Consider the curve represented by the parametric equations x(t)= 2+sin(t) and y(t)=1-cos(t) when answering the following questions.
A) Find Dy/Dx in terms of t
B) Find all values of t where the curve has a horizontal tangent.
C) Find all values of t where the curve has a vertical tangent.
D) Write an integral that represents the arc length of the curve on the interval 0 ≤ t ≤ 2π. Evaluate the integral.
a)
dy/dt = d(1-cos(t))/dt = sint
dx/dt = d(2+sin(t))/dt = cost
=> dy/dx = (dy/dt)/(dx/dt) = sint/cost = tan(t)
b)
Horizontal tangent implies that the slope of the line is 0
We already have the term that gives us the slope, which is dy/dx, which is tant.
For tan(t) = 0,
t = 0, pi, 2pi,.....
c)
Vertical tangent implies that the slope of the line is not defined, but for the given tan(t) slope this would imply that cos(t) is 0.
FOr cos(t) = 0,
t = (pi/2), (3pi/2),......
So for B and C, how can I show the answer if it continues on?
and could you help me with part D?
A) To find Dy/Dx in terms of t, we need to calculate the derivative of y with respect to x, which is given by the formula
Dy/Dx = (dy/dt) / (dx/dt).
Given that x(t) = 2 + sin(t) and y(t) = 1 - cos(t), let's find the derivatives dy/dt and dx/dt:
dy/dt = d/dt (1 - cos(t)) = 0 + sin(t) = sin(t),
dx/dt = d/dt (2 + sin(t)) = 0 + cos(t) = cos(t).
Now, substituting these derivatives into the formula, we get:
Dy/Dx = (dy/dt) / (dx/dt) = sin(t) / cos(t).
Thus, Dy/Dx in terms of t is sin(t) / cos(t).
B) To find all the values of t where the curve has a horizontal tangent, we need to find where the derivative Dy/Dx equals zero. Set Dy/Dx = 0:
sin(t) / cos(t) = 0.
For sin(t) = 0, we have t = 0, π, 2π, etc.
But for cos(t) = 0, we have t = π/2, 3π/2, etc. However, remember that division by zero is undefined.
Hence, the curve will have horizontal tangents at t = 0 and t = 2π.
C) To find all the values of t where the curve has a vertical tangent, we need to find where the derivative Dx/Dy is infinite. The derivative Dx/Dy is the reciprocal of Dy/Dx:
Dx/Dy = 1 / (sin(t) / cos(t)) = cos(t) / sin(t).
Setting Dx/Dy equal to infinity, we analyze where cos(t) = 0 and sin(t) = 0.
For cos(t) = 0, we have t = π/2, 3π/2, etc.
For sin(t) = 0, we have t = 0, π, 2π, etc.
However, we disregard t = π/2 and t = 3π/2 since division by zero is undefined.
Thus, the curve will have vertical tangents at t = 0 and t = π.
D) The formula for the arc length of a curve represented by parametric equations x(t) and y(t) on the interval a ≤ t ≤ b is given by the integral:
Arc length = ∫[a,b] √(dx/dt)^2 + (dy/dt)^2 dt.
Given that x(t) = 2 + sin(t) and y(t) = 1 - cos(t), we need to find dx/dt and dy/dt:
dx/dt = cos(t),
dy/dt = sin(t).
Now we can calculate the integral for the arc length on the interval 0 ≤ t ≤ 2π:
Arc length = ∫[0,2π] √(cos(t))^2 + (sin(t))^2 dt
= ∫[0,2π] √(cos^2(t) + sin^2(t)) dt
= ∫[0,2π] √1 dt
= ∫[0,2π] 1 dt
= t |[0,2π]
= 2π - 0
= 2π.
Therefore, the arc length of the curve on the interval 0 ≤ t ≤ 2π is 2π.