Question

(a) find the point on the curve at which the curvature \(K\) is a maximum and (b) find the limit of \(K\) as \(x \rightarrow \infty\)/. $$ y=e^{x} $$

Short answer

The curvature \(K\) does not reach a maximum for real \(x\). The limit of \(K\) as \(x\) approaches infinity is \(0\).

Step-by-step solution

Find the first and second derivatives of y

The first derivative of \(y=e^x\) is \(y'=e^x\). The second derivative is \(y''=e^x\).

Substitute in the equation for curvature

We can substitute these derivatives into the equation for \(K(x)\): \(K(x) = \frac{e^x}{(1 + e^{2x})^{3/2}}\).

Find where the curvature reaches its maximum

To find the maximum of \(K(x)\), we take its derivative and set it to zero. We find that this equation has no real solution, meaning \(K(x)\) has no maximum value for real \(x\).

Find the limit of the curvature as x approaches infinity

As \(x\) approaches infinity, both the numerator and denominator in \(K(x)\) approach infinity. By applying L'Hopital's rule, we find the limit to be \(0\).

Key concepts

Derivatives

Understanding derivatives is crucial for calculus, especially when analyzing the behavior of functions like exponential functions used in curvature problems.

Derivatives represent how a function changes as its input changes. Essentially, the derivative of a function at a certain point is the slope of the tangent line at that point on the graph of the function. It gives us a mathematical way to understand rates of change.

For the function y = e^x, which is an exponential function, the derivative with respect to x is simply y' = e^x. This fascinating property of exponential functions, where the derivative is the same as the function itself, is a cornerstone in calculus. The second derivative, or the derivative of the derivative, y'' = e^x, represents the rate of change of the rate of change, which can be associated with acceleration in physics.

L'Hopital's Rule

L'Hopital's rule is a powerful tool for solving limit problems in calculus, particularly when those limits are of the indeterminate form 0/0 or ∞/∞.

According to L'Hopital's rule, if the limit of functions f(x) and g(x) as x approaches a certain value results in an indeterminate form, then the limit of the ratio of their derivatives might give us a clear answer.

In practice, L'Hopital's rule tells us that the limit as x approaches some value of f(x)/g(x) can be found by taking the limit of their derivatives, provided certain conditions are met. This is exactly what was applied in the exercise to find the limit of the curvature as x approaches infinity.

Limits

In calculus, limits help us understand the behavior of functions as they approach a specific point or infinity. They are the foundation of both derivatives and integrals.

For instance, when we say that the limit of K(x) as x approaches infinity is 0, we mean that as x gets larger and larger, the curvature K(x) gets closer and closer to 0, even if it never actually reaches 0. This concept of approaching a value is essential for dealing with continuous functions and determining the behavior of curves, slopes, and areas.

Exponential Functions

Exponential functions like y = e^x have unique characteristics that make them widely applicable in various fields of study, including natural growth/decay processes, finance, and physics.

These functions are characterized by a constant rate of growth (or decay), proportional to their value. When we say y = e^x, the base e is an irrational and transcendental number approximately equal to 2.71828. The function grows faster as x increases because the rate of increase is always proportional to the current value of y.

The property that the derivative of e^x is itself e^x makes it very convenient to work with when finding derivatives and solving calculus problems related to growth and decay.