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
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Function f(x) = | x | − | x − 1 | is monotonically increasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
For every value of x, the function f(x) = `1/7^x` is ______
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
