Advertisements
Advertisements
प्रश्न
Solve the following : Find the approximate value of cos–1 (0.51), given π = 3.1416, `(2)/sqrt(3)` = 1.1547.
Advertisements
उत्तर
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
संबंधित प्रश्न
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
`sqrt(49.5)`
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
`(26)^(1/3)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(81.5)^(1/4)`
If the radius of a sphere is measured as 7 m with an error of 0.02m, then find the approximate error in calculating its volume.
If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating in surface area
The approximate change in the volume of a cube 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
Using differentials, find the approximate value of each of the following.
`(17/81)^(1/4)`
Show that the function given by `f(x) = (log x)/x` has maximum at x = e.
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
Find the approximate value of log10 (1016), given that log10e = 0⋅4343.
Find the approximate change in the volume ‘V’ of a cube of side x metres caused by decreasing the side by 1%.
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 ?
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 loge 4.04, it being given that log104 = 0.6021 and log10e = 0.4343 ?
Using differential, find the approximate value of the \[\frac{1}{\sqrt{25 . 1}}\] ?
Using differential, find the approximate value of the \[\sin\left( \frac{22}{14} \right)\] ?
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 \[\sqrt{0 . 48}\] ?
Using differential, find the approximate value of the \[\left( 82 \right)^\frac{1}{4}\] ?
Using differential, find the approximate value of the \[\sqrt{49 . 5}\] ?
Using differential, find the approximate value of the \[{25}^\frac{1}{3}\] ?
If the radius of a sphere is measured as 9 cm with an error of 0.03 m, find the approximate error in calculating its surface area ?
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 there is an error of a% in measuring the edge of a cube, then percentage error in its surface is
The height of a cylinder is equal to the radius. If an error of α % is made in the height, then percentage error in its volume is
A sphere of radius 100 mm shrinks to radius 98 mm, then the approximate decrease in its volume is
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
Find the approximate values of: `root(3)(28)`
Find the approximate values of sin (29° 30'), given that 1° = 0.0175°, `sqrt(3) = 1.732`.
Find the approximate values of : tan (45° 40'), given that 1° = 0.0175°.
Find the approximate values of : tan–1(0.999)
Find the approximate values of : cot–1 (0.999)
Find the approximate values of : e2.1, given that e2 = 7.389
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 + 5x2 – 7x + 10 at x = 1.12.
The approximate value of the function f(x) = x3 − 3x + 5 at x = 1.99 is ____________.
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
Find the approximate value of tan−1 (1.002).
[Given: π = 3.1416]
