For what values of theta, on the polar curve r=theta, 0<=theta<=2pi, are there horizontal tangent curves? Vertical?
I got one answer for the horizontal tangent. and it's theta = 0 but what are the other ones?/
since r = θ
x = r cosθ
y = r sinθ
dy/dθ = r' sinθ + r cosθ = sinθ + θcosθ
dx/dθ = r' cosθ - r sinθ = cosθ - θsinθ
dy/dx = (sinθ + θcosθ)/(cosθ - θsinθ)
You are correct that dy/dx = 0 when θ=0
It is also zero when x = 2.04 and 4.88
(approximately)
The tangent is vertical when the denominator is zero: x = 0.84 and 3.45
To determine the values of theta on the given polar curve r = theta, 0 <= theta <= 2pi, where there are horizontal tangent lines, we need to find the points on the curve where the slope of the tangent line is zero.
The slope of the tangent line on a polar curve can be calculated using the derivative of the polar equation, which is given by:
dy/dx = (dr/dtheta * sin(theta) + r * cos(theta)) / (dr/dtheta * cos(theta) - r * sin(theta))
In this case, r = theta, so we can substitute the value of r into the equation:
dy/dx = (d(theta)/dtheta * sin(theta) + theta * cos(theta)) / (d(theta)/dtheta * cos(theta) - theta * sin(theta))
To find the points where the slope is zero, we need to solve the equation:
dy/dx = 0
Let's find the derivative of theta:
d(theta)/dtheta = 1
Substituting this into the equation above, we get:
(sin(theta) + theta * cos(theta)) / (cos(theta) - theta * sin(theta)) = 0
To find the values of theta that satisfy this equation, you can set the numerator equal to zero and the denominator equal to zero separately.
Setting the numerator equal to zero:
sin(theta) + theta * cos(theta) = 0
This equation does not have any simple algebraic solution. However, it can be solved numerically using methods such as graphing the function or using numerical methods like Newton's method or bisection method.
For the denominator, setting it equal to zero:
cos(theta) - theta * sin(theta) = 0
This equation also does not have a simple algebraic solution. It can be solved numerically using the same methods mentioned above.
Using numerical methods, we can find multiple values of theta that satisfy these equations and correspond to points where the tangent lines are horizontal on the polar curve. You may use a graphing calculator or software to find these values more easily.
As for the vertical tangent lines, they occur when the derivative dy/dx is undefined. In this case, it happens when the denominator in the derived equation is equal to zero:
dr/dtheta * cos(theta) - r * sin(theta) = 0
Substituting r = theta:
d(theta)/dtheta * cos(theta) - theta * sin(theta) = 0
Similar to before, this equation does not have a simple algebraic solution, and you would need to solve it numerically to find the corresponding values of theta.