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Chapter 4: Problem 46

Rank the functions in order of how quickly they approach 0 as \(x \rightarrow \infty\) \(y=e^{-x}, \quad y=x e^{-x}, \quad y=e^{-x^{2}}, \quad y=x e^{-x^{2}}\)

Short Answer

Expert verified
Fastest to slowest: 1) \(e^{-x^2}\), 2) \(x e^{-x^2}\), 3) \(x e^{-x}\), 4) \(e^{-x}\).

Step by step solution

01

Analyze Each Function

Let's evaluate how each function behaves as \(x\) approaches infinity. For \(y = e^{-x}\), as \(x\) increases, \(e^{-x}\) decreases exponentially towards 0. Similarly, \(y = e^{-x^2}\) decreases even faster since the exponent \(-x^2\) reduces the value more rapidly than \(-x\).
02

Analyze Functions with \(x\) Multiplicative Term

For \(y = x e^{-x}\), the presence of \(x\) means we multiply a growing term with a rapidly decreasing term. As \(x\) becomes large, the exponential factor will dominate, driving the product towards 0. In \(y = x e^{-x^2}\), the term \(x e^{-x^2}\) includes a polynomial multiplied by a more rapidly decaying exponential, which will approach 0 even more quickly.
03

Rate of Decay Analysis

Compare exponential rates. \(e^{-x}\) decreases gradually, \(x e^{-x}\) decreases faster due to the extra polynomial \(x\) but eventually \(e^{-x}\) dominates. \(e^{-x^2}\) decreases extremely fast, and \(x e^{-x^2}\) also decreases extremely fast, but a bit less quickly than \(e^{-x^2}\) alone.
04

Rank the Functions

The order in which these functions approach 0 fastest to slowest as \(x\rightarrow fty\) is as follows: 1) \(e^{-x^2}\), 2) \(x e^{-x^2}\), 3) \(x e^{-x}\), 4) \(e^{-x}\). \(e^{-x^2}\) decreases the fastest due to the squared term, while \(e^{-x}\) is the slowest among them as it lacks the enhancing factor of \(x^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay
Exponential decay refers to a process where quantities decrease exponentially over time or distance. In mathematics, this is represented by an exponential function such as \(e^{-x}\). Here, as \(x\) increases, the value of the function decreases towards zero at an exponential rate.
  • For the function \(e^{-x}\), this means that no matter how large \(x\) becomes, \(e^{-x}\) will continue to get smaller, but it will never actually reach zero.
  • Another example is \(e^{-x^2}\), which decreases even faster than \(e^{-x}\) because the exponent of the exponential term involves a squared term, exacerbating the decrease as \(x\) grows.
Exponential decay is often used in real-world phenomena like radioactive decay or cooling processes, where things diminish at a rate proportional to their current value. Understanding this principle helps in analyzing how variables quickly approach their limits or zero.
Behavior at Infinity
The behavior of functions as they approach infinity helps us understand their long-term trends. This is generally referred to as 'behavior at infinity'. Particularly for exponential decay, it tells us how fast a function drops towards zero as \(x\) goes to infinity.
For instance:
  • In the function \(e^{-x}\), as \(x\) becomes very large, the function decreases towards zero. This is a classic example of an exponential function approaching its limit at infinity.
  • Contrast this with \(e^{-x^2}\). The squared term in the exponent hastens the descent, driving the function closer to zero at a much faster pace as \(x\) increases.
The concept of behavior at infinity is crucial in calculus and mathematical analysis because it gives us insights into how functions behave in extreme conditions, allowing for more informed predictions and problem-solving.
Comparing Rates of Decay
Comparing rates of decay involves analyzing which functions decrease faster as their input variable increases. In our example:
  • \(e^{-x}\) represents a pure exponential decay, providing a basic understanding of declining rates.
  • Introducing \(x\) in the function, as in \(x e^{-x}\), complicates the situation a bit, because there's a delicate balance between polynomials and exponentials.
  • When moving to \(e^{-x^2}\), we encounter faster decay due to the exponent \(-x^2\). This is even more pronounced when combined with a polynomial term as \(x e^{-x^2}\).
Essentially, when comparing these functions, consider:
  • Exponential terms typically dominate over polynomial terms as \(x\) tends towards infinity.
  • Adding squared terms to the exponent increases the rate of decay significantly.
By understanding these differences, we can more accurately rank the functions according to their speed of decay towards zero.

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Most popular questions from this chapter

Find the most general antiderivative of the function.(Check your answer by differentiation.) \(f(x)=5 x^{1 / 4}-7 x^{3 / 4}\)

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Find the most general antiderivative of the function.(Check your answer by differentiation.) \(f(x)=2 \sqrt{x}+6 \cos x\)
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