Advertisements
Advertisements
Question
Solve the following : Find the approximate value of cos–1 (0.51), given π = 3.1416, `(2)/sqrt(3)` = 1.1547.
Advertisements
Solution
Let f(x) = cos–1 x.
Then f'(x) = `d/dx(cos^-1 x) = (-1)/sqrt(1 - x^2)`
Take a = 0.5 and h = 0.01
Then f(a) = f(0.5)
= cos–1 (0.5)
= `cos^-1(cos pi/3)`
= `pi/(3)`
and
f'(a) = f'(0.5)
= `-(1)/sqrt(1 - (1/2)^2`
= `-(2)/sqrt(3)`
= – 1.1547
The formula for approximation is
f(a + h) ≑ f(a) + h.f'(a)
∴ cos–1 (0.51) = f(0.51)
= f(0.5 + 0.01)
≑ f(0.5) + (0.01)f'(0.5)
≑ `pi/(3) + 0.01 xx (-1.1547)`
≑ `(3.1416)/(3) - 0.011547`
≑ 1.0472 - 0.01157 = 1.035653
∴ cos–1 (0.51) ≑ 1.035653.
APPEARS IN
RELATED QUESTIONS
Find the approximate value of ` sqrt8.95 `
Using differentials, find the approximate value of the following up to 3 places of decimal
`sqrt(25.3)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(0.009)^(1/3)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(0.999)^(1/10)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(15)^(1/4)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(255)^(1/4)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(82)^(1/4)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(401)^(1/2)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(26.57)^(1/3)`
Find the approximate change in the volume V of a cube of side x metres caused by increasing side by 1%.
The normal to the curve x2 = 4y passing (1, 2) is
(A) x + y = 3
(B) x − y = 3
(C) x + y = 1
(D) x − y = 1
Find the approximate value of log10 (1016), given that log10e = 0⋅4343.
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 ?
Using differential, find the approximate value of the following: \[\left( 0 . 009 \right)^\frac{1}{3}\]
Using differential, find the approximate value of the following: \[\left( 0 . 007 \right)^\frac{1}{3}\]
Using differentials, find the approximate values of the cos 61°, it being given that sin60° = 0.86603 and 1° = 0.01745 radian ?
Using differential, find the approximate value of the \[\frac{1}{\sqrt{25 . 1}}\] ?
Using differential, find the approximate value of the \[\left( 33 \right)^\frac{1}{5}\] ?
Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2 ?
Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15 ?
Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1% ?
If the radius of a sphere is measured as 7 m with an error of 0.02 m, find the approximate error in calculating its volume ?
A piece of ice is in the form of a cube melts so that the percentage error in the edge of cube is a, then find the percentage error in its volume ?
If an error of k% is made in measuring the radius of a sphere, then percentage error in its volume is
While measuring the side of an equilateral triangle an error of k % is made, the percentage error in its area is
If loge 4 = 1.3868, then loge 4.01 =
The pressure P and volume V of a gas are connected by the relation PV1/4 = constant. The percentage increase in the pressure corresponding to a deminition of 1/2 % in the volume is
The approximate value of (33)1/5 is
Find the approximate value of f(3.02), up to 2 places of decimal, where f(x) = 3x2 + 5x + 3.
Find the approximate values of : `root(5)(31.98)`
Find the approximate values of : (3.97)4
Find the approximate values of : sin 61° , given that 1° = 0.0174c, `sqrt(3) = 1.732`
Find the approximate values of : tan (45° 40'), given that 1° = 0.0175°.
Find the approximate values of : tan–1 (1.001)
Find the approximate values of : e0.995, given that e = 2.7183.
Find the approximate values of : 32.01, given that log 3 = 1.0986
Find the approximate values of : loge(101), given that loge10 = 2.3026.
Find the approximate values of : f(x) = x3 – 3x + 5 at x = 1.99.
Find the approximate values of : f(x) = x3 + 5x2 – 7x + 10 at x = 1.12.
The approximate value of tan (44°30'), given that 1° = 0.0175c, is ______.
Using differentiation, approximate value of f(x) = x2 - 2x + 1 at x = 2.99 is ______.
Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm respectively
If y = x4 – 10 and if x changes from 2 to 1.99, what is the change in y ______.
If the radius of a sphere is measured as 9 m with an error of 0.03 m. the find the approximate error in calculating its surface area
