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
find dy/dx if x=e2t , y=`e^sqrtt`
If x=at2, y= 2at , then find dy/dx.
If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`
If `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`
If y =1 − cos θ, x = 1 − sin θ, then `dy/dx "at" θ =pi/4` is ______
If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `
If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find `dy/dx `
Derivatives of tan3θ with respect to sec3θ at θ=π/3 is
(A)` 3/2`
(B) `sqrt3/2`
(C) `1/2`
(D) `-sqrt3/2`
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 = (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 (cos θ + θ sin θ), y = a (sin θ – θ cos θ)
x = `"t" + 1/"t"`, y = `"t" - 1/"t"`
x = 3cosθ – 2cos3θ, y = 3sinθ – 2sin3θ
sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`
x = `(1 + log "t")/"t"^2`, y = `(3 + 2 log "t")/"t"`
If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`
Differentiate `x/sinx` w.r.t. sin x
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 `"x = a sin" theta "and y = b cos" theta, "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 ______.
