Answer to Question #166974 in Calculus for Hiba

Question #166974

i) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the x-axis.

ii) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the y-axis


1
Expert's answer
2021-02-26T04:53:04-0500

i) Use implicit differentiation, then set dy dx = 0.


ddx(x(x+y2))=dydx\dfrac{d}{dx}(x(x+y^2))=\dfrac{dy}{dx}

2x+y2+2xydydx=dydx2x+y^2+2xy\dfrac{dy}{dx}=\dfrac{dy}{dx}

dydx=2x+y212xy\dfrac{dy}{dx}=\dfrac{2x+y^2}{1-2xy}

dydx=0=>2x+y212xy=0\dfrac{dy}{dx}=0=>\dfrac{2x+y^2}{1-2xy}=0

2x+y2=0,12xy02x+y^2=0, 1-2xy\not=0

x=y22x=-\dfrac{y^2}{2}

Substitute


y22(y22+y2)=y-\dfrac{y^2}{2}(-\dfrac{y^2}{2}+y^2)=y

y44=y-\dfrac{y^4}{4}=y

y1=0,x1=0,Point(0,0)y_1=0, x_1=0, Point(0, 0)


y3=4y^3=-4

y2=43,x2=23,Point(23,43)y_2=-\sqrt[3]{4}, x_2=-\sqrt[3]{2}, Point (-\sqrt[3]{2}, -\sqrt[3]{4})


Point(0,0),Point(23,43)Point(0, 0), Point (-\sqrt[3]{2}, -\sqrt[3]{4})


ii) ) The points with vertical tangents are those where the denominator of dydx\dfrac{dy}{dx} is zero (making the slope undefined). From part (i), we have


dydx=2x+y212xy=>12xy=0\dfrac{dy}{dx}=\dfrac{2x+y^2}{1-2xy}=>1-2xy=0

x=12y,y0x=\dfrac{1}{2y}, y\not=0

Substitute


12y(12y+y2)=y\dfrac{1}{2y}(\dfrac{1}{2y}+y^2)=y

14y2=y2\dfrac{1}{4y^2}=\dfrac{y}{2}

y=432y=\dfrac{\sqrt[3]{4}}{2}

x=232x=\dfrac{\sqrt[3]{2}}{2}

Point(232,432)Point(\dfrac{\sqrt[3]{2}}{2}, \dfrac{\sqrt[3]{4}}{2})



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