We are analyzing the curve given by $y = e^x$, and we want to determine the following:

1. At what points does the curve reach maximum curvature?

2. What happens to the curve as $x \to \infty$?

1. Finding Maximum Curvature

The curvature $\kappa$ of a function $y = f(x)$ is given by the formula:

$\kappa(x) = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}$

For $y = e^x$:

• First derivative: $f'(x) = e^x$
• Second derivative: $f''(x) = e^x$

Now, applying these to the curvature formula:

$\kappa(x) = \frac{|e^x|}{(1 + (e^x)^2)^{3/2}} = \frac{e^x}{(1 + e^{2x})^{3/2}}$

To find the maximum curvature, we need to differentiate $\kappa(x)$ with respect to $x$ and solve $\kappa'(x) = 0$.

Let's compute $\kappa'(x)$ to find the critical points where the curvature could be maximized.

$\kappa'(x) = \frac{(e^x)'(1 + e^{2x})^{3/2} - e^x((1 + e^{2x})^{3/2})'}{(1 + e^{2x})^3}$

Rather than manually expanding this expression, we can notice that $e^x$ grows exponentially, while $1 + e^{2x}$ grows even faster. Hence, as $x \to \infty$, $\kappa(x)$ tends to zero, meaning that the curvature decreases as $x$ becomes large.

By observation, $\kappa(x)$ is highest when $x$ is small, particularly around $x = 0$.

Thus, the curve reaches maximum curvature around $x = 0$.

2. Behavior of the Curve as $x \to \infty$

As $x \to \infty$, the exponential function $y = e^x$ increases without bound. The curve steepens rapidly as $x$ increases, becoming almost vertical at large values of $x$. This means:

• The slope $f'(x) = e^x$ increases exponentially.
• The curvature $\kappa(x)$ tends to 0 because the curve flattens out in a relative sense, even though it is steep. The rate at which the curvature decreases far outweighs the increase in the steepness of the curve.

Summary

• The curve $y = e^x$ reaches its maximum curvature around $x = 0$.
• As $x \to \infty$, the curve becomes steeper, and the curvature tends to zero.

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Would you like more details on the curvature calculation, or have any other questions?

Here are 5 related questions for further exploration:

1. What is the curvature at $x = 0$ for the curve $y = e^x$?
2. How does the curvature of $y = e^x$ compare to other exponential functions like $y = 2^x$?
3. What is the second derivative test, and how does it relate to maximum curvature?
4. How does the curvature of $y = e^x$ change for negative values of $x$?
5. What happens to the curvature of the curve $y = \ln(x)$ as $x \to \infty$?

Tip: For steep curves like $y = e^x$, the curvature decreases even as the slope increases, as curvature depends on the change in slope rather than just its magnitude.