Advertisements
Advertisements
प्रश्न
Find the approximate values of : tan–1 (1.001)
Advertisements
उत्तर
Let f(x) = tan–1x
∴ f'(x) = `d/dx(tan^-1x) = (1)/(1 + x^2)`
Take a = 1 and h = 0.001
Then f(a) = f(1) = tan–11 = `pi/(4)`
and f'(a) = f'(1) = `(1)/(1 + 1^2) = (1)/(2)`
The formula for approximation is
f(a + h) ≑ f(a) + h.f'(a)
∴ tan–11 (1.001)
= f(1.001)
= f(1 + 0.001)
= f(1) + (0.001).f'(1)
≑ `pi/(4) + (0.001) xx (1)/(2)`
= `pi/(4) + 0.0005`
∴ tan–1 (1.001) ≑ `pi/(4) + 0.0005`.
Remark: the answer can also be given as :
tan–1 (1.001) ≑ f(1) + (0.001).f'(1)
≑ `pi/(4) + (0.001) xx (1)/(2)`
≑ `(3.1416)/(4) + 0.0005`
≑ 0.7854 + 0.0005
= 0.7859.
APPEARS IN
संबंधित प्रश्न
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
`(0.009)^(1/3)`
Using differentials, find the approximate value of the following up to 3 places of decimal
`(0.999)^(1/10)`
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 volume V of a cube of side x metres caused by increasing side by 1%.
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 0.02m, 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
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
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
Show that the function given by `f(x) = (log x)/x` has maximum at x = e.
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)`
Find the approximate change in the volume ‘V’ of a cube of side x metres caused by decreasing the side by 1%.
The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume ?
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 ?
The pressure p and the volume v of a gas are connected by the relation pv1.4 = const. Find the percentage error in p corresponding to a decrease of 1/2% in v .
1 Using differential, find the approximate value of the following:
\[\sqrt{25 . 02}\]
Using differential, find the approximate value of the following: \[\left( 0 . 009 \right)^\frac{1}{3}\]
Using differential, find the approximate value of the \[\frac{1}{(2 . 002 )^2}\] ?
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 loge 10.02, it being given that loge10 = 2.3026 ?
Using differential, find the approximate value of the \[\frac{1}{\sqrt{25 . 1}}\] ?
Using differential, find the approximate value of the \[\left( 66 \right)^\frac{1}{3}\] ?
Using differential, find the approximate value of the \[\sqrt{0 . 082}\] ?
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% ?
Find the approximate change in the value V of a cube of side x metres caused by increasing the side by 1% ?
If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area ?
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
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 =
If y = xn then the ratio of relative errors in y and x is
Find the approximate values of : `root(5)(31.98)`
Find the approximate values of : sin 61° , given that 1° = 0.0174c, `sqrt(3) = 1.732`
Find the approximate values of sin (29° 30'), given that 1° = 0.0175°, `sqrt(3) = 1.732`.
Find the approximate values of : tan–1(0.999)
Find the approximate values of : loge(9.01), given that log 3 = 1.0986.
Find the approximate values of : f(x) = x3 – 3x + 5 at x = 1.99.
The approximate value of tan (44°30'), given that 1° = 0.0175c, is ______.
The approximate value of the function f(x) = x3 − 3x + 5 at x = 1.99 is ____________.
Using differentiation, approximate value of f(x) = x2 - 2x + 1 at x = 2.99 is ______.
Using differentials, find the approximate value of `sqrt(0.082)`
If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximating error in calculating its volume.
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
