Chapter 6 Application of Derivatives Ex 6.4 Solutions
Question - 1 : - Using differentials, find the approximatevalue of each of the following up to 3 places of decimal
Answer - 1 : - (i) 
Consider
. Let x = 25 and Δx =0.3. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 0.03 + 5 = 5.03.
(ii) 
Consider. Let x = 49 and Δx =0.5. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 7 + 0.035 = 7.035.
(iii) 
Consider. Let x = 1 and Δx =− 0.4. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 1 + (−0.2) = 1 − 0.2 = 0.8.
(iv) 
Consider
. Let x = 0.008 and Δx =0.001. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 0.2 + 0.008 = 0.208.
(v) 
Consider
. Let x = 1 and Δx =−0.001. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 1 + (−0.0001) = 0.9999.
(vi) 
Consider
. Let x = 16 and Δx =−1. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 2 + (−0.03125) = 1.96875.
(vii) 
Consider
. Let x = 27 and Δx =−1. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 3 + (−0.0370) = 2.9629.
(viii) 
Consider
. Let x = 256 and Δx =−1. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 4 + (−0.0039) = 3.9961.
(ix) 
Consider. Let x = 81 and Δx =1. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 3 + 0.009 = 3.009.
(x) 
Consider
. Let x = 400 and Δx =1. Then,
Now, dy is approximatelyequal to Δy and is given by,
Hence, the approximatevalue of
is 20 + 0.025 = 20.025.
Question - 2 : - Find the approximate value of f (2.01),where f (x) = 4x2 +5x + 2
Answer - 2 : -
Let x = 2 and Δx =0.01. Then, we have:
f(2.01)= f(x + Δx) = 4(x + Δx)2 +5(x + Δx) + 2
Now, Δy = f(x +Δx) − f(x)
∴ f(x + Δx) = f(x)+ Δy
Hence, the approximate value of f (2.01)is 28.21.
Question - 3 : - Find the approximate value of f (5.001),where f (x) = x3 −7x2 + 15.
Answer - 3 : -
Let x = 5 and Δx =0.001. Then, we have:
Hence, the approximate value of f (5.001)is −34.995.
Question - 4 : - Find the approximate change in thevolume V of a cube of side x metres caused byincreasing side by 1%.
Answer - 4 : -
The volume of a cube (V) ofside x is given by V = x3.
Hence, the approximate change in the volumeof the cube is 0.03x3 m3.
Question - 5 : - Find the approximate change in the surfacearea of a cube of side x metres caused by decreasing the sideby 1%
Answer - 5 : -
The surface area of a cube (S) ofside x is given by S = 6x2.
Hence, the approximate change in the surfacearea of the cube is 0.12x2 m2.
Question - 6 : - If the radius of a sphere is measured as 7 m with an errorof 0.02m, then find the approximate error in calculating its volume.
Answer - 6 : -
Let r be the radius of thesphere and Δr be the error in measuring the radius.
Then,
r = 7 mand Δr = 0.02 m
Now, the volume V of thesphere is given by,
Hence, the approximate error in calculatingthe volume is 3.92 π m3.
Question - 7 : - If the radius of a sphere is measured as 9 m with an errorof 0.03 m, then find the approximate error in calculating in surface area.
Answer - 7 : -
Let r be the radius of thesphere and Δr be the error in measuring the radius.
Then,
r = 9 mand Δr = 0.03 m
Now, the surface area of the sphere (S) isgiven by,
S = 4πr2
Hence, the approximate error in calculatingthe surface area is 2.16π m2.
Question - 8 : - If f (x) = 3x2 +15x + 5, then the approximate value of f (3.02) is
A. 47.66 B. 57.66 C. 67.66 D. 77.66
Answer - 8 : -
Let x = 3 and Δx =0.02. Then, we have:
Hence, the approximate value of f(3.02)is 77.66.
The correct answer is D.
Question - 9 : - The approximate change in the volume of acube of side x metres caused by increasing the side by 3% is
A. 0.06 x3 m3 B. 0.6 x3 m3 C. 0.09 x3 m3 D. 0.9 x3 m3
Answer - 9 : -
The volume of a cube (V) ofside x is given by V = x3.
Hence, the approximate change in the volumeof the cube is 0.09x3 m3.
The correct answer is C.