

Describe the level curves f(x, y) =   .

Accepted Answer

A)  all parabolas in the xy-plane tangent to the x-axis at the origin 
B)  all parabolas in the xy-plane tangent to the y-axis at the origin 
C)  all parabolas in the xy-plane tangent to the y-axis 
D)  all parabolas in the xy-plane tangent to the x-axis 
E)  all parabolas in the xy-plane 
A)  all parabolas in the xy-plane tangent to the x-axis at the origin 
B)  all parabolas in the xy-plane tangent to the y-axis at the origin 
C)  all parabolas in the xy-plane tangent to the y-axis 
D)  all parabolas in the xy-plane tangent to the x-axis 
E)  all parabolas in the xy-plane

Question 19

Find the slope of the tangent line to the curve that is the intersection of the surface z = x2 - y2 with the plane x = 2 at the point (2, 1, 3).

Accepted Answer

A)  -2 
B)  2 
C)  0 
D)  -4 
E)  4 
A)  -2 
B)  2 
C)  0 
D)  -4 
E)  4

Question 20

Find an equation of the plane tangent to the surface xy3z2 = 2 at the point (2, 1, -1).

Accepted Answer

A)  x = (2 + t) i + (1 + 6t) j + (-1 - 4t) k, t    R 
B)  x + 6y + 4z = 4 
C)  x + 6y - 4z = 12 
D)  x + 6y - 4z = 0 
E)  x + 5y - 4z = 11 
A)  x = (2 + t) i + (1 + 6t) j + (-1 - 4t) k, t    R 
B)  x + 6y + 4z = 4 
C)  x + 6y - 4z = 12 
D)  x + 6y - 4z = 0 
E)  x + 5y - 4z = 11

Evaluate the limit.

Find and if z = f(x, y) = sin(x - 2y).

Find if z = sin(uv)cos(uv), where u = e^t and y = e^2t.

Let g be a function of a single variable having continuous second order derivatives, and letu(x, y) = g(y + mx) , for some constant real number m. Determine all values of m such that u(x, y) satisfies the partial differential equation - 10 + 24 = 0.

Describe the level curves of the function f(x,y) = , where a > 0.

Find the equation of the normal line to the surface z = at the point(2, -2, 1).

The graph below shows level curves f(x,y) = C for a function f and equally spaced values of C. Which of the following functions f is consistent with the graph? (a) f(x,y) = + , (b) f(x,y) = (c) f(x,y) =

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy² + x²y + z³ = 3 and x³ - y³ - xyz = -1.

Identify the function f(x,y) whose domain is the shaded region shown in the figure below.

Find an equation of the plane tangent to the surface z = x² - y² at the point (2, 1, 3).

Given z = f(x, y) = y - , find .

Assuming that the function f has continuous partial derivatives of orders 1 and 2, calculate and simplify f( y, x ).

Evaluate (3, 2, 1) if f(x, y, z) = x (yz).

Given that the relation y² + y = 14 - sin(xz²) + implicitly defines x as a differentiable function of y and z, find at the point (0, 3, 4).

Show that the function g(x,y) = is continuous at (x, y) = (0, 0).

Describe the level curves f(x, y) = .

Find the slope of the tangent line to the curve that is the intersection of the surface z = x² - y² with the plane x = 2 at the point (2, 1, 3).

Find an equation of the plane tangent to the surface xy³z² = 2 at the point (2, 1, -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

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 13: Partial Differentiation

Evaluate the limit.

Find and if z = f(x, y) = sin(x - 2y).

Find if z = sin(uv)cos(uv), where u = et and y = e2t.

Let g be a function of a single variable having continuous second order derivatives, and letu(x, y) = g(y + mx) , for some constant real number m. Determine all values of m such that u(x, y) satisfies the partial differential equation - 10 + 24 = 0.

Describe the level curves of the function f(x,y) = , where a > 0.

Find the equation of the normal line to the surface z = at the point(2, -2, 1).

The graph below shows level curves f(x,y) = C for a function f and equally spaced values of C. Which of the following functions f is consistent with the graph? (a) f(x,y) = + , (b) f(x,y) = (c) f(x,y) =

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy2 + x2y + z3 = 3 and x3 - y3 - xyz = -1.

Identify the function f(x,y) whose domain is the shaded region shown in the figure below.

Find an equation of the plane tangent to the surface z = x2 - y2 at the point (2, 1, 3).

Given z = f(x, y) = y - , find .

Assuming that the function f has continuous partial derivatives of orders 1 and 2, calculate and simplify f( y, x ).

Evaluate (3, 2, 1) if f(x, y, z) = x (yz).

Given that the relation y2 + y = 14 - sin(xz2) + implicitly defines x as a differentiable function of y and z, find at the point (0, 3, 4).

Show that the function g(x,y) = is continuous at (x, y) = (0, 0).

Describe the level curves f(x, y) = .

Find the slope of the tangent line to the curve that is the intersection of the surface z = x2 - y2 with the plane x = 2 at the point (2, 1, 3).

Find an equation of the plane tangent to the surface xy3z2 = 2 at the point (2, 1, -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

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Find if z = sin(uv)cos(uv), where u = e^t and y = e^2t.

Find the two unit vectors tangent at the point (1, 1, 1) to the curve of intersection of the surfaces xy² + x²y + z³ = 3 and x³ - y³ - xyz = -1.

Find an equation of the plane tangent to the surface z = x² - y² at the point (2, 1, 3).

Given that the relation y² + y = 14 - sin(xz²) + implicitly defines x as a differentiable function of y and z, find at the point (0, 3, 4).

Find the slope of the tangent line to the curve that is the intersection of the surface z = x² - y² with the plane x = 2 at the point (2, 1, 3).

Find an equation of the plane tangent to the surface xy³z² = 2 at the point (2, 1, -1).