This set of Differential and Integral Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Cauchy’s Mean Value Theorem”.
1. Cauchy’s Mean Value Theorem can be reduced to Lagrange’s Mean Value Theorem.
a) True
b) False
View Answer
Explanation: Cauchy’s Mean Value Theorem is the generalized form of Lagrange’s Mean Value Theorem and can be given by,
\(\frac{f'(a+θh)}{g'(a+θh)} = \frac{f(a+h)-f(a)}{g(a+h)-g(a)}, 0 < θ < 1\)
Hence, if g(x) = x, then CMV reduces to LMV.
2. Which of the following is not a necessary condition for Cauchy’s Mean Value Theorem?
a) The functions, f(x) and g(x) be continuous in [a, b]
b) The derivation of g'(x) be equal to 0
c) The functions f(x) and g(x) be derivable in (a, b)
d) There exists a value c Є (a, b) such that, \(\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}\)
View Answer
Explanation: Cauchy’s Mean Value theorem is given by, \(\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}\), where f(x) and g(x) be two functions which are derivable in [a, b] and g'(x)≠0 for any value of x in [a, b] and where c Є (a, b).
3. Cauchy’s Mean Value Theorem is also known as ‘Extended Mean Value Theorem’.
a) False
b) True
View Answer
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’.
4. The Mean Value Theorem was stated and proved by _______
a) Parameshvara
b) Govindasvami
c) Michel Rolle
d) Augustin Louis Cauchy
View Answer
Explanation: Augustin Louis Cauchy was a French Mathematician, Engineer and Physicist who first stated and proved the Mean Value Theorem.
5. Find the value of c which satisfies the Mean Value Theorem for the given function,
f(x)= x2+2x+1 on [1,2].
a) \(\frac{3}{2} \)
b) \(\frac{7}{2} \)
c) \(\frac{13}{2} \)
d) \(\frac{-13}{2} \)
View Answer
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
\(2c+2 = \frac{(4+4+1)-(1+2+1)}{2-1}=\frac{9-4}{1}= 5 \)
2c= 3
\(c= \frac{3}{2} \)
6. 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
View Answer
Explanation: From the Mean Value Theorem, we have, \(f'(c) = \frac{f(b)-f(a)}{b-a} \)
\(f'(c) = \frac{f(0)-f(-5)}{0-(-5)} \)
5f’ (c) = f(0)-8
f(0)=8 + 5f'(c) ≤ 8+5(2) = 18
f(0)=18
7. 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
View Answer
Explanation: From Cauchy’s Mean Value Theorem, we have, \(\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}\)
\(\frac{8-4}{12-3}=\frac{4}{6c}\)
\(6c=\frac{4*9}{4} \)
\(c=\frac{9}{6}=\frac{3}{2}=1.5\)
8. 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
View Answer
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.
9. 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
View Answer
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,
\(lim_{x→0}\frac{2}{1}=2\)
10. L’Hospital’s Rule was first discovered by Marquis de L’Hospital.
a) True
b) False
View Answer
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.
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