Question

Find the length of the curve

1) x=2sin t+2t, y=2cos t, 0≤t≤pi

2) x=6 cos t, y=6 sin t, 0≤t≤pi

3) x=7sin t- 7t cos t, y=7cos t+ 7 t sin t, 0≤t≤pi/4

Answer #1

find t?
x= 2cos(t)+ sin(2t)
y=2sin(t)+cos(2t)
when x= 0, y= -3

Find the length of the curve x = 3t^(2), y = 2t^(3) , 0 ≤ t ≤
1

1) Find the length of the parametric curve x=2 cos(t)
, y=2 sin(t) on the interval [0, pi].
2) A rope lying on the floor is 10 meters long and its
mass is 80 kg. How much work is required to raise one end of the
rope to a height of 15 meters?

Consider the curve c(t) = (t – 2sin(t), 1 – 2cos(t)).
(a) Find an equation for the tangent line to the curve at
t=π/6
(b) For 0 ≤ t ≤ 2π, find the interval(s) of t-values that make
dx/dt < 0.

7. For the parametric curve x(t) = 2 − 5 cos(t), y(t) = 1 + 3
sin(t), t ∈ [0, 2π) Part a: (2 points) Give an equation relating x
and y that represents the curve. Part b: (4 points) Find the slope
of the tangent line to the curve when t = π 6 . Part c: (4 points)
State the points (x, y) where the tangent line is horizontal

For the following exercises, find all exact solutions on [0,
2π)
23. sec(x)sin(x) − 2sin(x) = 0
25. 2cos^2 t + cos(t) = 1
31. 8sin^2 (x) + 6sin(x) + 1 = 0
32. 2cos(π/5 θ) = √3

Show that the curve
x = 7 cos(t), y = 6 sin(t) cos(t)
has two tangents at (0, 0) and find their equations.

Eliminate the parameter to find a Cartesian equation of the
curves:
a) x=3t+2,y=2t−3
b) x=21sin(t)−3,y=2cos(t)+5

Given the function f(t) = 1/2cos (pi/3 *t) +3pi^2. Which of the
following is equal to f'(t)
f'(t) = 1/2sin(pi/3 t) +6pi
f'(t) = -pi/3cos(pi/3 *t)
f'(t) = -pi/6 sin(pi/3 T)
f'(t) = pi/3 sin(pi/3 t) +6pit

Find the derivative of the parametric curve x=2t-3t2,
y=cos(3t) for 0 ≤ ? ≤ 2?.
Find the values for t where the tangent lines are horizontal on
the parametric curve. For the horizontal tangent lines, you do not
need to find the (x,y) pairs for these values of t.
Find the values for t where the tangent lines are vertical on
the parametric curve. For these values of t find the coordinates of
the points on the parametric curve.

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