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
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing 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)
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
