The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.
The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]
(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project ""The exponential Eiffel Tower"")
b. With a = 0 and c = 2, find the equations of the lines tangent to both curves at x = 0

Verified step by step guidance

1. Identify the two curves given: the upper curve is \(y = e^{-2x}\) and the lower curve is \(y = -e^{-2x}\), since \(a = 0\) and \(c = 2\).
2. Find the derivative of the upper curve \(y = e^{-2x}\) with respect to \(x\) to determine the slope of the tangent line at \(x = 0\). Use the chain rule: \(\frac{dy}{dx} = -2e^{-2x}\).
3. Evaluate the slope of the tangent line at \(x = 0\) by substituting \(x = 0\) into the derivative: \(m = -2e^{0} = -2\).
4. Find the point of tangency on the upper curve at \(x = 0\): \(y = e^{0} = 1\). So the point is \((0, 1)\).
5. Use the point-slope form of a line \(y - y_1 = m(x - x_1)\) with the point \((0, 1)\) and slope \(-2\) to write the equation of the tangent line to the upper curve. Repeat the process for the lower curve \(y = -e^{-2x}\), noting that its derivative is \(\frac{dy}{dx} = 2e^{-2x}\) and the point at \(x=0\) is \((0, -1)\).

Key Concepts

Exponential Functions and Their Properties

Exponential functions have the form y = e^(kx), where e is Euler's number and k is a constant. They exhibit rapid growth or decay depending on the sign of k. Understanding their behavior, especially at specific points like x = 0, is crucial for analyzing curves and their tangents.

Derivative and Tangent Line Equation

The derivative of a function at a point gives the slope of the tangent line there. To find the tangent line equation, compute the derivative at the point, then use the point-slope form: y - y₀ = m(x - x₀), where m is the slope and (x₀, y₀) is the point of tangency.

Evaluating Definite Integrals with Infinite Limits

Integrals with limits extending to infinity are improper integrals and require limits to evaluate. For functions like e^(-cx), these integrals converge if c > 0. Understanding how to compute these integrals is essential for problems involving areas, centers of mass, or average values over infinite intervals.