For a function $f(x,y)$ to be continuous at $(a,b)$, which condition must hold?

$\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b)$

$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}$

If $z=f(x,y)$, where $x=x(t)$ and $y=y(t)$, then $\frac{dz}{dt}$ is given by:

$\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$

$\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$

$\frac{\partial z}{\partial x}\frac{dy}{dt}$

If $z=f(x,y)$ and $x=x(u,v)$, $y=y(u,v)$, then $\frac{\partial z}{\partial u}$ equals:

$\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u}$

$\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$

$\frac{\partial x}{\partial u}+\frac{\partial y}{\partial u}$

$\frac{\partial z}{\partial x}\frac{\partial y}{\partial u}$

A function $f(x,y)$ is homogeneous of degree $n$ if:

$f(tx,ty)=f(x,y)$ for every $t$

MTH174 Unit 5: Multivariate Calculus - MCQ Practice | B. Tech. CSE

1 $For a function to be continuous at, which condition must hold?$

Limit, continuity and differentiability of functions of two variables Easy

A.

B.

C.

D.

2 $A function of two variables is differentiable at a point if it has a:$

Limit, continuity and differentiability of functions of two variables Easy

A.

Constant value at the point

B.

Zero limit at the point

C.

Tangent plane at the point

D.

Single partial derivative

3 $The limit of as exists only when the limiting value is:$

Limit, continuity and differentiability of functions of two variables Easy

A.

Equal to one

B.

Independent of the path

C.

Dependent on the path

D.

Equal to zero

4 $If, where and, then is given by:$

Chain rule Easy

A.

B.

C.

D.

5 $If and,, then equals:$

Chain rule Easy

A.

B.

C.

D.

6 $The chain rule is used when variables are related through:$

Chain rule Easy

A.

Independent equations only

B.

Constant numbers

C.

Zero derivatives

D.

Intermediate variables

7 $The main purpose of changing variables in a double integral is to:$

Change of variables Easy

A.

Make the integral divergent

B.

Simplify the region or integrand

C.

Change a double integral into a constant

D.

Remove all variables

8 $In polar coordinates, the Cartesian variables are represented as:$

Change of variables Easy

A.

,

B.

,

C.

,

D.

,

9 $The area element in polar coordinates becomes:$

Change of variables Easy

A.

B.

C.

D.

10 $A function is homogeneous of degree if:$

Euler's theorem for homogeneous equations Easy

A.

for every

B.

C.

D.

11 $If is homogeneous of degree, Euler's theorem states that:$

Euler's theorem for homogeneous equations Easy

A.

B.

C.

D.

12 $The function is homogeneous of degree:$

Euler's theorem for homogeneous equations Easy

A.

0

B.

2

C.

1

D.

3

13 $The Jacobian is defined as:$

Jacobians Easy

A.

B.

C.

D.

14 $If and, then is:$

Jacobians Easy

A.

B.

C.

D.

15 $For inverse transformations, the Jacobians satisfy:$

Jacobians Easy

A.

B.

C.

D.

16 $A necessary condition for an interior local extremum of is:$

Extrema of functions of two variables Easy

A.

and

B.

and

C.

only

D.

17 $The point where and is called a:$

Extrema of functions of two variables Easy

A.

Maximum value

B.

Boundary point

C.

Singular line

D.

Stationary point

18 $For the second-derivative test, the quantity commonly used is:$

Extrema of functions of two variables Easy

A.

B.

C.

D.

19 $Lagrange's method is used to find extrema subject to a:$

Lagrange's method of undetermined multipliers Easy

A.

Constant derivative

B.

Linear graph only

C.

Zero function only

D.

Constraint equation

20 $For optimizing subject to, the Lagrange equations include:$

Lagrange's method of undetermined multipliers Easy

A.

only

B.

only

C.

D.

21 $Evaluate the limit, if it exists: .$

Limit, continuity and differentiability of functions of two variables Medium

A.

B.

The limit does not exist because different paths give different values.

C.

D.

22 $Let for and . Which statement is correct?$

Limit, continuity and differentiability of functions of two variables Medium

A.

is differentiable but not continuous at the origin

B.

is continuous at the origin

C.

is discontinuous at the origin

D.

has limit at the origin

23 $Which statement about at is correct?$

Limit, continuity and differentiability of functions of two variables Medium

A.

It is differentiable with gradient

B.

It is continuous but not differentiable

C.

It is discontinuous but differentiable

D.

It has no limit at the origin

24 $If,, and, find .$

Chain rule Medium

A.

B.

C.

D.

25 $Let, where and . For fixed, find .$

Chain rule Medium

A.

B.

C.

D.

26 $Suppose, where and . Which expression gives ?$

Chain rule Medium

A.

B.

C.

D.

27 $Under the transformation and, what is the absolute value of ?$

Change of variables Medium

A.

B.

C.

D.

28 $In polar coordinates, the double integral transformation from to requires which area element?$

Change of variables Medium

A.

B.

C.

D.

29 $The transformation and maps the square region into which region in the -plane?$

Change of variables Medium

A.

B.

C.

D.

30 $If, what is the value of ?$

Euler's theorem for homogeneous equations Medium

A.

B.

C.

D.

31 $If, which relation follows from Euler's theorem?$

Euler's theorem for homogeneous equations Medium

A.

B.

C.

D.

32 $If is homogeneous of degree, which formula is correct for its second derivatives?$

Euler's theorem for homogeneous equations Medium

A.

B.

C.

D.

33 $For and, find at .$

Jacobians Medium

A.

B.

C.

D.

34 $If at a point and the transformation is locally invertible, what is there?$

Jacobians Medium

A.

B.

C.

D.

35 $Let and . At which points does the transformation fail to be locally invertible?$

Jacobians Medium

A.

Points satisfying

B.

Only the origin

C.

All points on the line

D.

Points satisfying

36 $Find the minimum value of .$

Extrema of functions of two variables Medium

A.

B.

C.

D.

37 $Classify the critical point of .$

Extrema of functions of two variables Medium

A.

Inconclusive point

B.

Local minimum

C.

Local maximum

D.

Saddle point

38 $Using the second derivative test, classify the critical point of .$

Extrema of functions of two variables Medium

A.

Strict local minimum

B.

The test gives a local minimum

C.

Saddle point

D.

Strict local maximum

39 $The maximum value of subject to occurs at which point?$

Lagrange's method of undetermined multipliers Medium

A.

B.

C.

D.

40 $For positive satisfying, where does attain its maximum?$

Lagrange's method of undetermined multipliers Medium

A.

B.

C.

D.

41 $For when and, which statement is correct?$

Limit, continuity and differentiability of functions of two variables Hard

A.

The limit does not exist

B.

The function is discontinuous only along the -axis

C.

The limit exists and equals

D.

The limit exists and equals

42 $Define for and . Which conclusion is valid at the origin?$

Limit, continuity and differentiability of functions of two variables Hard

A.

is neither continuous nor partially differentiable

B.

is discontinuous but has both partial derivatives

C.

is differentiable with derivative zero

D.

is continuous but not differentiable

43 $Let for and . Which statement is true?$

Limit, continuity and differentiability of functions of two variables Hard

A.

is continuous and differentiable at the origin

B.

is continuous at the origin but not differentiable

C.

has both partial derivatives at the origin but is not continuous

D.

has no limit at the origin, although both partial derivatives exist

44 $Suppose has continuous first partial derivatives in a neighborhood of, and . Which additional condition is sufficient to conclude that is a strict local minimum?$

Limit, continuity and differentiability of functions of two variables Hard

A.

and

B.

and

C.

and

D.

with positive diagonal entries

45 $Let, where and . If and, find .$

Chain rule Hard

A.

B.

C.

D.

46 $Let, where and . At a point satisfying and, what is ?$

Chain rule Hard

A.

B.

C.

D.

47 $Let with and . If and, determine at .$

Chain rule Hard

A.

B.

C.

D.

48 $Under the transformation and, which expression equals and what is the absolute Jacobian factor ?$

Change of variables Hard

A.

and

B.

and

C.

and

D.

and

49 $Evaluate, where is bounded by and .$

Change of variables Hard

A.

B.

C.

D.

50 $For and, determine the image of the circle under the transformation.$

Change of variables Hard

A.

B.

C.

D.

51 $If is homogeneous of degree and twice differentiable, which identity necessarily holds?$

Euler's theorem for homogeneous equations Hard

A.

B.

C.

D.

52 $Let for, where is twice differentiable. Which statement is correct?$

Euler's theorem for homogeneous equations Hard

A.

B.

C.

is homogeneous of degree

D.

53 $Suppose is homogeneous of degree and satisfies . At points where, which degree is possible?$

Euler's theorem for homogeneous equations Hard

A.

B.

C.

D.

54 $For and, calculate and identify where the transformation is locally singular.$

Jacobians Hard

A.

; on the coordinate axes

B.

; only at

C.

; only at

D.

; on the lines

55 $If and, then at a point with, what is ?$

Jacobians Hard

A.

B.

C.

D.

56 $Let and, with . Which expression is the Jacobian ?$

Jacobians Hard

A.

B.

C.

D.

57 $Classify the critical point of .$

Extrema of functions of two variables Hard

A.

Strict local maximum

B.

Degenerate point that cannot be classified

C.

Saddle point

D.

Strict local minimum

58 $For, which classification applies to the critical point ?$

Extrema of functions of two variables Hard

A.

Strict local maximum

B.

Saddle point

C.

Strict local minimum

D.

Inflection point with no extremum

59 $Find the maximum value of on the ellipse .$

Extrema of functions of two variables Hard

A.

B.

C.

D.

60 $Using Lagrange multipliers, find the minimum of subject to and .$

Lagrange's method of undetermined multipliers Hard

A.

at

B.

at

C.

at and

D.

at

Unit 5: Multivariate Calculus - Practice Quiz

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