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Unit 4 - Practice Quiz

MTH165

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 exists and equals 0
B. The limit exists but is infinite
C. The limit does not exist
D. The function is continuous at

3 Evaluate .

A. $1$
B. $5$
C. $7$
D. $0$

4 What is the value of ?

A. $0$
B. $1$
C.
D. Does not exist

5 A function is continuous at if:

A. exists
B. is defined
C.
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. It must be continuous at that point
B. It is not necessarily continuous at that point
C. Only partial derivatives exist, but limit may not
D. The limit depends on the path

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. $2$
C. $3$
D. $0$

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

A.
B.
C.
D.

18 If , the value of is:

A. $1$
B.
C. $0$
D.

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.
C. $1$
D. $0$

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.
D. and

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

A. Local Maximum
B. Local Minimum
C. Saddle Point
D. Point of inflection

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

A. Local Maximum
B. Local Minimum
C. Saddle Point
D. Test Inconclusive

25 If at a critical point, then:

A. It is a Maximum
B. It is a Minimum
C. It is a Saddle Point
D. The test is inconclusive

26 Find the critical point of .

A.
B.
C.
D.

27 The Lagrange Multiplier method is used to find:

A. Roots of a polynomial
B. Area under a curve
C. Maxima or minima subject to constraints
D. Partial derivatives

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

A. and
B.
C.
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. $0$
C.
D.

31 If and , find the Jacobian .

A.
B.
C. $1$
D.

32 If and are functionally dependent, their Jacobian is:

A. $1$
B.
C. $0$
D. Constant

33 For the function , the point is a:

A. Local Maximum
B. Local Minimum
C. Saddle Point
D. Global 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.
D. and

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. $1$
D.

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

A. The distance formula
B. The square of the distance
C. The equation of the plane
D. The gradient vector

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

A.
B.
C.
D.

44 Determine the limit .

A. $0$
B.
C. Does not exist
D. $1$

45 Given , equals:

A. $0$
B. $1$
C.
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 differentiable
C. Function is continuous but partial derivatives do not exist
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. $3$
B. $4$
C. $5$
D. $2$

50 The gradient vector points in the direction of:

A. Maximum rate of decrease
B. Maximum rate of increase
C. Zero change (along contour)
D. Any arbitrary direction