For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 66. x = 3 sin t, y = 3 cost, t = À 67. x = cos t, y = 8 sin t, t = 5 68. x = 2t, y = f, t = -1 69. x = 1+1, y=t-7, t = 1 70. x = Vt, y = 2t, t = 4

Answer and Step-by-step explanation:

(66) x = 3 sin t, y = 3cos t at t = π/4

dx/dt = 3cos t, dy/dt = - 3sin t

Slope of the tangent, m = dy/dx = (dy/dt)/(dx/dt) = -(sin t)/(cos t) = - tan t = - tan (π/4) = - 1

At this point

x = 3 sin (π/4) = (3√2)/2

y = 3 cos (π/4) = (3√2)/2

Equation of the tangent

(y - ((3√2)/2) = -1(x - (3√2)/2)

y + x = ((3√2)/2) - ((3√2)/2) = 0

y + x = 0

(67) x = cos t, y = 8 sin t, t = π/2

dx/dt = - sin t, dy/dt = 8 cos t

slope, m = dy/dx = (dy/dt)/(dx/dt) = - (8cos t)/(sin t) = -(8 cos (π/2))/(sin (π/2)) = 0

At this point,

x = cos π/2 = 0, y = sin π/2 = 1

Equation of the tangent,

y - 1 = 0(x - 0)

y = 1.

(68) x = 2t, y = t³, t = -1

dx/dt = 2, dy/dt = 3t²

slope, m = dy/dx = (dy/dt)/(dx/dt) = 3t²/2 = 3(-1)²/2 = 3/2

At this point,

x = 2(-1) = -2, y = (-1)³ = -1

Equation of the tangent,

y + 1 = (3/2)(x + 2)

2y - 3x = 4

(69) x = t + (1/t), y = t - (1/t), t = 1

dx/dt = 1 - (1/t²), dy/dt = 1 + (1/t²)

slope, m = dy/dx = (dy/dt)/(dx/dt) = (t² + 1)/(t - 1) = (1+1)/(1-1) = 2/0

At this point, t = 1

x = 2 = -2, y = 0

Equation of the tangent,

y - 0 = (2/0)(x - 2)

x - 2 = 0

x = 2

(70) x = √t, y = 2t, t = 4

dx/dt = -(1/2√t), dy/dt = 2

slope, m = dy/dx = (dy/dt)/(dx/dt) = -4√t = -4√4 = -8

At this point,

x = √4 = 2, y = 2×4 = 8

Equation of the tangent,

y - 8 = - 8(x - 2)

y + 8x = 16

ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga ooga booga

step-by-step explanation:

v v cj van config k tbs eng r.c r.c fcb fb gb frfr