r = 2(1 - sin(๐))(a) Find dy/dx (if it exists). (Use theta for ๐ as necessary.) If it does not exist, enter NONE.dy/dx = (b) Find the slope of the tangent line at (2, 0). If it does not exist, enter NONE.(c) Find the slope of the tangent line at (4, 3๐/2). If it does not exist, enter NONE.
Question
r = 2(1 - sin(๐))(a) Find dy/dx (if it exists). (Use theta for ๐ as necessary.) If it does not exist, enter NONE.dy/dx = (b) Find the slope of the tangent line at (2, 0). If it does not exist, enter NONE.(c) Find the slope of the tangent line at (4, 3๐/2). If it does not exist, enter NONE.
Solution
(a) To find dy/dx, we first need to convert the polar coordinates to Cartesian coordinates. The conversion formulas are x = rcos(๐) and y = rsin(๐). Substituting r = 2(1 - sin(๐)) into these formulas, we get:
x = 2(1 - sin(๐))cos(๐) y = 2(1 - sin(๐))sin(๐)
Next, we differentiate both x and y with respect to ๐:
dx/d๐ = -2sin(๐)cos(๐) - 2(1 - sin(๐))sin(๐) dy/d๐ = 2cos(๐)sin(๐) - 2(1 - sin(๐))cos(๐)
Finally, we find dy/dx by dividing dy/d๐ by dx/d๐:
dy/dx = (2cos(๐)sin(๐) - 2(1 - sin(๐))cos(๐)) / (-2sin(๐)cos(๐) - 2(1 - sin(๐))sin(๐))
(b) To find the slope of the tangent line at (2, 0), we substitute ๐ = 0 into the formula for dy/dx. However, this results in a division by zero, so the slope of the tangent line at (2, 0) does not exist.
(c) To find the slope of the tangent line at (4, 3๐/2), we substitute ๐ = 3๐/2 into the formula for dy/dx. However, this also results in a division by zero, so the slope of the tangent line at (4, 3๐/2) does not exist.
Similar Questions
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