Advertisements
Advertisements
Question
If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `at t = `pi/4`
Advertisements
Solution
We have,
x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 - cos 2t)
`therefore "dx"/"dt" = "a"["sin" "2t" ."d"/"dt" (1 + "cos" "2t") + (1 + "cos" 2"t") "d"/"dt" "sin" "2t"]`
`= "a" ["sin" 2"t" . (-2 "sin" "2t") + (1 + "cos" "2t") . 2 "cos" "2t"]`
`= -2 "a" "sin"^2 "2t" + 2"a" "cos" 2"t" (1 + "cos" "2t")`
`=> "dx"/"dt" = -2"a" ["sin"^2 "2t" - "cos" "2t" (1 + "cos" "2t")]` .....(1)
and `"dy"/"dt" = "b" ["cos" "2t" . (2 "sin" "2t") + (1 - "cos" "2t") + (1 - "cos" "2t") . "d"/"dt" "cos" "2t" . "d"/"dt" "cos" "2t"]`
`= "b" ["cos" "2t" . (2 "sin" "2t") + (1 - "cos" "2t") (-2 "sin " "2t")]`
`= "2b" ["sin" "2t" . "cos" "2t" - (1 - "cos" "2t") "sin" "2t"]`
`therefore "dy"/"dx" = ("dy"/"dt")/("dx"/"dt") = ("2b" ["sin" "2t" . "cos" "2t" - (1 - "cos" "2t") "sin" "2t"])/(-2"a" ["sin"^2 "2t" - "cos" "2t" (1 + "cos" "2t")])`
`=> ("dy"/"dx")_("t" = pi/4) = - "b"/"a" ["sin" pi/2 "cos" pi/2 - (1 - "cos" pi/2) "sin" pi/2]/["sin"^2 pi/2 - "cos" pi/2 (1 + "cos" pi/2)]`
`= -"b"/"a" . (0-1)/(1 - 0) = "b"/"a"`
APPEARS IN
RELATED QUESTIONS
If `log_10((x^3-y^3)/(x^3+y^3))=2 "then show that" dy/dx = [-99x^2]/[101y^2]`
If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`
If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `
If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1−cos 2t), show that `dy/dx=β/αtan t`
If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find `dy/dx `
Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = sin t, y = cos 2t
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = cos θ – cos 2θ, y = sin θ – sin 2θ
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = a (θ – sin θ), y = a (1 + cos θ)
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
`x = (sin^3t)/sqrt(cos 2t), y = (cos^3t)/sqrt(cos 2t)`
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = `a(cos t + log tan t/2)`, y = a sin t
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = a sec θ, y = b tan θ
If X = f(t) and Y = g(t) Are Differentiable Functions of t , then prove that y is a differentiable function of x and
`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`
Hence find `"dy"/"dx"` if x = a cos2 t and y = a sin2 t.
IF `y = e^(sin-1x) and z =e^(-cos-1x),` prove that `dy/dz = e^x//2`
The cost C of producing x articles is given as C = x3-16x2 + 47x. For what values of x, with the average cost is decreasing'?
x = `"t" + 1/"t"`, y = `"t" - 1/"t"`
x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`
x = 3cosθ – 2cos3θ, y = 3sinθ – 2sin3θ
sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`
If x = ecos2t and y = esin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`
Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan–1x, when x ≠ 0
If x = t2, y = t3, then `("d"^2"y")/("dx"^2)` is ______.
Derivative of x2 w.r.t. x3 is ______.
If y `= "Ae"^(5"x") + "Be"^(-5"x") "x" "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
Let a function y = f(x) is defined by x = eθsinθ and y = θesinθ, where θ is a real parameter, then value of `lim_(θ→0)`f'(x) is ______.
