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 cos (60° 30').
(Given: 1° = 0.0175c, sin 60° = 0.8660)
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
`(0.999)^(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
`(81.5)^(1/4)`
Using differentials, find the approximate value of each of the following.
`(17/81)^(1/4)`
The normal at the point (1, 1) on the curve 2y + x2 = 3 is
(A) x + y = 0
(B) x − y = 0
(C) x + y + 1 = 0
(D) x − y = 1
The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are
(A)`(4, +- 8/3)`
(B) `(4,(-8)/3)`
(C)`(4, +- 3/8)`
(D) `(+-4, 3/8)`
The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume ?
A circular metal plate expends 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.
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 ?
Show that the relative error in computing the volume of a sphere, due to an error in measuring the radius, is approximately equal to three times the relative error in the radius ?
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 differential, find the approximate value of the \[\left( 15 \right)^\frac{1}{4}\] ?
Using differential, find the approximate value of the log10 10.1, it being given that log10e = 0.4343 ?
Using differential, find the approximate value of the \[\cos\left( \frac{11\pi}{36} \right)\] ?
Using differential, find the approximate value of the \[\left( 66 \right)^\frac{1}{3}\] ?
Using differential, find the approximate value of the \[\sqrt{37}\] ?
Using differential, find the approximate value of the \[\sqrt{0 . 48}\] ?
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 ?
If there is an error of 2% in measuring the length of a simple pendulum, then percentage error in its period is
If there is an error of a% in measuring the edge of a cube, then percentage error in its surface is
If loge 4 = 1.3868, then loge 4.01 =
If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius is λ %, then the error in its volume is
The approximate value of (33)1/5 is
The circumference of a circle is measured as 28 cm with an error of 0.01 cm. The percentage error in the area 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 : `sqrt(8.95)`
Find the approximate values of: `root(3)(28)`
Find the approximate values of : `root(5)(31.98)`
Find the approximate values of : (3.97)4
Find the approximate values of sin (29° 30'), given that 1° = 0.0175°, `sqrt(3) = 1.732`.
Find the approximate values of : cos(60° 30°), given that 1° = 0.0175°, `sqrt(3) = 1.732`
Find the approximate values of : cot–1 (0.999)
Find the approximate values of : loge(9.01), given that log 3 = 1.0986.
Using differentials, find the approximate value of `sqrt(0.082)`
If y = x4 – 10 and if x changes from 2 to 1.99, what is the change in y ______.
The approximate value of f(x) = x3 + 5x2 – 7x + 9 at x = 1.1 is ______.
