Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider \(f(x)=n^{2} x e^{-n x}\). (a) Graph \(f(x)\) for \(n=1,2,3,4,5,6\) on \([0,1]\) in the same graph window. (b) For \(x>0\), find \(\lim _{n \rightarrow \infty} f(x)\). (c) Evaluate \(\int_{0}^{1} f(x) d x\) for \(n=1,2,3,4,5,6\). (d) Guess at \(\lim _{n \rightarrow \infty} \int_{0}^{1} f(x) d x\). Then justify your answer rigorously.

Step-by-Step Solution

Answer

(a) Plot shows steeper graph; (b) Limit: 0; (c) Values using formula; (d) Integral limit: 1.

Step 1: Graphing for Different n

To graph \(f(x) = n^2 x e^{-nx}\) on the interval \([0, 1]\) for \(n = 1, 2, 3, 4, 5, 6\), plot the function for each value of \(n\). Note that as \(n\) increases, the graph of \(f(x)\) becomes steeper and shifts toward the y-axis.

Step 2: Finding the Limit for x > 0

For \(x > 0\), evaluate the limit \(\lim_{n \to \infty} f(x) = \lim_{n \to \infty} n^2 x e^{-nx}\). As \(n\) becomes very large, the term \(e^{-nx}\) approaches zero faster than \(n^2 x\) increases. Therefore, the overall product tends to zero. Thus, \(\lim_{n \to \infty} f(x) = 0\).

Step 3: Evaluate the Integral for Given n

Compute the integral \(\int_{0}^{1} f(x) dx = \int_{0}^{1} n^2 x e^{-nx} dx\) for each \(n = 1, 2, 3, 4, 5, 6\). Using integration by parts, where \(u = x\) and \(dv = n^2 e^{-nx} dx\), the integral becomes \(-e^{-nx}(nx + 1)\) evaluated from 0 to 1. This results in \(1 - (n+1)e^{-n}\). Do this for each \(n\) to get the values.

Step 4: Guessing and Justifying the Limit of the Integral

As \(n \rightarrow \infty\), look at the expression \(1 - (n+1)e^{-n}\) from Step 3. Since \(e^{-n}\) approaches zero faster than \(n \rightarrow \infty\), the term \((n+1)e^{-n}\) also approaches zero. Thus, \(\lim_{n \rightarrow \infty} \int_{0}^{1} f(x) dx = 1\). Hence, the integral approaches 1.

Key Concepts

Graphing Functions

Graphing functions involves plotting a mathematical function to visually analyze its behavior and properties. Imagine the function \(f(x) = n^2 x e^{-nx}\) graphed over the interval \([0, 1]\) for different values of \(n\). A graph allows us to see how the function behaves as we vary \(n\), which helps in understanding its dynamics.
For our function, you can notice that as \(n\) increases from 1 to 6, the curve gets steeper and piles up closer to the y-axis. This graphical re-evaluation gives us an insight into how the function behaves or "stretches" when the parameter \(n\) changes.
Here are a few tips for graphing functions effectively:

• Choose the right window range; here, it's \([0, 1]\).
• Use different colors or line styles to distinguish each graph which represents a different value of \(n\).
• Look for trends or patterns, like how changes to \(n\) affect the function’s shape.

Integration by Parts

Integration by parts is a powerful technique used in calculus to integrate products of functions. It is particularly useful when standard integration methods don't apply easily. The formula for integration by parts is: \[\int u \, dv = uv - \int v \, du\]In the context of the function \(f(x) = n^2 x e^{-nx}\), we apply integration by parts to evaluate the definite integral \(\int_{0}^{1} f(x) dx\).
Let's choose:

• \(u = x\), making \(du = dx\)
• \(dv = n^2 e^{-nx} dx\), which upon integration gives \(v = -e^{-nx}\)

Applying the above choices in the integration by parts formula results in an expression that simplifies the evaluation of our integral. This process transforms the integral into a more manageable expression, leading to the solution \(-e^{-nx}(nx + 1)\) from 0 to 1.

Exponential Functions

Exponential functions are defined by expressions where the variable is in the exponent, typically in the form \(a^x\) where \(a\) is a constant. These functions play a critical role in many mathematical models, including growth and decay processes.
The function in consideration, \(f(x) = n^2 x e^{-nx}\), incorporates the exponential component \(e^{-nx}\). This part of the function decreases rapidly as \(n\) increases, especially for \(x > 0\), which significantly influences the behavior of \(f(x)\).
Understanding the properties of exponential functions can help explain why \(e^{-nx}\) approaches zero faster compared to the increase of \(n^2 x\) when \(n\) is large. This is why \(\lim_{n \to \infty} f(x) = 0\) for \(x > 0\). Exponential decay plays a key role here, overcoming linear and polynomial growth represented by \(n^2 x\).

Definite Integrals

Definite integrals calculate the accumulated quantity, or "net area", from one point to another under the curve of a function. This gives an exact numerical value representing the total accumulation on the given interval \([a, b]\).
For our function \(f(x) = n^2 x e^{-nx}\), we evaluated the integral over \([0, 1]\) for different \(n\). In each case, the outcome \(1 - (n+1)e^{-n}\) shows how we calculate the precise "net area" formed by the function's graph.
As \(n\) tends to infinity, understanding why this integral approaches 1 relies on the interplay between the exponential decay \(e^{-n}\) and the finite bounds of the interval.
Here's a short recap of tips for definite integrals:

• Understand the bounds and interval of integration - in our case, from 0 to 1.
• Evaluate the integral step by step, double checking calculations in intricate expressions.
• Consider how changes in the function's parameters affect the outcome, observing patterns as \(n\) increases.