A) Determine the equation of the tangent line to F(x) = (2 - x^2)/(1+x) at (0,2). write your answer in the form Y=mx+b? b) determine the equation of the normal to f(x) at (0,2)?

1 Answer
Jim G.
Apr 16, 2018

#y=-2x+2" and "y=1/2x+2#

Explanation:

#"the slope of the tangent is given by "f'(x)" at "x=0#

#"differentiate using the "color(blue)"quotient rule"#

#"Given "f(x)=(g(x))/(h(x))" then"#

#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#

#g(x)=2-x^2rArrg'(x)=-2x#

#h(x)=1+xrArrh'(x)=1#

#rArrf'(x)=(-2x(1+x)-(2-x^2))/(1+x)^2#

#color(white)(rArrf'(x))=(-x^2-2x-2)/(1+x)^2#

#rArrf'(0)=-2/1=-2#

#rArry=-2x+2larrcolor(red)"equation of tangent line"#

#"the normal line is perpendicular to the tangent line"#

#rArrm_(color(red)"normal")=-1/(-2)=1/2#

#rArry=1/2x+2larrcolor(red)"equation of normal line"#

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