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TN 12th Computer Applications மின்னணு செலுத்தல் முறைகள் Sample Question Papers Study Material - QB365 Set A

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12th Standard English Medium Maths Subject Application of Differential Calculus Creative 3 Mark Questions with Solution Part - II

Published on: 19/06/2021

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Std: 12th Standard - Maths

chapter: - Creative Questions

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Questions:

Generic 3

5 x 3 = 15

Write down the Taylor series expansion of the function cos x in ascending powers \(x-\frac { \pi }{ 4 } \) upto three non-zero terms.

Find the angle between two curves 2x²+y²=20 and x²-4y²+8=0

The volume of a cube is increasing at the rate of 8cm³/s.How fast is the surface area increasing when the length of an edge is 12cm?

Verify LMV theorem for f(x) = x³ - 2x² - x + 3 in [0, 1].

Prove that \(\frac { x }{ 1+x } \) < < log (1+ x) for x > 0.

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Answers Key:

Generic 3

5 x 3 = 15

\(\frac { 1 }{ \sqrt { 2 } } \left[ 1-\frac { \left( x-\frac { \pi }{ 4 } \right) }{ 1! } -\frac { \left( x-\frac { \pi }{ 4 } \right) ^{ 2 } }{ 2! } \right] \)

\(\theta =\frac { \pi }{ 2 } \)

\(\frac { 8 }{ 3 } { cm }^{ 2 }/sec\)

f(x) = x³-2x²-x+3
f'(x) = 3x² - 4x - 1
f'(c) = 3c² - 4c- 1
f(a) = f(0) = 3
f(b) = f(1) = 1³-2-1+3 = 1
Then, if atleast one C \(\in \) (0, 1) such that f'(c) = \(\frac { f(b)-f(a) }{ b-a } \)
⇒ 3c²-4c-1 = \(\frac { 1-3 }{ 1-0 } \)
⇒ 3c²-4c+1 = 0
⇒ c = \(\frac { 4\pm \sqrt { 16-4(3) } }{ 2(3) } \)
⇒ c = \(\frac { 4\pm 2 }{ 6 } =\frac { 6 }{ 6 } \) or \(\frac { 2 }{ 6 } \)
⇒ 1 or \(\frac { 1 }{ 3 } \).

Let f(x) = log(1+x)-\(\frac { x }{ 1+x } \)
f'(x) =\(\frac { 1 }{ 1+x } -\frac { (1+x)-x }{ (1+x)^{ 2 } } \)
=\(\frac { 1 }{ 1+x } -\frac { 1 }{ (1+x)^{ 2 } } \)
=\(\frac { x }{ (1+x)^{ 2 } } \) > 0 for x>0
∴ f(x) is strictly increasing in (0, ∞)
∴ x>0 ⇒ f(x) > f(0)
⇒ log (1+x) - \(\frac { x }{ 1+x } \) >0
⇒ log(1+x) > \(\frac { x }{ 1+x } \)
⇒ \(\frac { x }{ 1+x } \) < log (1+x).