1For a limit of a function of two variables to exist as , the limit must be:
limit, continuity and differentiability of functions of two variables
Easy
A.The same along any path approaching
B.Infinite along at least one path
C.Zero along all paths
D.Different along different paths
Correct Answer: The same along any path approaching
Explanation:
The limit of a function of two variables exists if and only if the function approaches the same value regardless of the path taken to reach the point .
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2A function is said to be continuous at a point if:
limit, continuity and differentiability of functions of two variables
Easy
A.Its partial derivatives evaluate to zero
B.
C.
D. is undefined
Correct Answer:
Explanation:
For continuity at a point, the limit of the function as it approaches the point must exist and equal the function's value at that exact point.
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3If a function is differentiable at a point , then at that point it must also be:
limit, continuity and differentiability of functions of two variables
Easy
A.Undefined
B.Equal to zero
C.A constant function
D.Continuous
Correct Answer: Continuous
Explanation:
Differentiability is a stronger condition than continuity. If a function is differentiable at a point, it is guaranteed to be continuous there.
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4If where and , what is the total derivative according to the chain rule?
chain rule
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The total derivative of with respect to is the sum of the partial derivatives of with respect to and multiplied by the respective ordinary derivatives of and with respect to .
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5If and , , how is the partial derivative expressed?
chain rule
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
By the chain rule for partial derivatives, we sum the products of the partial derivatives along all paths leading from to .
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6When changing variables in a double integral from to , the area element becomes:
change of variables
Easy
A.
B.
C.
D. where is the Jacobian
Correct Answer: where is the Jacobian
Explanation:
The absolute value of the Jacobian determinant, , acts as the scaling factor for the area element during a change of variables.
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7In the standard change of variables to polar coordinates, what are and substituted with?
change of variables
Easy
A.,
B.,
C.,
D.,
Correct Answer: ,
Explanation:
The standard transformation from Cartesian to polar coordinates is given by and .
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8A function is homogeneous of degree if for any scalar :
Euler’s theorem for homogeneous equations
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
This is the mathematical definition of a homogeneous function of degree .
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9According to Euler's theorem, if is a homogeneous function of degree , then is equal to:
Euler’s theorem for homogeneous equations
Easy
A.$0$
B.
C.
D.
Correct Answer:
Explanation:
Euler's theorem states that for a homogeneous function of degree , the sum of and equals .
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10What is the degree of homogeneity of the function ?
Euler’s theorem for homogeneous equations
Easy
A.3
B.1
C.2
D.4
Correct Answer: 2
Explanation:
. Therefore, the degree is 2.
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11The Jacobian of and with respect to and , denoted by , is defined as:
Jacobians
Easy
A.The matrix itself
B.The determinant of the matrix
C.The sum of the partial derivatives
D.The product of and
Correct Answer: The determinant of the matrix
Explanation:
The Jacobian is the determinant of the Jacobian matrix, containing all first-order partial derivatives.
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12If is the Jacobian of with respect to , and is the Jacobian of with respect to , then equals:
Jacobians
Easy
A.
B.-1
C.0
D.1
Correct Answer: 1
Explanation:
By the inverse function theorem for Jacobians, , provided .
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13What is the Jacobian for the polar coordinate transformation , ?
Jacobians
Easy
A.
B.
C.
D.$1$
Correct Answer:
Explanation:
The determinant of the partial derivatives matrix is .
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14A point is a critical (stationary) point of a differentiable function if:
extrema of functions of two variables
Easy
A. and
B. and
C.
D. and
Correct Answer: and
Explanation:
At a critical point, the first-order partial derivatives of the function with respect to both and must be zero.
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15Let , , and . For a critical point to be a local minimum, which conditions must hold?
extrema of functions of two variables
Easy
A. and
B. and
C.
D.
Correct Answer: and
Explanation:
By the second derivative test, if and , the critical point is a local minimum.
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16If at a critical point of , the point is called a:
extrema of functions of two variables
Easy
A.Local minimum
B.Point of inflection
C.Local maximum
D.Saddle point
Correct Answer: Saddle point
Explanation:
When the discriminant is less than zero at a critical point, the point is neither a maximum nor a minimum; it is a saddle point.
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17If at a critical point, what does the second derivative test conclude?
extrema of functions of two variables
Easy
A.The test is inconclusive
B.It is definitively a saddle point
C.It is definitively a local minimum
D.It is definitively a local maximum
Correct Answer: The test is inconclusive
Explanation:
When , the second derivative test fails and cannot determine the nature of the critical point; further analysis is needed.
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18Lagrange’s method of undetermined multipliers is primarily used for:
Lagrange’s method of undetermined multipliers
Easy
A.Calculating the Jacobian of a transformation
B.Finding the limit of a multivariable function
C.Finding extrema of a function subject to equality constraints
D.Testing for continuity
Correct Answer: Finding extrema of a function subject to equality constraints
Explanation:
Lagrange multipliers are a strategy for finding the local maxima and minima of a function subject to equation constraints.
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19To find the extrema of subject to the constraint , we form the Lagrangian function as:
Lagrange’s method of undetermined multipliers
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The standard Lagrangian function is defined as the objective function plus the constraint function multiplied by a scalar .
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20In the Lagrangian function , what is the variable called?
Lagrange’s method of undetermined multipliers
Easy
A.Euler constant
B.Critical variable
C.Lagrange multiplier
D.Jacobian determinant
Correct Answer: Lagrange multiplier
Explanation:
The scalar introduced in this method to bind the constraint to the objective function is known as the Lagrange multiplier.
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21Evaluate the limit: .
limit, continuity and differentiability of functions of two variables
Medium
A.$0$
B.Does not exist
C.
D.$1$
Correct Answer: $0$
Explanation:
Switching to polar coordinates , the expression becomes . As , the limit goes to $0$ regardless of .
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22Determine the limit along the path .
limit, continuity and differentiability of functions of two variables
Medium
A.
B.Does not exist
C.
D.$0$
Correct Answer:
Explanation:
Substitute : . The limit depends on , hence the overall limit does not exist, but along , it is .
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23If where and , find .
chain rule
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Notice that . Then Wait, . Let's recalculate: , so .
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24Let . Find .
chain rule
Medium
A.$0$
B.
C.$1$
D.$3$
Correct Answer: $0$
Explanation:
Let , , . Then , , and . Adding these gives .
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25If , find the value of .
Euler’s theorem for homogeneous equations
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Let . is homogeneous of degree 1. By Euler's theorem, . Since , LHS is . Thus .
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26Given , evaluate .
Euler’s theorem for homogeneous equations
Medium
A.$4$
B.
C.
D.$3$
Correct Answer: $3$
Explanation:
Let , which is homogeneous of degree 3. By Euler's theorem, . Since and , we have . Dividing by gives $3$.
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27If and , find the Jacobian .
Jacobians
Medium
A.
B.
C.
D.$1$
Correct Answer:
Explanation:
The Jacobian is the determinant of the matrix of partial derivatives: .
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28Given and , find the Jacobian .
Jacobians
Medium
A.
B.$1$
C.
D.
Correct Answer:
Explanation:
The Jacobian is .
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29Under the transformation , what does the differential operator become?
change of variables
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Using the chain rule, and . Hence .
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30Find the critical points of the function .
extrema of functions of two variables
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Set the partial derivatives to zero: and . The critical point is .
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31Determine the nature of the critical point for the function .
extrema of functions of two variables
Medium
A.Saddle point
B.Local minimum
C.Local maximum
D.Inconclusive
Correct Answer: Saddle point
Explanation:
We find , , and . The discriminant . Since , the critical point is a saddle point.
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32For a function , at a critical point , let , , and . Which condition guarantees a local maximum?
extrema of functions of two variables
Medium
A.
B.
C. and
D. and
Correct Answer: and
Explanation:
By the second derivative test, if and , the function has a local maximum at the critical point.
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33To find the extreme values of subject to , what are the Lagrange multiplier equations?
Lagrange’s method of undetermined multipliers
Medium
A.,
B.,
C.,
D.,
Correct Answer: ,
Explanation:
The equations are . Here and . So , , , and . This gives and .
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34Find the maximum value of subject to the constraint using Lagrange multipliers.
Lagrange’s method of undetermined multipliers
Medium
A.$50$
B.$100$
C.$20$
D.$25$
Correct Answer: $25$
Explanation:
Let . Then and , which implies . From , we get . The maximum value is .
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35Which of the following functions is continuous at ?
limit, continuity and differentiability of functions of two variables
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The function is a polynomial in and , which is continuous everywhere. The other options involve division by zero at the origin and their limits do not exist.
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36If , where and , express in terms of and .
chain rule
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
By the chain rule, . Since and , this becomes .
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37If is a homogeneous function of degree , what is the degree of homogeneity of ?
Euler’s theorem for homogeneous equations
Medium
A.$1$
B.
C.
D.
Correct Answer:
Explanation:
By definition, . Differentiating both sides with respect to gives , which implies , so it is homogeneous of degree .
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38Let and , . What is the value of the Jacobian ?
Jacobians
Medium
A.
B.$0$
C.
D.$1$
Correct Answer: $0$
Explanation:
Notice that . Since there is a functional relationship between , they are not independent, meaning their Jacobian determinant is identically zero.
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39For , finding critical points involves solving:
extrema of functions of two variables
Medium
A. and
B. and
C. and
D. and
Correct Answer: and
Explanation:
Setting partial derivatives to zero gives and .
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40When applying Lagrange multipliers to minimize subject to two constraints and , how many multipliers are needed?
Lagrange’s method of undetermined multipliers
Medium
A.Two
B.One
C.Three
D.None
Correct Answer: Two
Explanation:
Each constraint requires its own Lagrange multiplier. Thus, two constraints require two multipliers, leading to the equation .
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41Consider the function for and . Which of the following statements is true regarding its differentiability at ?
limit, continuity and differentiability of functions of two variables
Hard
A. is differentiable at .
B.The directional derivatives exist in all directions, but is not continuous at .
C.The directional derivatives do not exist in all directions at .
D. is continuous but not differentiable at .
Correct Answer: The directional derivatives exist in all directions, but is not continuous at .
Explanation:
Approaching along gives a limit of , which depends on , meaning the limit does not exist, so is not continuous. However, approaching along any straight line gives a limit of 0, and directional derivatives exist in all directions.
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42Let for , and $0$ otherwise. Which of the following holds at ?
limit, continuity and differentiability of functions of two variables
Hard
A. is differentiable and its partial derivatives are continuous.
B. is differentiable but its partial derivatives are not continuous.
C. is not continuous.
D. is continuous but not differentiable.
Correct Answer: is differentiable but its partial derivatives are not continuous.
Explanation:
is differentiable at by the definition since the remainder term goes to 0 fast enough. However, calculating the partial derivatives reveals terms like , which oscillate and have no limit at , meaning the partial derivatives are not continuous there.
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43Let , where and . If satisfies Laplace's equation , what does equal?
chain rule
Hard
A.
B.
C.$0$
D.
Correct Answer: $0$
Explanation:
By the chain rule, . Since satisfies Laplace's equation in and , the sum in the parenthesis is zero, making the whole expression zero.
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44If , what is the value of ?
chain rule
Hard
A.$0$
B.
C.
D.$1$
Correct Answer: $0$
Explanation:
Let and . Then . Applying the chain rule to find , , and , and substituting them into the given expression yields zero.
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45If , what is the value of ?
Euler’s theorem for homogeneous equations
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Let . is a homogeneous function of degree . By Euler's theorem, . For the second order, the expression equals .
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46Let . Evaluate .
Euler’s theorem for homogeneous equations
Hard
A.
B.$3$
C.
D.
Correct Answer: $3$
Explanation:
Let . Here is homogeneous of degree 3. By Euler's theorem, . Since , substituting gives , so the result is 3.
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47If , , and , find the Jacobian .
Jacobians
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
We have , , and . Calculating the determinant of the Jacobian matrix gives .
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48Given functions , , and , what is the relationship between ?
Jacobians
Hard
A.
B.They are functionally independent.
C.
D.
Correct Answer:
Explanation:
The Jacobian evaluates to zero, implying they are functionally dependent. Using algebraic identities, . We know , substituting this yields .
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49Find the nature of the critical point for the function .
extrema of functions of two variables
Hard
A.Inconclusive by the second derivative test, but it is a saddle point
B.Local minimum
C.Saddle point
D.Local maximum
Correct Answer: Inconclusive by the second derivative test, but it is a saddle point
Explanation:
At , , . The Hessian at is . The test is inconclusive. However, . For , . For , for small . Since takes both positive and negative values near , it is a saddle point.
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50What are the absolute extrema of on the closed domain ?
extrema of functions of two variables
Hard
A.Maximum is , Minimum is
B.Maximum is , Minimum is
C.Maximum is , Minimum is
D.Maximum is , Minimum is
Correct Answer: Maximum is , Minimum is
Explanation:
Critical points inside are where and , giving . On the boundary , substituting gives for . . Max and Min .
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51Using Lagrange multipliers, find the minimum distance from the origin to the surface , where .
Lagrange’s method of undetermined multipliers
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Minimize subject to . Lagrange equations yield , so . By symmetry, . Thus . The minimum distance squared is , so the distance is .
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52Find the maximum value of subject to two constraints: and .
Lagrange’s method of undetermined multipliers
Hard
A.
B.
C.
D.$3$
Correct Answer:
Explanation:
Substitute into , yielding . The problem reduces to maximizing subject to . Max is (when ). Thus, the maximum value is .
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53Under the transformation , the Laplacian operator transforms to:
change of variables
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
This is the standard transformation of the Laplacian into polar coordinates where and . The chain rule derivations explicitly show the inclusion of the first-derivative term .
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54Let and . Transform the differential equation into the coordinate system.
change of variables
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
By the chain rule, and . Thus, .
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55Let for and $0$ at . What is the relationship between the mixed partial derivatives at ?
limit, continuity and differentiability of functions of two variables
Hard
A. and
B.
C. and
D.The mixed partial derivatives do not exist at
Correct Answer: and
Explanation:
Calculating by definition, and . Then , and . The mixed partials are unequal.
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56Suppose defines implicitly as a function of and . What is the expression for ?
chain rule
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
From , . Differentiating again with respect to using the quotient rule and chain rule (where applies as ) yields the formula .
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57If is a homogeneous function of degree , what is the value of ?
Euler’s theorem for homogeneous equations
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Applying Euler's theorem successively: the first operator . Applying the operator three times gives the third-order expansion .
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58Let be the roots of the cubic equation . What is the absolute value of the Jacobian ?
Jacobians
Hard
A.
B.
C.$1$
D.
Correct Answer:
Explanation:
By Vieta's formulas, , , . The Jacobian of with respect to is the determinant of a Vandermonde-like matrix, which equals . The Jacobian of with respect to is the reciprocal.
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59Find the maximum volume of a rectangular parallelepiped inscribed in the ellipsoid with sides parallel to the coordinate axes.
extrema of functions of two variables
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The volume is . Maximize subject to . Using AM-GM on the terms , their maximum product occurs when they are all . Thus . Maximum volume is .
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60Given a triangle with constant perimeter . What are the lengths of the sides that maximize its area?
Lagrange’s method of undetermined multipliers
Hard
A.The maximum area cannot be uniquely determined by the perimeter alone.
B.
C.
D.
Correct Answer:
Explanation:
By Heron's formula, the square of the area is . Maximize this subject to . By Lagrange multipliers or the AM-GM inequality, the product is maximized when , which means .