Since the function is f(x) = exp(2-x), the derivative, f', is
f' = -exp(2-x) .
When evaluated at x = 1, this gives -e. This is the slope of the tangent line.
Since the slope and one point (1,e) are known, the point slope equation can be used to find the equation of the line.
y - e = (-e) ( x-1) This can be rearranged into the standard form
y = -e x + 2e. The y intercept of this line is clearly y = 2e. The x intercept (value of x that makes y zero) is x = 2.
This line, together with the coordinate axes, make a right triangle with base 2 and height 2e.
The area of a triangle is (1/2) base x height. So the area is 2e.
