given x=2cos(t)+t*sin(t) and y=2sin(t)-t*cost(t), with t:[pi, 2pi]
find all critical values.
by finding dy/dx and setting it equal to zero, i only find one value, t=6.121, within that domain. that gives you x=.987. I'm suppose to get three other answers. How do i do this?!
To find all critical values, you need to find values of t that make the derivative dy/dx equal to zero within the given domain.
First, let's find the derivative dy/dx.
To find dy/dx, we need to calculate dx/dt and dy/dt using the given equations for x and y, and then divide dy/dt by dx/dt.
Given:
x = 2cos(t) + t*sin(t)
y = 2sin(t) - t*cos(t)
Calculate dx/dt:
dx/dt = -2sin(t) + sin(t) + t*cos(t) = sin(t) + t*cos(t)
Calculate dy/dt:
dy/dt = 2cos(t) - cos(t) + t*sin(t) = cos(t) + t*sin(t)
Now, divide dy/dt by dx/dt to find dy/dx:
dy/dx = (cos(t) + t*sin(t)) / (sin(t) + t*cos(t))
Next, set dy/dx equal to zero and solve for t:
(cos(t) + t*sin(t)) / (sin(t) + t*cos(t)) = 0
To solve this equation, you can multiply both sides by (sin(t) + t*cos(t)) to eliminate the denominator:
cos(t) + t*sin(t) = 0
Now, solve this equation for t. Unfortunately, there is no simple algebraic solution for this equation. You will need to solve it numerically using methods such as graphing, iteration, or a numerical solver.
In your case, you found one solution t = 6.121 that satisfies the equation within the given domain. However, there may be additional solutions that you need to find.
To find the remaining solutions, you can use numerical methods. One approach is to graph the function cos(t) + t*sin(t) and find the x-intercepts within the given domain. Another approach is to use an iterative method (such as the Newton-Raphson method) to approximate the solutions.
By using these numerical methods, you can find the remaining critical values of t, which will give you additional corresponding values of x.