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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 ?

derivatives of standard functions and general rules of differentiation Easy
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B.
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D.

2 What is the derivative of with respect to ?

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

3 Which rule is used to differentiate the product ?

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

4 What is the derivative of ?

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

5 If and , what is for ?

derivatives of parametric forms Easy
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B.
C.
D.

6 For parametric equations and , which formula gives when ?

derivatives of parametric forms Easy
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B.
C.
D.

7 If , what is ?

derivatives of implicit functions Easy
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B.
C.
D.

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

derivatives of implicit functions Easy
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B.
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D.

9 Logarithmic differentiation is especially useful for differentiating which type of function?

logarithmic differentiation Easy
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B.
C.
D.

10 What is the derivative of for ?

logarithmic differentiation Easy
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B.
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D.

11 What is the second derivative of ?

higher order derivatives of simple functions Easy
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B.
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D.

12 What is the third derivative of ?

higher order derivatives of simple functions Easy
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C.
D.

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

Rolle's theorem Easy
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14 According to Rolle's theorem, there is at least one such that:

Rolle's theorem Easy
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C.
D.

15 According to the Lagrange mean value theorem, there exists such that:

mean value theorems Easy
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C.
D.

16 A Maclaurin series is a Taylor series expanded about which point?

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

17 What is the Maclaurin expansion of up to the term?

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

18 Which expression is an indeterminate form?

indeterminate forms Easy
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B.
C.
D.

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

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

20 If and , what does have at ?

maxima and minima Easy
A. A local maximum
B. A point of discontinuity
C. A local minimum
D. A vertical tangent

21 If , find .

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

22 Find the derivative of .

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

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

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

24 For and , find at .

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

25 For the curve , find at the point .

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

26 If , find at .

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

27 If , which expression equals ?

logarithmic differentiation Medium
A.
B.
C.
D.

28 For , find the derivative of .

logarithmic differentiation Medium
A.
B.
C.
D.

29 Find the fourth derivative of .

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

30 For , what is the th derivative of ?

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

31 For on , find the value of guaranteed by Rolle's theorem.

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

32 Why can Rolle's theorem not be applied to on ?

Rolle's theorem Medium
A. is not equal to
B. is not differentiable at
C. has no minimum on the interval
D. is not continuous at

33 For on , find the value of satisfying the Lagrange mean value theorem.

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

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

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

35 Which is the second-degree Taylor polynomial for about ?

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

36 Using the Maclaurin polynomial for through the cubic term, approximate .

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

37 Before evaluation, which indeterminate form is represented by ?

indeterminate forms Medium
A.
B.
C.
D.

38 Evaluate .

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

39 Evaluate .

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

40 A rectangle has perimeter units. What dimensions maximize its area?

maxima and minima Medium
A. units by units
B. units by units
C. units by units
D. units by units

41 For , find if

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. for every
D. for every

43 A curve is defined parametrically by where and . Find .

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

44 For the cycloid where and , determine .

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

45 The equation defines implicitly near . Find .

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

46 For the curve find at the point .

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

47 For , let Which expression equals ?

logarithmic differentiation Hard
A.
B.
C.
D.

48 For and an integer , determine

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

49 If , what is ?

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

50 For on , what is the complete set of points whose existence is identified by solving the Rolle condition ?

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

51 Suppose is twice differentiable on an interval containing and Which conclusion is guaranteed by repeated application of Rolle's theorem?

Rolle's theorem Hard
A. The function must have another zero in .
B. The equality must necessarily hold.
C. The derivative vanishes at each of .
D. There exists such that .

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 Find the smallest constant such that for every .

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

54 Let . Which option gives the fourth-degree Taylor polynomial for about and the sign of the remainder ?

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

55 Evaluate using a Maclaurin expansion:

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

56 For identify the indeterminate form before transformation and the value of .

indeterminate forms Hard
A. and
B. and
C. and
D. and

57 Evaluate

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

58 Determine

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

59 For , classify the extrema of

maxima and minima Hard
A. A global maximum occurs at , while a global minimum occurs at .
B. A unique global maximum occurs at with value ; no global minimum exists.
C. A unique global minimum occurs at with value ; no global maximum exists.
D. A unique global maximum occurs at with value ; no global minimum exists.

60 Find the absolute maximum and absolute minimum of on the interval .

maxima and minima Hard
A. Maximum at ; minimum at
B. Maximum at ; minimum at
C. Maximum at ; minimum at
D. Maximum at ; minimum at
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