91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis
on the interval [b, ∞).
c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.

Verified step by step guidance

1. First, understand the region R is bounded by the curve \(y = e^{-a \cdot x}\) and the x-axis over the interval \([b, \infty)\). The area \(A(a,b)\) of this region is given by the integral of the function from \(b\) to infinity: \[A(a,b) = \int_{b}^{\infty} e^{-a \cdot x} \, dx\]
2. Next, compute the integral \(A(a,b)\). Since \(a > 0\), the integral converges and can be evaluated as an improper integral: \[A(a,b) = \lim_{t \to \infty} \int_{b}^{t} e^{-a \cdot x} \, dx\] Use the antiderivative of \(e^{-a x}\), which is \(-\frac{1}{a} e^{-a x}\).
3. Evaluate the definite integral: \[A(a,b) = \lim_{t \to \infty} \left[-\frac{1}{a} e^{-a x} \right]_{x=b}^{x=t} = \lim_{t \to \infty} \left(-\frac{1}{a} e^{-a t} + \frac{1}{a} e^{-a b} \right)\] Since \(a > 0\), \(e^{-a t} \to 0\) as \(t \to \infty\), so the area simplifies to \[A(a,b) = \frac{1}{a} e^{-a b}\]
4. The problem asks to find the minimum value \(b^*\) such that for any \(b > b^*\), there exists some \(a > 0\) with \(A(a,b) = 2\). Set up the equation: \[\frac{1}{a} e^{-a b} = 2\] Rewrite it as \[e^{-a b} = 2a\]
5. To find \(b^*\), consider the function \(f(a) = 2a e^{a b}\). For fixed \(b\), the equation \(e^{-a b} = 2a\) can be rearranged to \$1 = 2a e^{a b}\(. We want to find the smallest \)b\( such that this equation has a positive solution \)a\(. Analyze the function \)g(a) = 2a e^{a b}\( for \)a > 0\( and find its minimum value with respect to \)a\(. Then, find \)b^*\( such that the minimum of \)g(a)\( equals 1. This involves taking the derivative of \)g(a)\( with respect to \)a\(, setting it to zero to find critical points, and solving for \)b\( in terms of \)a\(. This will give the minimal \)b^*$.

Key Concepts

Definite Integral and Area Under a Curve

The definite integral calculates the area between a curve and the x-axis over a specified interval. For y = e^{-a·x}, integrating from b to infinity gives the total area under the exponential decay curve, which is essential for expressing A(a,b) and analyzing its behavior.

Improper Integrals and Convergence

Integrals with infinite limits, like from b to ∞, are improper and require evaluating limits to determine convergence. Understanding when the integral converges and how it depends on parameters a and b is crucial to finding values where the area equals a specific number.

Parameter Dependence and Optimization

The problem involves finding a minimum b* such that for b > b*, there exists an a > 0 making the area equal to 2. This requires analyzing how the integral's value changes with parameters a and b, and using algebraic or calculus methods to solve for the critical threshold b*.