1. Cauchy’s Mean Value Theorem is also known as ‘Extended Mean Value Theorem’.
a) False
b) True
Explanation: Mean Value Theorem is given by, \(\frac{f(b)-f(a)}{b-a} = f'(c),\) where c Є (a, b).
This theorem can be generalized to Cauchy’s Mean Value Theorem and hence CMV is also known as ‘Extended’ or ‘Second Mean Value Theorem’.
2. The Mean Value Theorem was stated and proved by _______
a) Parameshvara
b) Govindasvami
c) Michel Rolle
d) Augustin Louis Cauchy
Explanation: Augustin Louis Cauchy was a French Mathematician, Engineer and Physicist who first stated and proved the Mean Value Theorem.
3. Find the value of c which satisfies the Mean Value Theorem for the given function,
f(x)= x2+2x+1 on [1,2].
a) \(\frac{-7}{2} \)
b) \(\frac{7}{2} \)
c) \(\frac{13}{2} \)
d) \(\frac{-13}{2} \)
Explanation: Given function is, f(x)= x2+2x+1.
According to Mean Value Theorem,
\(f'(c) = \frac{f(b)-f(a)}{b-a} \)
f'(c)=2c+2
\(2c+2 = \frac{(1+2+1)-(4+4+1)}{2-1}=\frac{4-9}{1}= -5 \)
2c= -7
\(c= \frac{-7}{2} \)
4. What is the largest possible value of f(0), where f(x) is continuous and differentiable on the interval [-5, 0], such that f(-5)= 8 and f'(c)≤2.
a) 2
b) -2
c) 18
d) -18
Explanation: From the Mean Value Theorem, we have, \(f'(c) = \frac{f(b)-f(a)}{b-a} \)
\(f'(c) = \frac{f(0)-f(-5)}{-5-0} \)
-5f’ (c) = f(0)-8
f(0)=8 – 5f'(c) ≤ 8-5(2) = -2
f(0)=-2
5. What is the value of c which lies in [1, 2] for the function f(x)=4x and g(x)=3x2?
a) 1.6
b) 1.5
c) 1
d) 2
Explanation: From Cauchy’s Mean Value Theorem, we have, \(\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}\)
\(\frac{8-4}{12-3}=\frac{4}{6c}\)
\(6c=\frac{4*9}{4} \)
\(c=\frac{9}{6}=\frac{3}{2}=1.5\)
6. Which of the following method is used to simplify the evaluation of limits?
a) Cauchy’s Mean Value Theorem
b) Rolle’s Theorem
c) L’Hospital Rule
d) Fourier Transform
Explanation: L’Hospital’s Rule is used as a definitive way of simplification. The L’Hospital’s Rule does not directly evaluate the limits but only simplifies the evaluation.
7. What is the value of the given limit, \(\lim_{x\to 0}\frac{2}{x}\)?
a) 2
b) 0
c) 1/2
d) 3/2
Explanation: Given: \(\lim_{x\to 0}\frac{2}{x}\)
Using L’Hospital’s Rule, by differentiating both the numerator and denominator with respect to x,
\(lim_{x→0}\frac{2}{1}=2\)
8. L’Hospital’s Rule was first discovered by Marquis de L’Hospital.
a) True
b) False
Explanation: The L’Hospital’s Rule was first published in Marquis de L’Hospital’s book ‘Analyse des Infiniment Petits’, but the rule was discovered by Swiss Mathematician Johann Bernoulli.
9. 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.
10. 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, \(\frac{x^n}{n!} f^{(n)}(θx).\)