Advertisements
Advertisements
प्रश्न
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Advertisements
उत्तर १
`x=cos^2θ and y=cot θ`
`(dx)/(dθ)=d/(dθ) (cos^2θ)`
`dx/(dθ)=-2cosθ sin θ`
`dy/dθ=-cosec^2θ`
`dy/dx=dy/(dθ)/dx/(dθ)`
= `(-cosec^2θ)/(-2cosθ sinθ)`
=`1/(2sin^3 θ cos θ)`
=`(1/2sin^3θ cos θ)θ=pi/4`
`(dy/dx)_θ=pi/4`
=`1/2(1/sqrt2)^3 1/sqrt2`
=`1/(2 1/4)=2`
उत्तर २
`x=cos^2θ and y=cot θ`
`(dx)/(dθ)=2 cosθ (-sinθ )`
`dx/(dθ)=-2cosθ sin θ`
y = cotθ
`dy/dθ=-cosec^2θ`
`dy/dx=(dy/(dθ))/(dx/(dθ))`
= `(-cosec^2θ)/(-2cosθ sinθ)`
`((dy)/(dx))_(0=π/4) = (cosec^2 π/4)/(2.sin π/4. cos π/4) `
= `2/(2 xx 1/sqrt2 xx 1/sqrt2`
= 2
उत्तर ३
`x=cos^2θ and y=cot θ`
`(dx)/(dθ)=2 cosθ (-sinθ )`
`dx/(dθ)=-2cosθ sin θ`
y = cotθ
`dy/dθ=-cosec^2θ`
`dy/dx=(dy/(dθ))/(dx/(dθ))`
= `(-cosec^2θ)/(-2cosθ sinθ)`
`((dy)/(dx))_(0=π/4) = (cosec^2 π/4)/(2.sin π/4. cos π/4) `
= `2/(2 xx 1/sqrt2 xx 1/sqrt2`
= 2
APPEARS IN
संबंधित प्रश्न
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
The function f(x) = x9 + 3x7 + 64 is increasing on
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
The function f(x) = 9 - x5 - x7 is decreasing for
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
Show that function f(x) = tan x is increasing in `(0, π/2)`.
A function f is said to be increasing at a point c if ______.
The function f(x) = x3 + 3x is increasing in interval ______.
