Use the given equation to answer the following questions. (Enter your answers from smallest to largest.)
r = 8cos(¦È)
0 ¡Ü ¦È < ¦Ð
(a) Find the points on the given curve where the tangent line is horizontal.
theta=
theta=
(b) Find the points on the given curve where the tangent line is vertical.
theta=
theta=
To find the points on the given curve where the tangent line is horizontal, we need to find the values of ¦È for which the derivative of r with respect to ¦È, dr/d¦È, is equal to zero.
Given that r = 8cos(¦È), we can find the derivative dr/d¦È using the chain rule:
dr/d¦È = -8sin(¦È)
Setting dr/d¦È equal to zero:
-8sin(¦È) = 0
sin(¦È) = 0
For this equation, sin(¦È) = 0, we know that it will be true for ¦È values of 0, ¦Ð, 2¦Ð, 3¦Ð, and so on.
Therefore, the points on the given curve where the tangent line is horizontal are:
(a) theta = 0, theta = pi, theta = 2pi, theta = 3pi, and so on.
To find the points on the given curve where the tangent line is vertical, we need to find the values of ¦È for which the derivative of r with respect to ¦È, dr/d¦È, is undefined.
The derivative dr/d¦È = -8sin(¦È) is undefined when sin(¦È) is undefined, which occurs when ¦È = pi/2 and ¦È = 3pi/2.
Therefore, the points on the given curve where the tangent line is vertical are:
(b) theta = pi/2 and theta = 3pi/2.
To find the points on the curve where the tangent line is horizontal or vertical, we need to determine the values of ¦È for which the derivative of r with respect to ¦È is equal to 0 or undefined, respectively.
(a) The tangent line is horizontal when the derivative of r with respect to ¦È is equal to 0.
1. Find the derivative, dr/d¦È, by taking the derivative of r with respect to ¦È:
dr/d¦È = -8sin(¦È)
2. Set the derivative equal to 0 and solve for ¦È:
-8sin(¦È) = 0
sin(¦È) = 0
3. Find the values of ¦È within the given range (0 ¡Ü ¦È < ¦Ð) that satisfy sin(¦È) = 0. These values correspond to the points on the curve where the tangent line is horizontal. The solutions to sin(¦È) = 0 are:
¦È = 0, ¦È = ¦Ð
Therefore, the points on the curve where the tangent line is horizontal are ¦È = 0 and ¦È = ¦Ð.
(b) The tangent line is vertical when the derivative of r with respect to ¦È is undefined. This occurs when the cosine function approaches 0.
1. Find the values of ¦È within the given range (0 ¡Ü ¦È < ¦Ð) where cos(¦È) = 0. These values correspond to the points on the curve where the tangent line is vertical. The solutions to cos(¦È) = 0 are:
¦È = ¦Ð/2, ¦È = 3¦Ð/2
Therefore, the points on the curve where the tangent line is vertical are ¦È = ¦Ð/2 and ¦È = 3¦Ð/2.