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Unit 4: Multivariate differentiation - Practice Quiz

MTH165 — Mathematics For Engineers 60 Questions
0 Correct 0 Wrong 60 Left
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1 Evaluate .

Limits and continuity Easy
A.
B.
C.
D.

2 A function is continuous at if which condition holds?

Limits and continuity Easy
A.
B. is always an integer
C. is always positive
D.

3 Which function is continuous at every point in ?

Limits and continuity Easy
A.
B.
C.
D.

4 For , find .

Partial derivatives and total derivative Easy
A.
B.
C.
D.

5 For , find .

Partial derivatives and total derivative Easy
A.
B.
C.
D.

6 If , which expression gives its total differential?

Partial derivatives and total derivative Easy
A.
B.
C.
D.

7 For , what is the mixed partial derivative ?

Partial derivatives and total derivative Easy
A.
B.
C.
D.

8 If , , and , what is ?

Chain rule Easy
A.
B.
C.
D.

9 If , where and , which formula represents the chain rule?

Chain rule Easy
A.
B.
C.
D.

10 Let , , and . What is ?

Chain rule Easy
A.
B.
C.
D.

11 A function is homogeneous of degree if which relation holds?

Euler's theorem for homogeneous functions Easy
A.
B.
C.
D.

12 What is the degree of the homogeneous function ?

Euler's theorem for homogeneous functions Easy
A.
B.
C.
D.

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

Euler's theorem for homogeneous functions Easy
A.
B.
C.
D.

14 At an interior stationary point of a differentiable function , which conditions usually hold?

Maxima and minima for a function of two variables Easy
A. and
B. and
C. and
D. and

15 Which point is the minimum point of ?

Maxima and minima for a function of two variables Easy
A.
B.
C.
D.

16 For the second derivative test, the discriminant at a stationary point is:

Maxima and minima for a function of two variables Easy
A.
B.
C.
D.

17 If and at a stationary point, the function has a:

Maxima and minima for a function of two variables Easy
A. Local minimum
B. Local maximum
C. Saddle point
D. Discontinuity

18 The Lagrange multiplier method is mainly used to find:

Lagrange method of multiplier Easy
A. Partial derivatives
B. Constrained extrema
C. Ordinary limits
D. Taylor coefficients

19 To optimize subject to , the basic Lagrange condition is:

Lagrange method of multiplier Easy
A.
B.
C.
D.

20 For the constraint , which function may be used as in the equation ?

Lagrange method of multiplier Easy
A.
B.
C.
D.

21 Evaluate the limit

Limits and continuity Medium
A. The limit does not exist because different paths approaching the origin produce different values
B. The limit is
C. The limit is
D. The limit is

22 Let For which value of is continuous at the origin?

Limits and continuity Medium
A.
B.
C. No value of
D.

23 Find

Limits and continuity Medium
A.
B.
C. The limit does not exist
D.

24 For , find the total differential at .

Partial derivatives and total derivative Medium
A.
B.
C.
D.

25 If , what is ?

Partial derivatives and total derivative Medium
A.
B.
C.
D.

26 Find the tangent plane to at the point .

Partial derivatives and total derivative Medium
A.
B.
C. , since both partial derivatives equal the coordinates at the given point
D.

27 For , determine the mixed partial derivative .

Partial derivatives and total derivative Medium
A.
B.
C.
D.

28 Let , where and . Find .

Chain rule Medium
A. , obtained by differentiating both transformed variables without combining terms
B.
C.
D.

29 If , , and , find at .

Chain rule Medium
A.
B.
C.
D.

30 Suppose , where and . Which expression equals ?

Chain rule Medium
A.
B.
C.
D.

31 Let . According to Euler's theorem, what is ?

Euler's theorem for homogeneous functions Medium
A.
B.
C.
D.

32 For , where defined, evaluate .

Euler's theorem for homogeneous functions Medium
A.
B.
C.
D.

33 If is twice differentiable and homogeneous of degree , what is

Euler's theorem for homogeneous functions Medium
A.
B. , after applying Euler's theorem separately to every second derivative
C.
D.

34 Classify the stationary point of .

Maxima and minima for a function of two variables Medium
A. An inconclusive stationary point
B. A saddle point
C. A strict local minimum
D. A strict local maximum

35 For , which statement about the stationary point is correct?

Maxima and minima for a function of two variables Medium
A. It is a saddle point with value
B. It is a local minimum with value
C. It is a local maximum with value
D. It is a local minimum with value

36 Determine the nature of the stationary point of .

Maxima and minima for a function of two variables Medium
A. A local minimum at
B. A saddle point at
C. A saddle point at
D. A local maximum at

37 Find the minimum value of .

Maxima and minima for a function of two variables Medium
A. at
B. at
C. at
D. at

38 Using Lagrange multipliers, find the maximum value of subject to .

Lagrange method of multiplier Medium
A.
B.
C.
D.

39 Find the minimum value of subject to the constraint .

Lagrange method of multiplier Medium
A.
B.
C.
D.

40 For and , determine the constrained minimum of subject to .

Lagrange method of multiplier Medium
A. The minimum is at
B. The minimum is at
C. The minimum is approached as one variable tends to zero and the other increases without bound
D. The minimum is at

41 Evaluate the limit

Limits and continuity Hard
A. The limit does not exist
B.
C. The function is unbounded near
D.

42 For both iterated limits at are zero. Which statement about the two-variable limit is correct?

Limits and continuity Hard
A. The limit exists and is zero, but continuity fails only because the function is undefined at the origin
B. The limit does not exist because different straight-line paths give different values
C. The limit is because both iterated limits agree
D. The limit is because the numerator and denominator have equal degree

43 Define and, for , For which real values of is continuous at the origin?

Limits and continuity Hard
A.
B.
C.
D.

44 Let Which statement is correct at ?

Partial derivatives and total derivative Hard
A. is continuous, but neither partial derivative exists
B. is differentiable with total derivative
C. Only the partial derivative with respect to exists
D. Both partial derivatives exist, but is not differentiable

45 For with , what is the total derivative applied to an increment ?

Partial derivatives and total derivative Hard
A.
B.
C.
D.

46 Define Which assertion is correct?

Partial derivatives and total derivative Hard
A. The mixed partial derivatives fail to exist because their defining one-variable limits approach unequal finite values
B. is twice differentiable with zero Hessian at the origin
C. All second-order partial derivatives exist at the origin, but is not twice differentiable there
D. is not differentiable at the origin

47 Let , where is twice continuously differentiable. If and , what is ?

Chain rule Hard
A.
B.
C.
D.

48 Let . Suppose . Find the directional derivative of at in the direction of the unit vector .

Chain rule Hard
A.
B.
C.
D.

49 Let , where is twice differentiable. If and , what is ?

Chain rule Hard
A.
B.
C.
D.

50 If is twice continuously differentiable and homogeneous of degree , which identity must hold?

Euler's theorem for homogeneous functions Hard
A.
B.
C.
D.

51 Let on a region where this expression is twice differentiable and nonzero. What is

Euler's theorem for homogeneous functions Hard
A.
B.
C.
D.

52 Suppose is twice differentiable and homogeneous of degree , and let . Assuming is differentiable, which identity is satisfied by ?

Euler's theorem for homogeneous functions Hard
A.
B.
C.
D.

53 For , which classification of its stationary points is correct?

Maxima and minima for a function of two variables Hard
A. Both and are local minima
B. is a local maximum and is a saddle point
C. is inconclusive and is a local maximum
D. is a saddle point and is a local minimum

54 At the stationary point of , the second derivative test has determinant zero. What is the correct classification?

Maxima and minima for a function of two variables Hard
A. Saddle point
B. Strict local minimum
C. Non-strict local minimum
D. Strict local maximum

55 On the domain , determine the global minimum of

Maxima and minima for a function of two variables Hard
A. The minimum is , attained at
B. No global minimum exists because the domain has four disconnected components
C. The infimum is , but it is not attained in the domain
D. The minimum is , attained at all four points

56 For , which statement describes all stationary points?

Maxima and minima for a function of two variables Hard
A. is a local minimum, while and are saddles
B. is a saddle, while and are global minima
C. The origin is the only stationary point because the quartic terms dominate the mixed term near infinity
D. All three stationary points are local minima with different function values

57 Using the constraint , what are the maximum and minimum values of ?

Lagrange method of multiplier Hard
A. Maximum and minimum
B. Maximum and minimum
C. Maximum and minimum
D. Maximum and minimum

58 Consider optimizing subject to . Which conclusion is correct?

Lagrange method of multiplier Hard
A. There is a maximum of at and , but no minimum
B. The multiplier equations locate every extremum, and because they have one solution the function has exactly one constrained maximum
C. There is a minimum of at , but no maximum
D. There are both a minimum of and a maximum of

59 Minimize subject to . Which statement correctly describes the role of Lagrange multipliers?

Lagrange method of multiplier Hard
A. The minimum is at , but no multiplier satisfies there
B. Every real multiplier works because both gradients vanish at the feasible point
C. No minimum exists because the multiplier equations have no solution
D. The minimum is at with the unique multiplier

60 Find the squared minimum distance from to the parabola .

Lagrange method of multiplier Hard
A.
B.
C.
D.
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