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Unit 3: Application of Derivatives - Practice Quiz

MTH165 — Mathematics For Engineers 50 Questions
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1 Which of the following is NOT a necessary condition for Rolle's Theorem to apply to a function on an interval ?

A. is continuous on the closed interval
B.
C. must be a polynomial function
D. is differentiable on the open interval

2 Find the value of that satisfies Rolle's Theorem for the function on the interval .

A. 1.5
B. 0
C. 2.5
D. 2

3 Geometrically, Rolle's Theorem implies that there is at least one point in where the tangent to the curve is:

A. Parallel to the y-axis
B. Parallel to the x-axis
C. Perpendicular to the x-axis
D. Passing through the origin

4 Lagrange's Mean Value Theorem states that if a function is continuous on and differentiable on , then there exists at least one such that:

A.
B.
C.
D.

5 If on , how many values of satisfy Rolle's Theorem?

A. 1
B. 3
C. 0
D. 2

6 Calculate the value of for Lagrange's Mean Value Theorem for on the interval .

A. 3
B. 2.5
C. 3.5
D.

7 Which theorem is a specific case of Lagrange's Mean Value Theorem where ?

A. Taylor's Theorem
B. Maclaurin's Theorem
C. Cauchy's Mean Value Theorem
D. Rolle's Theorem

8 Cauchy's Mean Value Theorem relates two functions and . The formula is given by:

A.
B.
C.
D.

9 Why is Rolle's Theorem not applicable to on the interval ?

A. The function is not continuous at
B. The function is not differentiable at
C. The function is not defined at
D.

10 The Maclaurin series expansion corresponds to the Taylor series expansion about the point:

A.
B.
C.
D.

11 What is the coefficient of the term in the Maclaurin series expansion of ?

A.
B. 2
C. 1
D.

12 Lagrange's form of the remainder in Taylor's theorem is given by:

A.
B.
C.
D.

13 The first non-zero term in the Maclaurin expansion of is:

A.
B.
C. 1
D.

14 Which of the following is the Maclaurin series for ?

A.
B.
C.
D.

15 Evaluate the limit using L'Hospital's rule: .

A. 0
B. -1
C. 1
D.

16 L'Hospital's Rule can be directly applied to which of the following indeterminate forms?

A.
B.
C. or
D.

17 Find .

A. 1
B. 0
C. 2
D. Undefined

18 Identify the indeterminate form of .

A.
B.
C.
D.

19 What is the value of ?

A.
B. 1
C.
D. 0

20 Which of the following functions does NOT satisfy the conditions of Rolle's Theorem on ?

A.
B.
C.
D.

21 Determine the maximum value of the function .

A. -1
B. 0
C. 1
D. -5

22 A function has a relative maximum at if:

A. and
B. and
C. and
D. and

23 Find the critical points of .

A.
B.
C.
D.

24 If at a point , and the concavity changes sign passing through , then is called:

A. A local minimum
B. A local maximum
C. A point of inflection
D. A saddle point

25 Calculate the value of for Cauchy's Mean Value Theorem for and on .

A. 1.5
B. 1.25
C. 2
D. 1.75

26 Evaluate .

A. 1
B. e
C.
D. 0

27 The Taylor series for cannot be expanded about (Maclaurin series) because:

A. is undefined
B. is always positive
C. The derivatives do not repeat
D. is not continuous at

28 What is the third term (coefficient of ) in the Taylor series expansion of about ?

A. 0
B.
C.
D. 1

29 For the function for , find the minimum value.

A. 2
B. 1
C. -2
D. 0

30 Evaluate .

A.
B.
C. 0
D. 1

31 Which term represents the error approximations when truncating a Taylor series after terms?

A. Remainder Term
B. Constant Term
C. Leading Term
D. Differential Term

32 Find the value of satisfying Rolle's theorem for on .

A.
B.
C.
D.

33 What is the condition for a function to be strictly increasing on an interval ?

A.
B.
C.
D.

34 Evaluate .

A.
B. 0
C. 3
D. 1

35 Which expansion correctly represents ?

A.
B.
C.
D.

36 Consider . At , the function has:

A. A local minimum
B. A vertical asymptote
C. A local maximum
D. A point of inflection

37 In the Taylor expansion of about , the term involving the second derivative is:

A.
B.
C.
D.

38 To evaluate , one should first:

A. Differentiate immediately
B. Substitute to get
C. Combine fractions to form a single rational expression
D. Apply the product rule

39 If is continuous on and , then:

A. is constant
B. There is a minimum between and
C. There is a maximum between and
D. There is a root between and

40 Find the value of for Lagrange's MVT for on .

A. 2
B. 0.5
C. 1
D. 4

41 The Maclaurin series for is:

A.
B.
C.
D.

42 Evaluate .

A. 1
B.
C.
D. 0

43 A stationary point is a point where:

A.
B.
C. The function is undefined
D.

44 What is the maximum area of a rectangle with perimeter 20?

A. 20
B. 10
C. 100
D. 25

45 Cauchy's form of the remainder in Taylor's theorem uses:

A. Complex numbers
B. where
C. Only integer values
D. where

46 Evaluate .

A. 1
B.
C.
D. 0

47 If , determine the nature of the point .

A. Local Minimum
B. Discontinuity
C. Point of Inflection
D. Local Maximum

48 The Taylor series expansion of exists only if:

A. is periodic
B. is infinitely differentiable at the point of expansion
C. is a polynomial
D. is continuous

49 Given , finding the maxima and minima involves solving:

A.
B.
C.
D.

50 Which of the following is true for on regarding Rolle's Theorem?

A. It fails because it is not continuous
B. It fails because
C. It satisfies all conditions
D. It fails because it is not differentiable
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