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?/
To find the values of theta for which there are horizontal tangent lines on the polar curve r = theta, we need to determine when the slope of the curve is equal to zero.
The slope of a polar curve can be found using the derivative with respect to theta:
dr/dtheta = d(theta)/dtheta = 1
When the slope is equal to zero, we have:
dr/dtheta = 0
1 = 0
This is not possible, as the equation is contradictory. Therefore, there are no other values of theta for which there is a horizontal tangent line on the polar curve r = theta, apart from theta = 0.
As for the vertical tangent lines, we need to find the values of theta for which the derivative of the polar curve, dr/dtheta, is undefined (or tends to infinity).
In this case, the derivative of r with respect to theta is:
dr/dtheta = d(theta)/dtheta = 1
This derivative is defined and has a constant value of 1 for all values of theta in the interval 0 <= theta <= 2pi. Hence, there are no vertical tangent lines on the given polar curve.
Therefore, the only value of theta for which there is a horizontal tangent line on the polar curve r = theta is theta = 0, and there are no vertical tangent lines.
To find the values of theta for which there are horizontal tangent curves on the polar curve r = theta, we need to find the values where the derivative of r with respect to theta, dr/dtheta, equals zero.
Let's take the derivative of r with respect to theta and set it equal to zero:
dr/dtheta = 1 = d(theta)/d(theta)
Since d(theta)/d(theta) is always equal to 1, the derivative is never equal to zero for any value of theta other than 0.
Therefore, theta = 0 is the only value for which there is a horizontal tangent on the polar curve r = theta.
Now let's find the values of theta for which there are vertical tangent curves. For a vertical tangent, the derivative of r with respect to theta, dr/dtheta, must be undefined (or approach infinity).
To check for vertical tangents, we need to see if the derivative dr/dtheta becomes undefined at any value of theta. However, since the derivative is always equal to 1, there are no values of theta for which there are vertical tangent curves on the polar curve r = theta.
In conclusion, the only value of theta for which there is a horizontal tangent on the polar curve r = theta is theta = 0. There are no values of theta for which there are vertical tangent curves.