Advertisements
Advertisements
Question
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Advertisements
Solution 1
`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`
Solution 2
`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
Solution 3
`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
RELATED QUESTIONS
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
