A. To find the exact area of the region bounded by the curves y = f(x) = 2^x - 1 and y = x in the first quadrant, we need to calculate the integral of the difference of the two functions over the interval where they intersect.
First, we need to find the x-coordinate of the point of intersection. Set f(x) = 2^x - 1 equal to x: 2^x - 1 = x. Rearranging this equation, we have 2^x - x - 1 = 0.
Unfortunately, there is no straightforward algebraic way to solve this equation, so we'll need to use numerical methods or a graphical approach to find an approximation of the x-coordinate. Once we have the x-coordinate, we can integrate the difference of the two functions to find the area.
B. To find the inverse function y = f^(-1)(x), we need to swap the roles of x and y in the equation y = 2^x - 1 and solve for x.
Start by rewriting the equation as x = 2^y - 1. Then, solve for y by isolating the exponential term: 2^y = x + 1. Taking the logarithm of both sides, we have y = log2(x + 1).
Therefore, the inverse function of f(x) = 2^x - 1 is f^(-1)(x) = log2(x + 1).
C. Since f(x) and f^(-1)(x) are inverse functions, the area bounded by their curves in the first quadrant is symmetric with the area bounded by the curves y = f(x) = 2^x - 1 and y = x.
This means that the area bounded by the curves y = f^(-1)(x) and y = x in the first quadrant is the same as the area bounded by the curves y = f(x) = 2^x - 1 and y = x.
Therefore, the exact area of the region in the first quadrant bounded by the curves y = f^(-1)(x) and y = x is the same as the area found in part A.
D. The average value of a function f(x) on an interval [a, b] is given by the formula (1 / (b - a)) * ∫[a,b] f(x)dx.
We can set up the integral to represent the average value of f(x) on the interval [0, a] and solve for a.
The average value is 1, so we have the equation 1 = (1 / a) * ∫[0,a] (2^x - 1)dx.
Integrating the function, we get 1 = (1 / a) * (2^(a+1) / ln(2) - a).
To find the value of a, we'll need to use a calculator or numerical methods to solve this equation.