The curve defined by the equation: x^(2/3) + y^(2/3) = 1 is an example of an astroid (shown below). In this problem, we will use and compare two methods for finding all points on the astroid at which the tangent lines have slope −1. (a) Method 1: Solving for y. i. Rearrange the equation of the astroid to find an equation for the top half of the astroid (i.e. solve for y where y ≥ 0). ii. Use this formula to calculate dy/dx and use this derivative formula to find the point(s) on the top half of the curve at which the tangent line has slope −1. iii. Use the symmetry of the curve to find the point(s) on the bottom half of the astroid at which the tangent line has slope −1. (b) Method 2: Use Implicit Differentiation. Use implicit differentiation to find dy/dx and use your calculation to find all of the points on
The curve defined by the equation:
x^(2/3) + y^(2/3) = 1
is an example of an astroid (shown below).
In this problem, we will use and compare two methods for finding all points on the astroid at which the tangent lines have slope −1.
(a) Method 1: Solving for y.
i. Rearrange the equation of the astroid to find an equation for the top half of the astroid (i.e. solve for y where y ≥ 0).
ii. Use this formula to calculate dy/dx and use this derivative formula to find the point(s) on the top half of the curve at which the tangent line has slope −1.
iii. Use the symmetry of the curve to find the point(s) on the bottom half of the astroid at which the tangent line has slope −1.
(b) Method 2: Use Implicit Differentiation. Use implicit differentiation to find dy/dx and use your calculation to find all of the points on the astroid at which the tangent line has slope −1. Does this answer agree with what you found in part (a)?
(c) Which method do you prefer? Why?
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