Differential Calculus Questions and Answers – Cauchy’s Mean Value Theorem

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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

Answer: a
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.
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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

Answer: b
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

Answer: b
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’.
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4. The Mean Value Theorem was stated and proved by _______
a) Parameshvara
b) Govindasvami
c) Michel Rolle
d) Augustin Louis Cauchy
View Answer

Answer: d
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) \(\frac{-7}{2} \)
b) \(\frac{7}{2} \)
c) \(\frac{13}{2} \)
d) \(\frac{-13}{2} \)
View Answer

Answer: a
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
\(c= \frac{-7}{2} \)
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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

Answer: b
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

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

Answer: b
Explanation: From Cauchy’s Mean Value Theorem, we have, \(\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}\)
\(\frac{8-4}{12-3}=\frac{4}{6c}\)
\(6c=\frac{4*9}{4} \)
\(c=\frac{9}{6}=\frac{3}{2}=1.5\)
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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

Answer: c
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

Answer: a
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,
\(lim_{x→0}⁡\frac{2}{1}=2\)
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10. L’Hospital’s Rule was first discovered by Marquis de L’Hospital.
a) True
b) False
View Answer

Answer: b
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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