You’ve given two equations:

1. $y = x e^x$
2. $y = e^x$

along with the conditions: $x = 0$ and $x = 1$. Based on the wording, it seems like you want to explore these two equations between the limits $x = 0$ and $x = 1$.

Let’s break it down:

1. Evaluating at $x = 0$:

For the first equation $y = x e^x$, when $x = 0$:

$y = 0 \times e^0 = 0$

For the second equation $y = e^x$, when $x = 0$:

$y = e^0 = 1$

2. Evaluating at $x = 1$:

For the first equation $y = x e^x$, when $x = 1$:

$y = 1 \times e^1 = e$

For the second equation $y = e^x$, when $x = 1$:

$y = e^1 = e$

Summary:

• At $x = 0$:

  • $y = 0$ for $y = x e^x$
  • $y = 1$ for $y = e^x$

• At $x = 1$:

  • $y = e$ for $y = x e^x$
  • $y = e$ for $y = e^x$

Let me know if you want to explore anything further like plotting or analyzing their behavior between $x = 0$ and $x = 1$, or anything else!

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Here are a few related questions:

1. How do the functions $y = x e^x$ and $y = e^x$ compare on an interval like $[0, 1]$?
2. What is the derivative of $y = x e^x$?
3. How do exponential growth and multiplication by $x$ affect the behavior of these functions?
4. Can you compute the area between the curves $y = x e^x$ and $y = e^x$ over the interval $[0, 1]$?
5. What is the general behavior of the function $y = x e^x$ for larger values of $x$?

Tip: When evaluating an exponential function like $e^x$, remember that $e^0 = 1$, and the exponential function grows faster as $x$ increases.