Question

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

Answer 1

Solution:

Since the derivative of y with respect to x is

 $y^{\prime}(x)=\frac{d}{d x}\left[5 x^{-1}-\frac{1}{5} x\right]=\frac{d}{d x}\left[5 x^{-1}\right]-\frac{d}{d x}\left[\frac{1}{5} x\right]=-5 x^{-2}-\frac{1}{5}$  

the slope of the tangent line at the point (5,0) is $y^{\prime}(5)=-\frac{2}{5}$ . Thus, the equation of the tangent line at this point is

 $y-0=-\frac{2}{5}(x-5) \text { or equivalently } \quad y=-\frac{2}{5} x+2$

Since the y-intercept of this line is 2 , the right triangle formed from the coordinate axes and the tangent line has legs of length 5 and 2, so its area is $\frac{1}{2}(5)(2)=5$