Answer to Question #251985 in Calculus for Steve

Question

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).

Answer 1

Let us find the area of the triangle formed from the coordinate axes and the tangent line to the curve $y = \frac{5}{x}−\frac{x}5$ at the point (5,0). It follows that $y' = -\frac{5}{x^2}−\frac{1}5,$ and hence $y'(5)=-\frac{2}5.$ The equation of the tangent to the curve at the point (5, 0) is $y=-\frac{2}5(x-5).$ If $x=0,$ then $y=2,$ and hence the point (0, 2) is the y-intercept. We conclude that the area of the triangle is equal to $\frac{1}2\cdot 5\cdot 2=5$ squared units.