Question

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.e. Does this pattern continue? Is it true that A(1, ln b) = a² * A(a, (ln b)/a)?

Accepted Answer

First, understand the function given: $$y = x e$$^{-a x}$$, where a$ is a parameter that changes the shape of the curve. The graph shows this function for a = 1$$, 2, 3$$. Next, identify what A(a$$, b)$$ represents in the problem. Typically, A(a$$, b)$$ might denote the value of the function or an area related to the function at specific points. Here, it likely means the value of the function y = x$$ $$e^{-a x}$$ evaluated at $$x = b$$ for a given $$a$$, so $$A(a, b) = b e$$^{-a b}$$. To check the given relation A(1$$, \ln b) = $$a^2$$ \cdot A(a, \frac{\ln b}{a})$$, substitute the definitions: 
- Compute A(1$$, \ln b) = (\ln b) $$e^{-1 \cdot \ln b}.
- $$Compute $$A(a, \frac{\ln b}{a}) = \left(\frac{\ln b}{a}\right) e$$^{-a \cdot \frac{\ln b}$${a}}. $$Simplify the exponentials using the property $$e^{\ln x} = x$ and e$$^{-\ln x}$$ = \frac{1}{x}$$:
- $$e^{-\ln b}$$ = \frac{1}{b}$$,
- e$$^{-a \cdot \frac{\ln b}$${a}} = e$$^{-\ln b}$$ = \frac{1}{b}. $$After simplification, compare both sides of the equation:
- Left side: $$A(1, \ln b) = (\ln b) \cdot \frac{1}{b} = \frac{\ln b}{b}$$,
- Right side: $$a^2$$ \cdot \left( \frac{\ln b}{a} \cdot \frac{1}{b} \right) = $$a^2$$ \cdot \frac{\ln b}{a b} = a \cdot \frac{\ln b}{b}$$.

Since the left side is $$\frac{\ln b}{b}$$ and the right side is a$$ \cdot \frac{\ln b}{b}$$, the equality holds only if a = 1$. Therefore, the pattern does not continue for all a$.

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

e. Does this pattern continue? Is it true that A(1, ln b) = a² * A(a, (ln b)/a)?

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Concetti chiave

Exponential Functions and Their Properties

Area Under a Curve and Definite Integrals

Change of Variables in Integration