Answer to Question #155280 in Calculus for Pia

Question #155280

Find dy/dx and d^2y/dx^2 without eliminating the parameter.


1. x=1-t^2 ; y=1+t


1
Expert's answer
2021-01-13T19:34:34-0500

We have "x = 1 - t^2" and "y=1+t"

We must find  "\\large\\frac{dy}{dx}" and "\\large\\frac{d^2y}{dx^2}" without eliminating the parameter t

"\\large\\frac{dy}{dx} = \\large\\frac{(1+t)'}{(1-t^2)} = \\large\\frac{1}{-2t}"

from this  "x = 1 - t^2" , we find "t = \\sqrt{1-x}"

"\\large\\frac{dy}{dx} = -\\large\\frac{1}{2\\sqrt{1-x}}"

"\\large\\frac{d^2y}{dx^2} =" "( -\\large\\frac{1}{2\\sqrt{1-x}})' = \\large\\frac{1}{4\\sqrt{(1-x)^3}}"


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