RD Sharma Class 12 Ex 14.1 Solutions Chapter 14 Differentials, Errors, and Approximations

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Textbook	NCERT
Class	Class 12th
Subject	Maths
Chapter	14
Exercise	14.1
Category	RD Sharma Solutions

Table of Contents

RD Sharma Class 12 Ex 14.1 Solutions Chapter 14 Differentials, Errors, and Approximations

Question 1: If y=sin x and x changes from π/2 to 22/14, what is the approximate change in y?

Solution:

According to the given condition,
x = π/2, and
x+△x = 22/14
△x = 22/14-x = 22/14 – π/2
As, y = sin x
$\frac{dy}{dx}$ = cos x
$(\frac{dy}{dx})_{x=\frac{\pi}{2}}$ = cos (π/2) = 0
△y = $(\frac{dy}{dx})_{x=\frac{\pi}{2}}$ △x
△y = 0 △x
△y = 0 (22/14 – π/2)
△y = 0
Hence, there will be no change in y.

Question 2: The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume?

Solution:

According to the given condition,
Let’s take radius as x
x = 10, and
Let △x be the error in the radius and △y be the error in the volume
x+△x = 9.8
△x = 9.8-x = 9.8-10 = -0.2
As, Volume of sphere = $\frac{4}{3}\pi x^3$
$\frac{dy}{dx} = \frac{4}{3}(3 \pi x^2)$ = 4πx²
$(\frac{dy}{dx})_{x=10}$ = 4π(10)² = 400 π
△y = $(\frac{dy}{dx})_{x=10}$ △x
△y = (400 π) (-0.2)
△y = -80 π
Hence, approximate decrease in its volume will be -80 π cm³

Question 3: The circular metal plate expands under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

Solution:

According to the given condition,
Let’s take radius as x
x = 10, and
Let △x be the error in the radius and △y be the error in the surface area
△x/x × 100 = k
△x = (k × 10)/100 = k/10
As, Area of circular metal = πx²
$\frac{dy}{dx}$ = π(2x) = 2πx
$(\frac{dy}{dx})_{x=10}$ = 2π(10) = 20 π
△y = $(\frac{dy}{dx})_{x=10}$ △x
△y = (20 π) (k/10)
△y = 2kπ
Hence, approximate increase in the area of the plate is 2kπ cm²

Question 4: Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube.

Solution:

According to the given condition,
Let △x be the error in the length and △y be the error in the surface area
Let’s take length as x
△x/x × 100 = 1
△x = x/100
x+△x = x+(x/100)
As, surface area of the cube = 6x²
$\frac{dy}{dx}$ = 6(2x) = 12x
△y = $(\frac{dy}{dx})$ △x
△y = (12x) (x/100)
△y = 0.12 x²
So, △y/y = 0.12 x²/6 x²= 0.02
Percentage change in y = △y/y × 100 = 0.02 × 100 = 2
Hence, the percentage error in calculating the surface area of a cubical box is 2%

Question 5: If there is an error of 0.1% in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere.

Solution:

According to the given condition,
As, Volume of sphere = $\frac{4}{3}\pi x^3$
Let △x be the error in the radius and △y be the error in the volume
△x/x × 100 = 0.1
△x/x = 1/1000
As, y = $\frac{4}{3}\pi x^3$
$\frac{dy}{dx} = \frac{4}{3}(3 \pi x^2)$ = 4πx²
dy = 4πx²dx
△y = (4πx²) △x
Change in volume,
△y/y = $\frac{(4\pi x^2) \triangle x}{\frac{4}{3}\pi x^3}$
△y/y = $\frac{3}{x} \triangle x$
△y/y = $3(\frac{\triangle x}{x} )$ = 3(0.001) = 0.003
Percentage change in y = △y/y × 100 = 0.003 × 100 = 0.3
Hence, approximately the percentage error in the calculation of the volume of the sphere is 0.3%

Question 6: The pressure p and the volume v of a gas are connected by the relation pv^1.4 = constant. Find the percentage error in p corresponding to a decrease of 1/2% in v.

Solution:

According to the given condition,

$\frac{\triangle v}{v} \times 100$ = – 1/2%

pv^1.4 = constant = k(say)

Taking log on both sides, we get

log(pv^1.4) = log (k)

log(p)+log(v^1.4) = log k

log(p) + 1.4 log(v) = log k

Differentiating wrt v, we get

$\frac{1}{p} \frac{dp}{dv} + 1.4 (\frac{1}{p}) = 0$

$\frac{dp}{p} = \frac{-1.4 dv}{v}$

Percentage change in p = △p/p × 100 = $\frac{-1.4 \triangle v}{v}$ × 100 = -1.4 $(\frac{\triangle v}{v} \times 100)$

= -1.4 $(\frac{-1}{2})$

= 0.7 %

Hence, percentage error in p is 0.7%.

Question 7: The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase?

Solution:

According to the given condition,
Let h be the height, y be the surface area. V be the volume, l be the slant height and r be the radius of the cone.
Let △h be the change in the height. △r be the change in the radius of base and △l be the change in slant height.
Semi-vertical angle remaining the same.
△h/h = △r/r = △l/l
and,
△h/h × 100 = k
△h/h × 100 = △r/r × 100 = △l/l × 100 = k

(i) in total surface area, and

Solution:

Total surface area of the cone
y = πrl + πr²
Differentiating both the sides wrt r, we get
$\frac{dy}{dr}$ = πl + πr $\frac{dl}{dr}$ + 2πr
$\frac{dy}{dr}$ = πl + πr $\frac{l}{r}$ + 2πr
$\frac{dy}{dr}$ = πl + πl + 2πr
$\frac{dy}{dr}$ = 2πl + 2πr = 2π(r+l)
△y = $(\frac{dy}{dr})$ △r
△y = (2π(r+l)) $(\frac{kr}{100}) = \frac{2\pi kr(r+l)}{100}$
Percentage change in y = △y/y × 100 = $\frac{\frac{2\pi kr(r+l)}{100}}{\pi r(l+r)}$ × 100
= 2k %
Hence, percentage increase in total surface area of cone 2k%.

(ii) in the volume assuming that k is small?

Solution:

Volume of cone (y) = $\frac{1}{3}\pi r^2h$
Differentiating both the sides wrt h, we get
$\frac{dy}{dh} = \frac{1}{3}\pi$ (r² + h(2r $\frac{dr}{dh}$ )
$\frac{dy}{dh} = \frac{1}{3}\pi$ (r² + h(2r $\frac{r}{h}$ )
$\frac{dy}{dh} = \frac{1}{3}\pi$ (r² + 2r²)
$\frac{dy}{dh}$ = πr²
△y = $(\frac{dy}{dh})$ △h
△y = (πr²) $(\frac{kh}{100}) = \frac{kh \pi r^2}{100}$
Percentage change in y = △y/y × 100 = $\frac{\frac{kh \pi r^2}{100}}{\frac{1}{3}\pi r^2h}$ × 100
= 3k %
Hence, percentage increase in the volume of cone 3k%.

Question 8: Show that the relative error in computing the volume of a sphere, due to an error in measuring the radius, is approximately equal to the three times the relative error in the radius.

Solution:

According to the given condition,
Let △x be the error in the radius and △y be the error in volume.
Volume of cone (y) = $\frac{4}{3}\pi x^3$
Differentiating both the sides wrt x, we get
$\frac{dy}{dx} = \frac{4}{3}\pi$ (3x²)
$\frac{dy}{dx}$ = 4πx²
△y = $(\frac{dy}{dx})$ △x
△y = (4πx²) (△x)
△y/y = $\frac{(4 \pi x^2) (\triangle x)}{\frac{4}{3}\pi x^3}$
△y/y = $3 \frac{\triangle x}{x}$
Hence proved!!

Question 9: Using differentials, find the approximate values of the following:

(i) $\sqrt{25.02}$

Solution:

Considering the function as

y = f(x) = $\sqrt{x}$

Taking x = 25, and

x+△x = 25.02

△x = 25.02-25 = 0.2

$y = \sqrt{x}$

$y_{(x=25)} = \sqrt{25} = 5$

$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$

$(\frac{dy}{dx})_{x=25} = \frac{1}{2\sqrt{25}} = \frac{1}{10}$

△y = dy = $(\frac{dy}{dx})_{x=25}$ dx

△y = $(\frac{1}{10})$ △x

△y = $(\frac{1}{10})$ (0.02) = 0.002

Hence, $\sqrt{25.02}$ = y+△y = 5 + 0.002 = 5.002

(ii) $(0.009)^{\frac{1}{3}}$

Solution:

Considering the function as

y = f(x) = $(x)^{\frac{1}{3}}$

Taking x = 0.008, and

x+△x = 0.009

△x = 0.009-0.008 = 0.001

$y = (x)^{\frac{1}{3}}$

$y_{(x=0.008)} = (0.008)^{\frac{1}{3}} = 0.2$

$\frac{dy}{dx} = \frac{1}{3(x)^{\frac{2}{3}}}$

$(\frac{dy}{dx})_{x=0.008} = \frac{1}{3(0.008)^{\frac{2}{3}}} = \frac{1}{0.12}$

△y = dy = $(\frac{dy}{dx})_{x=0.008}$ dx

△y = $(\frac{1}{0.12})$ △x

△y = $(\frac{1}{0.12})$ (0.001) = $\frac{1}{120}$ = 0.008333

Hence, $(0.009)^{\frac{1}{3}}$ = y+△y = 0.2 + 0.008333 = 0.208333

(iii) $(0.007)^{\frac{1}{3}}$

Solution:

Considering the function as

y = f(x) = $(x)^{\frac{1}{3}}$

Taking x = 0.008, and

x+△x = 0.007

△x = 0.007-0.008 = -0.001

$y = (x)^{\frac{1}{3}}$

$y_{(x=0.008)} = (0.008)^{\frac{1}{3}}$ = 0.2

$\frac{dy}{dx} = \frac{1}{3(x)^{\frac{2}{3}}}$

$(\frac{dy}{dx})_{x=0.008} = \frac{1}{3(0.008)^{\frac{2}{3}}} = \frac{1}{0.12}$

△y = dy = $(\frac{dy}{dx})_{x=0.008}$ dx

△y = $(\frac{1}{0.12})$ △x

△y = $(\frac{1}{0.12})$ (-0.001) = $\frac{-1}{120}$ = -0.008333

Hence, $(0.007)^{\frac{1}{3}}$ = y+△y = 0.2 + (-0.008333) = 0.191667

(iv) $\sqrt{401}$

Solution:

Considering the function as

y = f(x) = $\sqrt{x}$

Taking x = 400, and

x+△x = 401

△x = 401-400 = 1

$y = \sqrt{x}$

$y_{(x=400)} = \sqrt{400} = 20$

$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$

$(\frac{dy}{dx})_{x=400} = \frac{1}{2\sqrt{400}} = \frac{1}{40}$

△y = dy = $(\frac{dy}{dx})_{x=400}$ dx

△y = $(\frac{1}{40})$ △x

△y = $(\frac{1}{40})$ (1) = 0.025

Hence, $\sqrt{401}$ = y+△y = 20 + 0.025 = 20.025

(v) $(15)^{\frac{1}{4}}$

Solution:

Considering the function as

y = f(x) = $(x)^{\frac{1}{4}}$

Taking x = 16, and

x+△x = 15

△x = 15-16 = -1

$y = (x)^{\frac{1}{4}}$

$y_{(x=16)} = (16)^{\frac{1}{4}} = 2$

$\frac{dy}{dx} = \frac{1}{4(x)^{\frac{3}{4}}}$

$(\frac{dy}{dx})_{x=16} = \frac{1}{4(16)^{\frac{3}{4}}} = \frac{1}{32}$

△y = dy = $(\frac{dy}{dx})_{x=16}$ dx

△y = $(\frac{1}{32})$ △x

△y = $(\frac{1}{32})$ (-1) = $\frac{-1}{32}$ = -0.03125

Hence, $(15)^{\frac{1}{4}}$ = y+△y = 0.2 + (-0.03125) = 1.96875

(vi) $(255)^{\frac{1}{4}}$

Solution:

Considering the function as

y = f(x) = $(x)^{\frac{1}{4}}$

Taking x = 256, and

x+△x = 255

△x = 255-256 = -1

$y = (x)^{\frac{1}{4}}$

$y_{(x=256)} = (256)^{\frac{1}{4}}$ = 4

$\frac{dy}{dx} = \frac{1}{4(x)^{\frac{3}{4}}}$

$(\frac{dy}{dx})_{x=256} = \frac{1}{4(256)^{\frac{3}{4}}} = \frac{1}{256}$

△y = dy = $(\frac{dy}{dx})_{x=256}$ dx

△y = $(\frac{1}{256})$ △x

△y = $(\frac{1}{256})$ (-1) = $\frac{-1}{256}$ = -0.003906

Hence, $(255)^{\frac{1}{4}}$ = y+△y = 0.2 + (-0.003906) = 3.9961

(vii) $\frac{1}{(2.002)^2}$

Solution:

Considering the function as

y = f(x) = $\frac{1}{x^2}$

Taking x = 2, and

x+△x = 2.002

△x = 2.002-2 = 0.002

$y = \frac{1}{x^2}$

$y_{(x=2)} = \frac{1}{2^2} = \frac{1}{4}$

$\frac{dy}{dx} = \frac{-2}{x^3}$

$(\frac{dy}{dx})_{x=2} = \frac{-2}{2^3} = \frac{-1}{4}$

△y = dy = $(\frac{dy}{dx})_{x=2}$ dx

△y = $(\frac{-1}{4})$ △x

△y = $(\frac{-1}{4})$ (0.002) = -0.0005

Hence, $\frac{1}{(2.002)^2}$ = y+△y = $\frac{1}{4}$ + (-0.005) = 0.2495

(viii) log_e 4.04, it being given that log₁₀ 4=0.6021 and log₁₀ e=0.4343

Solution:

Considering the function as

y = f(x) = log_e x

Taking x = 4, and

x+△x = 4.04

△x = 4-4.04 = 0.04

y = log_e x

$y_{(x=4)}$ = log_e 4 = $\frac{log_{10}4}{log_{10}e} = \frac{0.6021}{0.4343}$ = 1.386368

$\frac{dy}{dx} = \frac{1}{x}$

$(\frac{dy}{dx})_{x=4} = \frac{1}{4}$

△y = dy = $(\frac{dy}{dx})_{x=4}$ dx

△y = $(\frac{1}{4})$ △x

△y = $(\frac{1}{4})$ (0.04) = 0.01

Hence, log_e 4.04 = y+△y = 1.386368 + 0.01 = 1.396368

(ix) log_e 10.02, it being given that log_e 10=2.3026

Solution:

Considering the function as

y = f(x) = log_e x

Taking x = 10, and

x+△x = 10.02

△x = 10.02-10 = 0.02

y = log_e x

$y_{(x=10)}$ = log_e 10 = 2.3026

$\frac{dy}{dx} = \frac{1}{x}$

$(\frac{dy}{dx})_{x=10} = \frac{1}{10}$

△y = dy = $(\frac{dy}{dx})_{x=10}$ dx

△y = $(\frac{1}{10})$ △x

△y = $(\frac{1}{10})$ (0.02) = 0.002

Hence, log_e 10.02 = y+△y = 2.3026 + 0.002 = 2.3046

(x) log₁₀ 10.1, it being given that log₁₀ e=0.4343

Solution:

Considering the function as

y = f(x) = log₁₀ x

Taking x = 10, and

x+△x = 10.1

△x = 10.1-10 = 0.1

y = log₁₀ x = $\frac{log_ex}{log_e10}$

$y_{(x=10)}$ = log₁₀ 10 = 1

$\frac{dy}{dx} = \frac{1}{2.3025x}$

$(\frac{dy}{dx})_{x=10} = \frac{1}{23.025}$

△y = dy = $(\frac{dy}{dx})_{x=10}$ dx

△y = $(\frac{1}{23.025})$ △x

△y = $(\frac{1}{23.025})$ (0.1) = 0.004343

Hence, log_e 10.1 = y+△y = 1 + 0.004343 = 1.004343

(xi) cos 61°, it being given that sin 60°=0.86603 and 1°=0.01745 radian.

Solution:

Considering the function as
y = f(x) = cos x
Taking x = 60°, and
x+△x = 61°
△x = 61°-60° = 1° = 0.01745 radian
y = cos x
$y_{(x=60 \degree)}$ = cos 60° = 0.5
$\frac{dy}{dx}$ = – sin x
$(\frac{dy}{dx})_{x=60\degree}$ = – sin 60° = -0.86603
△y = dy = $(\frac{dy}{dx})_{x=60\degree}$ dx
△y = (-0.86603) △x
△y = (-0.86603) (0.01745) = -0.01511
Hence, cos 61° = y+△y = 0.5 + (-0.01511) = 0.48489

(xii) $\frac{1}{\sqrt{25.1}}$

Solution:

Considering the function as

y = f(x) = $\frac{1}{\sqrt{x}}$

Taking x = 25, and

x+△x = 25.1

△x = 25.1-25 = 0.1

$y = \frac{1}{\sqrt{x}}$

$y_{(x=25)} = \frac{1}{\sqrt{25}} = \frac{1}{5}$

$\frac{dy}{dx} = \frac{-1}{2(x)^{\frac{3}{2}}}$

$(\frac{dy}{dx})_{x=25} = \frac{-1}{2(25)^{\frac{3}{2}}} = \frac{-1}{250}$

△y = dy = $(\frac{dy}{dx})_{x=25}$ dx

△y = $(\frac{-1}{250})$ △x

△y = $(\frac{-1}{250})$ (0.1) = $\frac{-1}{2500}$ = -0.0004

Hence, $\frac{1}{\sqrt{25.1}}$ = y+△y = $\frac{1}{5}$ + (-0.0004) = 0.1996

(xiii) $sin (\frac{22}{14})$

Solution:

Considering the function as
y = f(x) = sin x
Taking x = 22/7, and
x+△x = 22/14
△x = 22/14-22/7 = -22/14
sin (-22/14) = -1
y = sin x
$y_{(x=22/7)}$ = sin (22/7) = 0
$\frac{dy}{dx}$ = cos x
$(\frac{dy}{dx})_{x=22/7}$ = cos (22/7)= -1
△y = dy = $(\frac{dy}{dx})_{x=22/7}$ dx
△y = (-1) △x
△y = (-1) (-1) = 1
Hence, sin(22/14) = 0+1 = 1
Question 9: Using differentials, find the approximate values of the following:
(xiv) $cos (\frac{11\pi}{36})$
Solution:

Considering the function as
y = f(x) = cos x
Taking x = π/3, and
x+△x = 11π/36
△x = 11π/36-π/3 = -π/36
y = cos x
$y_{(x=\pi/3)}$ = cos (π/3) = 0.5
$\frac{dy}{dx}$ = – sin x
$(\frac{dy}{dx})_{x=\pi/3}$ = – sin (π/3) = -0.86603
△y = dy = $(\frac{dy}{dx})_{x=\pi/3}$ dx
△y = (-0.86603) (-π/36)
△y = 0.0756
Hence, $cos (\frac{11\pi}{36})$ = 0.5+0.0756 = 0.5755
(xv) $(80)^{\frac{1}{4}}$
Solution:
Considering the function as
y = f(x) = $(x)^{\frac{1}{4}}$
Taking x = 81, and
x+△x = 80
△x = 80-81 = -1
$y = (x)^{\frac{1}{4}}$
$y_{(x=81)} = (81)^{\frac{1}{4}}= 3$
$\frac{dy}{dx} = \frac{1}{4(x)^{\frac{3}{4}}}$
$(\frac{dy}{dx})_{x=81} = \frac{1}{4(81)^{\frac{3}{4}}} = \frac{1}{108}$
△y = dy = $(\frac{dy}{dx})_{x=81}$ dx
△y = $(\frac{1}{108})$ △x
△y = $(\frac{1}{108})$ (-1) = $\frac{-1}{108}$ = -0.009259
Hence,
$(80)^{\frac{1}{4}}$ = y+△y = 3 + (-0.009259) = 2.99074
(xvi) $(29)^{\frac{1}{3}}$
Solution:
Considering the function as
y = f(x) = $(x)^{\frac{1}{3}}$
Taking x = 27, and
x+△x = 29
△x = 29-27 = 2
$y = (x)^{\frac{1}{3}}$
$y_{(x=27)} = (27)^{\frac{1}{3}}= 3$
$\frac{dy}{dx} = \frac{1}{3(x)^{\frac{2}{3}}}$
$(\frac{dy}{dx})_{x=27} = \frac{1}{3(27)^{\frac{2}{3}}} = \frac{1}{27}$
△y = dy = $(\frac{dy}{dx})_{x=27}$ dx
△y = $(\frac{1}{27})$ △x
△y = $(\frac{1}{27})$ (2) = 0.074
Hence,
$(29)^{\frac{1}{3}}$ = y+△y = 3+0.074 = 3.074
(xvii) $(66)^{\frac{1}{3}}$
Solution:
Considering the function as
y = f(x) = $(x)^{\frac{1}{3}}$
Taking x = 64, and
x+△x = 66
△x = 66-64 = 2
$y = (x)^{\frac{1}{3}}$
$y_{(x=64)} = (64)^{\frac{1}{3}}= 4$
$\frac{dy}{dx} = \frac{1}{3(x)^{\frac{2}{3}}}$
$(\frac{dy}{dx})_{x=64} = \frac{1}{3(64)^{\frac{2}{3}}} = \frac{1}{48}$
△y = dy = $(\frac{dy}{dx})_{x=64}$ dx
△y = $(\frac{1}{48})$ △x
△y = $(\frac{1}{48})$ (2) = 0.042
Hence,
$(66)^{\frac{1}{3}}$ = y+△y = 4+0.042 = 4.042
(xviii) $\sqrt{26}$
Solution:
Considering the function as
y = f(x) = $\sqrt{x}$
Taking x = 25, and
x+△x = 26
△x = 26-25 = 1
$y = \sqrt{x}$
$y_{(x=25)} = \sqrt{25} = 5$
$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
$(\frac{dy}{dx})_{x=25} = \frac{1}{2\sqrt{25}} = \frac{1}{10}$
△y = dy = $(\frac{dy}{dx})_{x=25}$ dx
△y = $(\frac{1}{10})$ △x
△y = $(\frac{1}{10})$ (1) = 0.1
Hence,
$\sqrt{26}$ = y+△y = 5 + 0.1 = 5.1
(xix) $\sqrt{37}$
Solution:
Considering the function as
y = f(x) = $\sqrt{x}$
Taking x = 36, and
x+△x = 37
△x = 37-36 = 1
$y = \sqrt{x}$
$y_{(x=36)} = \sqrt{36} = 6$
$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
$(\frac{dy}{dx})_{x=36} = \frac{1}{2\sqrt{36}} = \frac{1}{12}$
△y = dy = $(\frac{dy}{dx})_{x=36}$ dx
△y = $(\frac{1}{12})$ △x
△y = $(\frac{1}{12})$ (1) = 0.0833
Hence,
$\sqrt{26}$ = y+△y = 6 + 0.0833 = 6.0833
(xx) $\sqrt{0.48}$
Solution:
Considering the function as
y = f(x) = $\sqrt{x}$
Taking x = 0.49, and
x+△x = 0.48
△x = 0.48-0.49 = -0.01
$y = \sqrt{x}$
$y_{(x=0.49)} = \sqrt{0.49} = 0.7$
$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
$(\frac{dy}{dx})_{x=0.49} = \frac{1}{2\sqrt{0.49}} = \frac{1}{1.4}$
△y = dy = $(\frac{dy}{dx})_{x=0.49}$ dx
△y = $(\frac{1}{1.4})$ △x
△y = $(\frac{1}{1.4})$ (-0.01) = -0.007143
Hence,
$\sqrt{0.48}$ = y+△y = 0.7 + (-0.007143) = 0.693
(xxi) $(82)^{\frac{1}{4}}$
Solution:
Considering the function as
y = f(x) = $(x)^{\frac{1}{4}}$
Taking x = 81, and
x+△x = 82
△x = 82-81 = 1
$y = (x)^{\frac{1}{4}}$
$y_{(x=81)} = (81)^{\frac{1}{4}}= 3$
$\frac{dy}{dx} = \frac{1}{4(x)^{\frac{3}{4}}}$
$(\frac{dy}{dx})_{x=81} = \frac{1}{4(81)^{\frac{3}{4}}} = \frac{1}{108}$
△y = dy = $(\frac{dy}{dx})_{x=81}$ dx
△y = $(\frac{1}{108})$ △x
△y = $(\frac{1}{108})(1) = \frac{1}{108}$ = 0.009259
Hence,
$(82)^{\frac{1}{4}}$ = y+△y = 3 + 0.009259 = 3.009259
(xxii) $(\frac{17}{81})^{\frac{1}{4}}$
Solution:
Considering the function as
y = f(x) = $(x)^{\frac{1}{4}}$
Taking x = 16/81, and
x+△x = 17/81
△x = 17/81-16/81 = 1/81
$y = (x)^{\frac{1}{4}}$
$y_{(x=16/81)} = (16/81)^{\frac{1}{4}}= 2/3$
$\frac{dy}{dx} = \frac{1}{4(x)^{\frac{3}{4}}}$
$(\frac{dy}{dx})_{x=16/81} = \frac{1}{4(16/81)^{\frac{3}{4}}} = \frac{27}{32}$
△y = dy = $(\frac{dy}{dx})_{x=16/81}$ dx
△y = $(\frac{27}{32})$ △x
△y = $(\frac{27}{32})(\frac{2}{3}) = \frac{1}{96}$ = 0.01042
Hence,
$(\frac{17}{81})^{\frac{1}{4}}$ = y+△y = 2/3 + 0.01042 = 0.6771
(xxiii) $(33)^{\frac{1}{5}}$
Solution:
Considering the function as
y = f(x) = $(x)^{\frac{1}{5}}$
Taking x = 32, and
x+△x = 33
△x = 33-32 = 1
$y = (x)^{\frac{1}{5}}$
$y_{(x=32)} = (32)^{\frac{1}{5}}= 2$
$\frac{dy}{dx} = \frac{1}{5(x)^{\frac{4}{5}}}$
$(\frac{dy}{dx})_{x=32} = \frac{1}{5(32)^{\frac{4}{5}}} = \frac{1}{80}$
△y = dy = $(\frac{dy}{dx})_{x=32}$ dx
△y = $(\frac{1}{80})$ △x
△y = $(\frac{1}{80})(1) = \frac{1}{80}$ = 0.0125
Hence,
$(32)^{\frac{1}{5}}$ = y+△y = 2 + 0.0125 = 2.0125
(xxiv) $\sqrt{36.6}$
Solution:
Considering the function as
y = f(x) = $\sqrt{x}$
Taking x = 36, and
x+△x = 36.6
△x = 36.6-36 = 0.6
$y = \sqrt{x}$
$y_{(x=36)} = \sqrt{36} = 6$
$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
$(\frac{dy}{dx})_{x=36} = \frac{1}{2\sqrt{36}} = \frac{1}{12}$
△y = dy = $(\frac{dy}{dx})_{x=36}$ dx
△y = $(\frac{1}{12})$ △x
△y = $(\frac{1}{12})$ (0.6) = 0.05
Hence,
$\sqrt{26}$ = y+△y = 6 + 0.05 = 6.05
(xxv) $(25)^{\frac{1}{3}}$
Solution:
Considering the function as
y = f(x) = $(x)^{\frac{1}{3}}$
Taking x = 27, and
x+△x = 25
△x = 25-27 = -2
$y = (x)^{\frac{1}{3}}$
$y_{(x=27)} = (27)^{\frac{1}{3}}= 3$
$\frac{dy}{dx} = \frac{1}{3(x)^{\frac{2}{3}}}$
$(\frac{dy}{dx})_{x=27} = \frac{1}{3(27)^{\frac{2}{3}}} = \frac{1}{27}$
△y = dy = $(\frac{dy}{dx})_{x=27}$ dx
△y = $(\frac{1}{27})$ △x
△y = $(\frac{1}{27})$ (-2) = -0.07407
Hence,
$(25)^{\frac{1}{3}}$ = y+△y = 3+(-0.07407) = 2.9259
(xxvi) $\sqrt{49.5}$
Solution:
Considering the function as
y = f(x) = $\sqrt{x}$
Taking x = 49, and
x+△x = 49.5
△x = 49.5-49 = 0.5
$y = \sqrt{x}$
$y_{(x=49)} = \sqrt{49} = 7$
$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
$(\frac{dy}{dx})_{x=49} = \frac{1}{2\sqrt{49}} = \frac{1}{14}$
△y = dy = $(\frac{dy}{dx})_{x=49}$ dx
△y = $(\frac{1}{14})$ △x
△y = $(\frac{1}{14})$ (0.5) = 0.0357
Hence,
$\sqrt{49.5}$ = y+△y = 7 + 0.0357 = 7.0357
Question 10: Find the appropriate value of f(2.01), where f(x) = 4x²+5x+2
Solution:
Considering the function as
y = f(x) = 4x²+5x+2
Taking x = 2, and
x+△x = 2.01
△x = 2.01-2 = 0.01
y = 4x²+5x+2
$y_{(x=2)}$ = 4(2)²+5(2)+2 = 28
$\frac{dy}{dx}$ = 8x+5
$(\frac{dy}{dx})_{x=2}$ = 8(2)+5 = 21
△y = dy = $(\frac{dy}{dx})_{x=2}$ dx
△y = (21) △x
△y = (21) (0.01) = 0.21
Hence,
f(2.01) = y+△y = 28 + 0.21 = 28.21
Question 11: Find the appropriate value of f(5.001), where f(x) = x³-7x²+15
Solution:
Considering the function as
y = f(x) = x³-7x²+15
Taking x = 5, and
x+△x = 5.001
△x =5.001-5 = 0.001
y = x³-7x²+15
$y_{(x=5)}$ = (5)³-7(5)²+15 = -35
$\frac{dy}{dx}$ = 3x²-14x
$(\frac{dy}{dx})_{x=5}$ = 3(5)²-14(5) = 5
△y = dy = $(\frac{dy}{dx})_{x=5}$ dx
△y = (5) △x
△y = (5) (0.001) = 0.005
Hence,
f(5.001) = y+△y = -35 + 0.005 = -34.995
Question 12: Find the appropriate value of log₁₀ 1005, given that log₁₀ e=0.4343
Solution:
Considering the function as
y = f(x) = log₁₀ x
Taking x = 1000, and
x+△x = 1005
△x =1005-1000 = 5
y = log₁₀ x = $\frac{log_ex}{log_e10}$
$y_{(x=1000)}$ = log₁₀ 1000 = 3
$\frac{dy}{dx} = \frac{0.4343}{x}$
$(\frac{dy}{dx})_{x=1000} = \frac{0.4343}{1000}$ = 0.0004343
△y = dy = $(\frac{dy}{dx})_{x=1000}$ dx
△y = (0.0004343) △x
△y = (0.0004343) (5) = 0.0021715
Hence, log₁₀ 1005 = y+△y = 3 + 0.0021715 = 3.0021715
Question 13: If the radius of a sphere is measured as 9cm with an error of 0.03m, find the approximate error in calculating its surface area.
Solution:
According to the given condition,
As, Surface area = 4πx²
Let △x be the change in the radius and △y be the change in the surface area
x = 9
△x = 0.03m = 3cm
x+△x = 9+3 = 12cm
$y_{(x=9)}$ = 4πx² = 4π(9)² = 324 π
$\frac{dy}{dx}$ = 8πx
$(\frac{dy}{dx})_{x=9}$ = 8π(9) = 72π
△y = dy = $(\frac{dy}{dx})_{x=9}$ dx
△y = (72π) △x
△y = (72π) (3) = 216 π
Hence, approximate error in surface area of the sphere is 216 π cm²
Question 14: Find the approximate change in the surface area of a cube as side x meters caused by decreasing the side by 1%.
Solution:
According to the given condition,
As, Surface area = 6x²
Let △x be the change in the length and △y be the change in the surface area
△x/x × 100 = 1
$\frac{dy}{dx}$ = 6(2x) = 12x
△y = $(\frac{dy}{dx})$ △x
△y = (12x) (x/100)
△y = 0.12 x²
Hence, the approximate change in the surface area of a cubical box is 0.12 x² m²
Question 15: If the radius of a sphere is measured as 7m with an error of 0.02m, find the approximate error in calculating its volume.
Solution:
According to the given condition,
As, Volume of sphere = $\frac{4}{3}$ πx³
Let △x be the error in the radius and △y be the error in the volume
x = 7
△x = 0.02 cm
$\frac{dy}{dx} = \frac{4}{3}$ π(3x²) = 4πx²
$(\frac{dy}{dx})_{x=7}$ = 4π(7)² = 196π
△y = dy = $(\frac{dy}{dx})_{x=7}$ dx
△y = (196π) △x
△y = (196π) (0.02) = 3.92 π
Hence, approximate error in volume of the sphere is 3.92 π cm²
Question 16: Find the approximate change in the volume of a cube as side x meters caused by increasing the side by 1%.
Solution:
According to the given condition,
As, Volume of cube = x³
Let △x be the change in the length and △y be the change in the volume
△x/x × 100 = 1
$\frac{dy}{dx}$ = 3x²
△y = $(\frac{dy}{dx})$ △x
△y = (3x²) (x/100)
△y = 0.03 x³
Hence, the approximate change in the volume of a cubical box is 0.03 x³ m³

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RD Sharma Class 12 Ex 14.1 Solutions Chapter 14 Differentials, Errors, and Approximations

Question 1: If y=sin x and x changes from π/2 to 22/14, what is the approximate change in y?

Question 2: The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume?

Question 3: The circular metal plate expands under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

Question 4: Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube.

Question 5: If there is an error of 0.1% in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere.

Question 6: The pressure p and the volume v of a gas are connected by the relation pv1.4 = constant. Find the percentage error in p corresponding to a decrease of 1/2% in v.

Question 7: The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase?

(i) in total surface area, and

(ii) in the volume assuming that k is small?

Question 8: Show that the relative error in computing the volume of a sphere, due to an error in measuring the radius, is approximately equal to the three times the relative error in the radius.

Question 9: Using differentials, find the approximate values of the following:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii) loge 4.04, it being given that log10 4=0.6021 and log10 e=0.4343

(ix) loge 10.02, it being given that loge 10=2.3026

(x) log10 10.1, it being given that log10 e=0.4343

(xi) cos 61°, it being given that sin 60°=0.86603 and 1°=0.01745 radian.

(xii)

(xiii)

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Question 6: The pressure p and the volume v of a gas are connected by the relation pv^1.4 = constant. Find the percentage error in p corresponding to a decrease of 1/2% in v.

(i) $\sqrt{25.02}$

(ii) $(0.009)^{\frac{1}{3}}$

(iii) $(0.007)^{\frac{1}{3}}$

(iv) $\sqrt{401}$

(v) $(15)^{\frac{1}{4}}$

(vi) $(255)^{\frac{1}{4}}$

(vii) $\frac{1}{(2.002)^2}$

(viii) log_e 4.04, it being given that log₁₀ 4=0.6021 and log₁₀ e=0.4343

(ix) log_e 10.02, it being given that log_e 10=2.3026

(x) log₁₀ 10.1, it being given that log₁₀ e=0.4343

(xii) $\frac{1}{\sqrt{25.1}}$

(xiii) $sin (\frac{22}{14})$