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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. is always constant
B.
C. is always positive
D.

3 Find .

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

4 If , what is ?

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

5 If , what is ?

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

6 For , which expression gives the total differential ?

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

7 If , what is its total differential?

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 ?

chain rule Easy
A.
B.
C.
D.

10 Let , where and . Find .

chain rule Easy
A.
B.
C.
D.

11 If is homogeneous of degree , what does Euler's theorem state?

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 For , what is ?

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

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

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

15 What type of point is for ?

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

16 For the second derivative test, let . If and , what is the stationary point?

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

17 If at a stationary point, how is the point classified?

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

18 To find extrema of subject to , which Lagrange multiplier equation is used?

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

19 In the Lagrange method, what does represent?

Lagrange method of multiplier Easy
A. A second derivative
B. A Lagrange multiplier
C. A constraint variable
D. A partial derivative

20 Subject to , at which point is minimized?

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

21 Evaluate

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

22 Let Which statement is correct?

limits and continuity Medium
A. is continuous at
B. has no limit at the origin
C. has limit at the origin
D. is discontinuous along

23 For which pair of paths proves that the limit at the origin does not exist?

limits and continuity Medium
A. and
B. and
C. and
D. and

24 If , which pair gives and ?

partial derivatives and total derivative Medium
A. ,
B. ,
C. ,
D. ,

25 For , what is the total differential at ?

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

26 Use linearization at to approximate for .

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

27 For , estimate the change at when and .

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

28 If , where and , find at .

chain rule Medium
A.
B.
C.
D.

29 Let , where and . Find at .

chain rule Medium
A.
B.
C.
D.

30 Given , , and , find at .

chain rule Medium
A.
B.
C.
D.

31 For , evaluate .

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

32 Let . What is ?

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

33 If is twice differentiable and homogeneous of degree , which identity is valid?

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

34 Find the minimum value of .

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

35 Classify the stationary point of .

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

36 For , how is the stationary point classified?

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

37 Find the global maximum of on .

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 point on the line that is closest to the origin.

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

40 Find the maximum of subject to .

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

41 Consider Which statement about the limit as is correct?

limits and continuity Hard
A. The limit diverges to infinity along every parabolic path.
B. The limit does not exist because along it equals .
C. The limit is because that value occurs along .
D. The limit is because it is along every straight line.

42 For real , define For which values of is continuous at the origin?

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

43 Evaluate the behavior of as .

limits and continuity Hard
A. The joint limit exists and equals .
B. The joint limit exists and equals .
C. Both iterated limits fail to exist.
D. The joint limit fails, although both iterated limits equal .

44 Define Which statement is correct at the origin?

partial derivatives and total derivative Hard
A. is differentiable with total derivative .
B. is continuous and has every directional derivative, but is not differentiable.
C. is continuous, but its directional derivative along does not exist.
D. is discontinuous, although both partial derivatives exist.

45 Near , the equation defines as a function of and . What is its total differential at ?

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

46 Let At the origin, what are respectively?

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

47 Let , where , , and has continuous second derivatives. Which identity is correct?

chain rule Hard
A.
B.
C.
D.

48 Let , where has continuous second derivatives. Which expression equals , with all derivatives of evaluated at ?

chain rule Hard
A.
B.
C.
D.

49 The variables are transformed successively by What is the Jacobian determinant ?

chain rule Hard
A.
B.
C.
D.

50 If is twice continuously differentiable and homogeneous of degree , which second-order identity follows from Euler's theorem?

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

51 Suppose is homogeneous of degree on a cone excluding the origin, and define Which relation must satisfy?

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

52 Let be twice continuously differentiable and homogeneous of degree . What must be true of its Hessian at every point ?

Euler's theorem for homogeneous functions Hard
A. Its determinant must equal .
B. It must be the zero matrix.
C. Its determinant must equal .
D. Its trace must equal .

53 For which classification of its critical points is correct?

maxima and minima for a function of two variables Hard
A. All three critical points are non-strict local minima.
B. The origin is a local minimum, while and are saddles.
C. The origin is a local maximum, while and are minima.
D. The origin is a saddle, while and are global minima.

54 For the parameterized function how is the origin classified?

maxima and minima for a function of two variables Hard
A. It is a strict local minimum for and a saddle otherwise.
B. It is a non-strict local minimum for and a strict minimum for .
C. It is a strict local minimum for , non-strict for , and a saddle for .
D. It is a strict local minimum for every real .

55 The function has a critical point at the origin, where the Hessian test is inconclusive. What is the correct classification?

maxima and minima for a function of two variables Hard
A. A saddle point
B. A strict local maximum
C. A non-strict local minimum
D. A strict local minimum

56 Classify the origin for

maxima and minima for a function of two variables Hard
A. It is a strict local maximum.
B. It is a saddle point.
C. It is a non-strict local minimum.
D. It is a strict local minimum.

57 Find the constrained extrema of subject to .

Lagrange method of multiplier Hard
A. The maximum is and the minimum is .
B. The maximum is and the minimum is .
C. The maximum is and the minimum is .
D. The maximum is and the minimum is .

58 Minimize subject to the cusp constraint . Which statement is correct at the origin?

Lagrange method of multiplier Hard
A. The origin is not an extremum because no Lagrange multiplier exists.
B. The origin is a strict constrained minimum, but the standard multiplier equation fails there.
C. The origin is a strict constrained maximum detected by a zero multiplier.
D. The origin is a non-strict constrained minimum detected by a unique multiplier.

59 Determine the maximum and minimum of subject to

Lagrange method of multiplier Hard
A. The maximum is at , and the minimum is at .
B. The maximum is at , and the minimum is at .
C. The maximum is at , and the minimum is at .
D. The maximum is at , and the minimum is at .

60 Find the global minimum of subject to .

Lagrange method of multiplier Hard
A. , attained at
B. , attained at
C. , attained at only
D. , attained at
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