This set of Differential and Integral Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Differential Calculus Questions and Answers – Generalized Mean Value Theorem”.

1. Taylor’s theorem was stated by the mathematician _____________

a) Brook Taylor

b) Eva Germaine Rimington Taylor

c) Sir Geoffrey Ingram Taylor

d) Michael Eugene Taylor

View Answer

Explanation:

- Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician, best known for Taylor’s theorem and the Taylor series.
- Eva Germaine Rimington Taylor (1879–1966) was an English geographer and historian of science.
- Sir Geoffrey Ingram Taylor (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory.
- Michael Eugene Taylor (born 1946) is an American mathematician who is working in partial differential equations.

2. Lagrange’s Remainder for Maclaurin’s Theorem is given by _____________

a) \(\frac{x^n}{(n-1)!}f^{(n)}(θx) \)

b) \(\frac{x^n}{n!} f^{(n)}(θx)\)

c) \(\frac{x^{n-1}}{n!} f^{(n)}(θx)\)

d) \(\frac{x^n}{n!}f^{(n-1)}(θx)\)

View Answer

Explanation: Maclaurin’s Theorem is a special case of Taylor’s Theorem; hence Schlomilch’s Remainder for Maclaurin’s Theorem is given by, \(\frac{x^n(1-θ)^{n-p}}{(n-1)!p} f^{(n)}(θx).\) To obtain Lagrange’s Remainder for Maclaurin’s Theorem, we put p=n, which gives us, \(\frac{x^n}{n!} f^{(n)}(θx).\)

3. Cauchy’s Remainder for Maclaurin’s Theorem is given by \(\frac{x^n(1-θ)^{n-1}}{(n-1)!}f^{(n)}(θx)\).

a) True

b) False

View Answer

Explanation: Schlomilch’s Remainder for Maclaurin’s Theorem is given by, \(\frac{x^n(1-θ)^{n-p}}{(n-1)!p} f^{(n)}(θx).\) To obtain Cauchy’s Remainder for Maclaurin’s Theorem, we put p=1, which gives us,

\(\frac{x^n(1-θ)^{n-1}}{(n-1)!}f^{(n)}(θx).\)

4. What is the Taylor series expansion of f(x)= x^{2}-x+1 about the point x=-1?

a) f(x) = -3-3(x+1)+(x+1)^{2}

b) f(x) = -3-(x+1)+(x+1)^{2}

c) f(x) = -3-3(x+1)+2(x+1)^{2}

d) f(x) = -1-3(x+1)+2(x+1)^{2}

View Answer

Explanation: Given, f(x)= x

^{2}-x+1 , f(-1)=1+1+1=3

f'(x) = 2x-1,f'(-1)=2(-1)-1= -3

f”(x) = 2, f”(-1)=2

f

^{n}(x)=0, for n > 2

The Taylor series expansion of f(x) about x=a is,

f(x) = f(a)+(x-a) f'(a)+(x-a)

^{2}\(\frac{f”}{2!}\) (a)+⋯

Here a=-1,

f(x) = f(-1)+(x+1) f'(-1)+(x+1)

^{2}\(\frac{f”}{2!}\) (-1) since,for n > 2, f

^{n}(x) = 0.

f(x) = -3-3(x+1)+(x+1)

^{2}

5. What is the first term in the Taylor series expansion of f(x) = 8x^{5}-3x^{2}-5x about x=2?

a) 232

b) 244

c) 234

d) 222

View Answer

Explanation: Given, f(x) = 8x

^{5}-3x

^{2}-5x

First term in the Taylor series for f(x) is f(a). Here, a=2. Therefore,

f(2) = 8(2)

^{5}-3(2)

^{2}-5(2)=8(32)-3(4)-10=234.

6. In recurrence relation, each further term of a sequence or array is defined as a function of its succeeding terms.

a) True

b) False

View Answer

Explanation: A recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of its

**preceding terms**.

7. The initial condition for the recurrence relation of a factorial is ___________

a) 0!=0

b) 0!=1

c) 1!=1

d) 1!=0

View Answer

Explanation: Factorial of a number is defined by the recurrence relation as, n!=n(n-1)! for n>0 and the initial condition is 0!=1.

8. For the power series of the form, \(∑_{i=0}^∞ a_i z^i,\) which one of the following may not be true?

a) The series converges only for z=0

b) The series converges absolutely for all z

c) The series converges absolutely for all z in some finite open interval (-R, R) and diverges if z<-R or z>R

d) At the points z=R and z=-R, the series will diverge

View Answer

Explanation: A power series in a variable z is in the form of an infinite sum as given, i.e., \(∑_{i=0}^∞ a_i z^i,\), where ai are integers, real numbers, complex numbers, or any other quantities of a given type.

For any power series, one of the following is true:

i. The series converges only for z=0.

ii. The series converges absolutely for all z.

iii. The series converges absolutely for all z in some finite open interval (-R, R) and diverges if z<-R or z>R. At the points z=R and z=-R, the series may converge absolutely, converge conditionally, or diverge.

9. Maclaurin Series is named after ______________

a) Colin Maclaurin

b) Normand Maclaurin

c) Ian Maclaurin

d) Richard Cockburn Maclaurin

View Answer

Explanation:

- Colin Maclaurin (February 1698 – 14 June 1746), was a Scottish Mathematician. The Maclaurin Series was named after him.
- Normand Maclaurin (10 December 1835 – 24 August 1914) was a Scottish-born physician, company director, Australian politician and university administrator.
- Ian Maclaurin, born 30 March 1937, is a British businessman, who has been chairman of Vodafone and chairman and chief executive of Tesco.
- Richard Cockburn Maclaurin (June 5, 1870 – January 15, 1920) was a Scottish-born U.S. educator and mathematical physicist.

10. Maclaurin’s Theorem is a special type of Taylor’s Theorem.

a) False

b) True

View Answer

Explanation: The Taylor series expansion of f(x) about x=a is,

f(x)= f(a)+(x-a) f'(a)+(x-a)

^{2}\(\frac{f”}{2!}\) (a)+⋯

If we put x=0, we get the Maclaurin series which is given by, f(x)= f(0)+xf'(0)+x

^{2}\(\frac{f”}{2!}\) (0)+⋯

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