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.

is differentiable on the open interval

C.

D.

must be a polynomial function

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

A.

1.5

B.

2

C.

2.5

D.

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

A.

Perpendicular to the x-axis

B.

Parallel to the x-axis

C.

Parallel to the y-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.

B.

1

C.

2

D.

3

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

A.

2.5

B.

3

C.

3.5

D.

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

A.

Cauchy's Mean Value Theorem

B.

Taylor's Theorem

C.

Rolle's Theorem

D.

Maclaurin'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.

B.

The function is not continuous at

C.

The function is not differentiable at

D.

The function is not defined at

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.

1

B.

C.

2

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.

1

B.

C.

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.

B.

1

C.

D.

-1

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

A.

B.

C.

or

D.

17 $Find .$

A.

B.

1

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.

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.

1

C.

-5

D.

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 saddle point

B.

A point of inflection

C.

A local maximum

D.

A local minimum

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

A.

1.5

B.

1.25

C.

1.75

D.

2

26 $Evaluate .$

A.

B.

1

C.

D.

e

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

A.

is always positive

B.

is undefined

C.

is not continuous at

D.

The derivatives do not repeat

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

A.

B.

C.

D.

1

29 $For the function for, find the minimum value.$

A.

1

B.

2

C.

D.

-2

30 $Evaluate .$

A.

1

B.

C.

D.

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

A.

Leading Term

B.

Remainder Term

C.

Differential Term

D.

Constant 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.

3

B.

C.

D.

1

35 $Which expansion correctly represents ?$

A.

B.

C.

D.

36 $Consider . At, the function has:$

A.

A local maximum

B.

A local minimum

C.

A point of inflection

D.

A vertical asymptote

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.

Apply the product rule

C.

Combine fractions to form a single rational expression

D.

Substitute to get $0$

39 $If is continuous on and, then:$

A.

There is a root between and

B.

There is a maximum between and

C.

There is a minimum between and

D.

is constant

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

A.

1

B.

2

C.

4

D.

0.5

41 $The Maclaurin series for is:$

A.

B.

C.

D.

42 $Evaluate .$

A.

B.

C.

1

D.

43 $A stationary point is a point where:$

A.

B.

C.

D.

The function is undefined

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

A.

20

B.

10

C.

25

D.

100

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

A.

where

B.

where

C.

Only integer values

D.

Complex numbers

46 $Evaluate .$

A.

B.

1

C.

D.

47 $If, determine the nature of the point .$

A.

Local Maximum

B.

Local Minimum

C.

Point of Inflection

D.

Discontinuity

48 $The Taylor series expansion of exists only if:$

A.

is continuous

B.

is infinitely differentiable at the point of expansion

C.

is a polynomial

D.

is periodic

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 satisfies all conditions

B.

It fails because

C.

It fails because it is not continuous

D.

It fails because it is not differentiable

Unit 3 - Practice Quiz