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Unit 2: Differential calculus and its applications - Practice Quiz

MTH165 — Mathematics For Engineers 60 Questions
0 Correct 0 Wrong 60 Left
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1 What is the derivative of with respect to ?

Derivatives of standard functions and general rules of differentiation Easy
A.
B.
C.
D.

2 What is ?

Derivatives of standard functions and general rules of differentiation Easy
A.
B.
C.
D.

3 According to the product rule, what is ?

Derivatives of standard functions and general rules of differentiation Easy
A.
B.
C.
D.

4 If and , what is for ?

Derivatives of parametric forms Easy
A.
B.
C.
D.

5 For parametric equations and , which formula gives when ?

Derivatives of parametric forms Easy
A.
B.
C.
D.

6 If , what is ?

Derivatives of implicit functions Easy
A.
B.
C.
D.

7 When differentiating implicitly with respect to , what is the result?

Derivatives of implicit functions Easy
A.
B.
C.
D.

8 Which first step is used in logarithmic differentiation of for ?

Logarithmic differentiation Easy
A.
B.
C.
D.

9 What is the derivative of for ?

Logarithmic differentiation Easy
A.
B.
C.
D.

10 What is the second derivative of ?

Higher order derivatives of simple functions Easy
A.
B.
C.
D.

11 What is the third derivative of ?

Higher order derivatives of simple functions Easy
A.
B.
C.
D.

12 If the conditions of Rolle's theorem hold for on , what must be true for some ?

Rolle's theorem Easy
A.
B.
C.
D.

13 Which endpoint condition is required by Rolle's theorem on ?

Rolle's theorem Easy
A.
B.
C.
D.

14 According to the Lagrange mean value theorem, what equation holds for some ?

Mean value theorems Easy
A.
B.
C.
D.

15 What is the coefficient of in the Taylor expansion of about ?

Taylor's theorems and Maclaurin theorems Easy
A.
B.
C.
D.

16 Which expression gives the Maclaurin series of ?

Taylor's theorems and Maclaurin theorems Easy
A.
B.
C.
D.

17 Which of the following is an indeterminate form?

Indeterminate forms Easy
A.
B.
C.
D.

18 Which expression represents another common indeterminate form?

Indeterminate forms Easy
A.
B.
C.
D.

19 Using L'Hospital's rule, what is ?

L'Hospital's rule Easy
A.
B.
C.
D.

20 If and , what does have at ?

Maxima and minima Easy
A. A local minimum
B. A local maximum
C. A point of discontinuity
D. A vertical asymptote

21 If , what is the value of ?

Derivatives of standard functions and general rules of differentiation Medium
A.
B.
C.
D.

22 Find the derivative of for values of where the function is defined.

Derivatives of standard functions and general rules of differentiation Medium
A.
B.
C.
D.

23 A curve is given parametrically by and . Find at .

Derivatives of parametric forms Medium
A.
B.
C.
D.

24 For the parametric curve and , find at .

Derivatives of parametric forms Medium
A.
B.
C.
D.

25 If , find at the point .

Derivatives of implicit functions Medium
A.
B.
C.
D.

26 For the circle , find at the point .

Derivatives of implicit functions Medium
A.
B.
C.
D.

27 If for , find at .

Logarithmic differentiation Medium
A.
B.
C.
D.

28 Let . What is at ?

Logarithmic differentiation Medium
A.
B.
C.
D.

29 If , find .

Higher order derivatives of simple functions Medium
A.
B.
C.
D.

30 Find the fourth derivative of .

Higher order derivatives of simple functions Medium
A.
B.
C.
D.

31 For on , which value of satisfies the conclusion of Rolle's theorem?

Rolle's theorem Medium
A.
B.
C. No such value exists because the function has two distinct roots at the endpoints
D.

32 Apply the Lagrange mean value theorem to on . Find the corresponding value of .

Mean value theorems Medium
A.
B.
C.
D.

33 For and on , Cauchy's mean value theorem guarantees a point satisfying . Find .

Mean value theorems Medium
A.
B.
C.
D.

34 Using the Maclaurin expansion of through the term in , approximate .

Taylor's theorems and Maclaurin theorems Medium
A.
B.
C.
D.

35 In the Taylor expansion of about , what is the coefficient of ?

Taylor's theorems and Maclaurin theorems Medium
A.
B.
C.
D.

36 Which limit has the indeterminate form before evaluation?

Indeterminate forms Medium
A.
B.
C.
D.

37 Evaluate .

L'Hospital's rule Medium
A.
B. The limit does not exist because both numerator and denominator approach zero
C.
D.

38 Evaluate using L'Hospital's rule.

L'Hospital's rule Medium
A.
B.
C.
D.

39 For , which statement correctly classifies its stationary points?

Maxima and minima Medium
A. The function has stationary points at and , but neither point is an extremum
B. A local minimum of at and a local maximum of at
C. Local maxima occur at both and
D. A local maximum of at and a local minimum of at

40 Find the absolute minimum and maximum values of on the closed interval .

Maxima and minima Medium
A. Minimum at ; maximum at
B. Minimum at ; maximum at
C. Minimum at ; maximum at
D. Minimum at ; maximum at and

41 For , let . What is at every point where the displayed expression is differentiable?

Derivatives of standard functions and general rules of differentiation Hard
A.
B.
C.
D.

42 Let . What is for ?

Derivatives of standard functions and general rules of differentiation Hard
A.
B.
C.
D.

43 A curve is given parametrically by and . What is at the point corresponding to ?

Derivatives of parametric forms Hard
A.
B.
C.
D.

44 For the parametric curve and , where and , determine .

Derivatives of parametric forms Hard
A.
B.
C.
D.

45 The equation defines locally as a function of near . What is at ?

Derivatives of implicit functions Hard
A.
B.
C.
D.

46 Suppose defines as a twice differentiable function of near . Determine .

Derivatives of implicit functions Hard
A.
B.
C.
D.

47 For , define . Which expression equals ?

Logarithmic differentiation Hard
A.
B.
C.
D.

48 Let . What is ?

Logarithmic differentiation Hard
A.
B.
C.
D.

49 For a positive integer , which formula gives the th derivative of ?

Higher order derivatives of simple functions Hard
A.
B.
C.
D.

50 Let and . Which expression is the th derivative of ?

Higher order derivatives of simple functions Hard
A.
B.
C.
D.

51 For on , which statement correctly describes the applicability of Rolle's theorem?

Rolle's theorem Hard
A. The theorem does not apply because is not differentiable at , and no satisfies .
B. The theorem applies separately on and , producing .
C. The theorem does not apply because is not continuous at , although .
D. The theorem applies because , and the required point is .

52 Apply Cauchy's mean value theorem to and on . What value of satisfies the theorem's conclusion?

Mean value theorems Hard
A.
B.
C.
D.

53 Let be continuous on and differentiable on , with and . Which conclusion is guaranteed by the mean value theorem?

Mean value theorems Hard
A. There exists such that .
B. There exists such that .
C. There exists such that .
D. There exists a unique such that .

54 What is the coefficient of in the Maclaurin expansion of ?

Taylor's theorems and Maclaurin theorems Hard
A.
B.
C.
D.

55 The fourth-degree Maclaurin polynomial for is used to approximate . Using the Lagrange remainder and the maximum of on , which bound follows for the absolute error?

Taylor's theorems and Maclaurin theorems Hard
A.
B.
C.
D.

56 The expression has the indeterminate form as . What is its limit?

Indeterminate forms Hard
A.
B.
C.
D.

57 Evaluate .

L'Hospital's rule Hard
A.
B.
C.
D.

58 Evaluate .

L'Hospital's rule Hard
A.
B.
C.
D.

59 Consider on . Which statement correctly classifies its stationary points and global extrema?

Maxima and minima Hard
A. is not an extremum, while is the global minimum with value .
B. is a local maximum, while is the global minimum with value .
C. is not an extremum, while is a local maximum with value .
D. is a local minimum, while is the global maximum with value .

60 For constants , the function is defined on . At which point is its global maximum attained, and what is the maximum value?

Maxima and minima Hard
A. , with maximum
B. , with maximum
C. , with maximum
D. , with maximum
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