# Differential Calculus Questions and Answers – Generalized Mean Value Theorem

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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

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) $$\frac{x^n}{n!}f^{(n-1)}(θx)$$

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

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)= x2-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

Explanation: Given, f(x)= x2-x+1 , f(-1)=1+1+1=3
f'(x) = 2x-1,f'(-1)=2(-1)-1= -3
f”(x) = 2, f”(-1)=2
fn (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, fn (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) = 8x5-3x2-5x about x=2?
a) 232
b) 244
c) 234
d) 222

Explanation: Given, f(x) = 8x5-3x2-5x
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

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

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

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

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

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)+x2 $$\frac{f”}{2!}$$ (0)+⋯

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