1 $What is the domain of the function ?$

A.

The set of all points such that

B.

The set of all points such that

C.

The set of all points such that

D.

The set of all points such that

2 $If depends on the path along which approaches, then:$

A.

The limit does not exist

B.

The function is continuous at

C.

The limit exists and equals 0

D.

The limit exists but is infinite

3 $Evaluate .$

A.

$5$

B.

$7$

C.

$1$

D.

$0$

4 $What is the value of ?$

A.

$1$

B.

C.

$0$

D.

Does not exist

5 $A function is continuous at if:$

A.

B.

is defined

C.

exists

D.

and exist at

6 $For, find the partial derivative .$

A.

B.

C.

D.

7 $For, find .$

A.

B.

C.

D.

8 $Clairaut's Theorem (or Schwarz's Theorem) states that under continuity conditions of second partial derivatives:$

A.

B.

C.

D.

9 $If, then is:$

A.

B.

C.

D.

10 $If is differentiable at a point, then:$

A.

The limit depends on the path

B.

It must be continuous at that point

C.

Only partial derivatives exist, but limit may not

D.

It is not necessarily continuous at that point

11 $The total differential of a function is given by:$

A.

B.

C.

D.

12 $If, calculate the approximate change in if changes from 2 to 2.01 and changes from 3 to 2.98.$

A.

$0.01$

B.

C.

$0.05$

D.

13 $Given where and, the Chain Rule for is:$

A.

B.

C.

D.

14 $If,,, what is ?$

A.

B.

C.

$0$

D.

$2$

15 $A function is homogeneous of degree if:$

A.

B.

C.

D.

16 $What is the degree of homogeneity of the function ?$

A.

$1$

B.

$0$

C.

$3$

D.

$2$

17 $Euler's Theorem for a homogeneous function of degree states that:$

A.

B.

C.

D.

18 $If, the value of is:$

A.

B.

C.

$0$

D.

$1$

19 $If is a homogeneous function of degree in and, then equals:$

A.

B.

C.

D.

20 $Determine the degree of the homogeneous function .$

A.

B.

$1$

C.

$0$

D.

21 $If, what is the value of ?$

A.

B.

C.

D.

22 $Which of the following is a stationary point condition for ?$

A.

and

B.

and

C.

and

D.

23 $Let at a critical point . If and, then is a:$

A.

Point of inflection

B.

Local Maximum

C.

Saddle Point

D.

Local Minimum

24 $Let at a critical point . If, then is a:$

A.

Test Inconclusive

B.

Local Maximum

C.

Saddle Point

D.

Local Minimum

25 $If at a critical point, then:$

A.

It is a Minimum

B.

It is a Saddle Point

C.

The test is inconclusive

D.

It is a Maximum

26 $Find the critical point of .$

A.

B.

C.

D.

27 $The Lagrange Multiplier method is used to find:$

A.

Area under a curve

B.

Partial derivatives

C.

Roots of a polynomial

D.

Maxima or minima subject to constraints

28 $To find the extrema of subject to using Lagrange multiplier, we solve the system generated by:$

A.

B.

C.

and

D.

29 $The auxiliary function (Lagrange function) for maximizing subject to is:$

A.

B.

C.

D.

30 $Given, the value of is:$

A.

$1$

B.

C.

$0$

D.

31 $If and, find the Jacobian .$

A.

$1$

B.

C.

D.

32 $If and are functionally dependent, their Jacobian is:$

A.

Constant

B.

C.

$0$

D.

$1$

33 $For the function, the point is a:$

A.

Global Maximum

B.

Local Minimum

C.

Saddle Point

D.

Local Maximum

34 $The equation of the tangent plane to at is:$

A.

B.

C.

D.

35 $If, then is:$

A.

B.

C.

D.

36 $If, then is:$

A.

B.

C.

D.

37 $A function has a local maximum at if:$

A.

and

B.

and

C.

and

D.

38 $Which of the following functions is homogeneous?$

A.

B.

C.

D.

39 $If is differentiable, can be written as:$

A.

B.

C.

D.

40 $Compute for .$

A.

B.

C.

D.

41 $If and, calculate the Jacobian .$

A.

B.

C.

D.

$1$

42 $When finding the shortest distance from a point to a plane using Lagrange multipliers, the objective function is usually taken as:$

A.

The square of the distance

B.

The equation of the plane

C.

The gradient vector

D.

The distance formula

43 $Implicit differentiation: If defines as a function of, then ?$

A.

B.

C.

D.

44 $Determine the limit .$

A.

Does not exist

B.

$1$

C.

$0$

D.

45 $Given, equals:$

A.

B.

$1$

C.

$0$

D.

46 $Which of the following is NOT an indeterminate form for limits?$

A.

B.

C.

D.

47 $For the function, at the origin :$

A.

Partial derivatives exist

B.

Function is continuous but partial derivatives do not exist

C.

Function is differentiable

D.

Function is discontinuous

48 $If is homogeneous of degree, then equals:$

A.

B.

C.

D.

49 $In the method of Lagrange multipliers for a function of three variables subject to one constraint, how many equations must be solved?$

A.

$2$

B.

$3$

C.

$5$

D.

$4$

50 $The gradient vector points in the direction of:$

A.

Zero change (along contour)

B.

Maximum rate of increase

C.

Any arbitrary direction

D.

Maximum rate of decrease

Unit 4 - Practice Quiz