Please help me with this calculus problem please!!

Question

The line tangent to y=e ^2-x at the point (1,e) intersects both axes. What is the area of the triangle formed by the tangent line and the axes?

Answer 1

y' = -e^2-x so slope m = y'(1) = -e

 

tangent at (1,e) has equation y - e = -e(x-1) or

y = -xe + 2e

 

They x intercept is 2 and the y intercept is 2e; this triangle in first quadrant (draw it!)  has 

Area = ½•2•2e = 2e.

 

Check all of this work!