Answer to Question #252563 in Calculus for Mary

Question #252563

Find the area of the triangle formed from the coordinate axes and the tangent line to the curve y = 5x^(−1) −x/5 at the point (5,0).


1
Expert's answer
2021-10-20T02:21:42-0400
y=5xx5y=\dfrac{5}{x}-\dfrac{x}{5}

Find the first derivative


y=5x215y'=-\dfrac{5}{x^2}-\dfrac{1}{5}

Point (5,0)(5,0)


slope=m=y(5)=55215=25slope=m=y'(5)=-\dfrac{5}{5^2}-\dfrac{1}{5}=-\dfrac{2}{5}

The equation of the tangent to the curve at the point (5, 0) in point-slope form


y0=25(x5)y-0=-\dfrac{2}{5}(x-5)

The equation of the tangent to the curve at the point (5, 0) in slope-intercept form


y=25x+2y=-\dfrac{2}{5}x+2

yy-intercept: x=0=>y=25(0)+2=2.x=0=>y=-\dfrac{2}{5}(0)+2=2. Point (0,2).(0, 2).


xx-intercept: 0=0=>y=25x+2=>x=5.0=0=>y=-\dfrac{2}{5}x+2=>x=5. Point (5,0).(5, 0).


The area of the right triangle is equal to


A=12(2)(5)=5(units2)A=\dfrac{1}{2}(2)(5)=5({units}^2)

The area of the triangle formed from the coordinate axes and the tangent line to the curve y=5x1x5y=5x^{-1}-\dfrac{x}{5} at the point (5,0) is 5 squared units.



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