Problem 54 - Calculus Early Transcendentals

Question

Using L'Hôpital's rule (Section 3.6) one can verify that $$ \lim _{x \rightarrow+\infty} \frac{e^{x}}{x}=+\infty, \quad \lim _{x \rightarrow+\infty} \frac{x}{e^{x}}=0, \quad \lim _{x \rightarrow-\infty} x e^{x}=0 $$ In these exercises: (a) Use these results, as necessary, to find the limits of \(f(x)\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). (b) Sketch a graph of \(f(x)\) and identify all relative extrema, inflection points, and asymptotes (as appropriate). Check your work with a graphing utility. $$ f(x)=x^{3} e^{x-1} $$

Step-by-Step Solution

Answer

As \(x \to +\infty\), \(f(x) \to +\infty\). As \(x \to -\infty\), \(f(x) \to 0\). Critical points occur at \(x = 0\) and \(x = -3\).

Step 1: Analyze the function as x approaches infinity

Consider \(f(x) = x^3 e^{x-1}\). To analyze the behavior as \(x \to +\infty\), note that \(e^{x-1} = \frac{e^x}{e}\). So, \(f(x) = \frac{x^3 e^x}{e}\). As \(x \to +\infty\), \(e\) is a constant, so we focus on \(x^3 e^x\). Since \(e^x\) grows faster than any polynomial, \(\lim_{x \to +\infty} f(x) = +\infty\).

Step 2: Analyze the function as x approaches negative infinity

Consider \(f(x) = x^3 e^{x-1}\). As \(x \to -\infty\), \(e^{x-1}\) approaches 0 more rapidly than any polynomial \(x^3\) as shown by the expression \(x^3 e^{x}\). From the provided limit, \(\lim_{x \to -\infty} x e^x = 0\), implying that higher powers of \(x\) with \(e^x\) also tend toward zero. Therefore, \(\lim_{x \to -\infty} f(x) = 0\).

Step 3: Identify key features for sketching the graph

Since \(f(x)\) is an exponential polynomial, determine critical points by finding \(f'(x)\). Use the product rule: \(f'(x) = 3x^2 e^x + x^3 e^x\). For stationary points, set \(f'(x) = 0\), which implies \(x^2 e^x (3 + x) = 0\). Thus, \(x = 0\) or \(x = -3\) solve this equation. Calculate the second derivative \(f''(x)\) for concavity and inflection points. Use a graphing utility to aid in precision.

Key Concepts

Limits

In calculus, limits are a fundamental concept that describe the behavior of a function as the input approaches a specified value. Limits are essential in analyzing function behavior, especially when dealing with infinity or situations where direct substitution is not possible. They provide insight into function tendencies and help in identifying asymptotes and other key characteristics of functions. For instance, with the function \( f(x) = x^3 e^{x-1} \), we utilize limits to understand how the function behaves as \( x \) approaches both positive and negative infinity.

- *As \( x \rightarrow +fty \)*: We see that exponential functions like \( e^x \) grow much faster than polynomial expressions such as \( x^3 \). Hence, the limit \( \lim_{x \to +\infty} x^3 e^x = +\infty \) emphasizes the dominance of the exponential term, leading the whole function to diverge to infinity.

- *As \( x \rightarrow -fty \)*: The exponential component \( e^{x-1} \) approaches zero more rapidly than the polynomial \( x^3 \) can increase due to its negative powers. Consequently, \( \lim_{x \to -\infty} x^3 e^{x-1} = 0 \), showing that the exponential factor significantly reduces the function value to zero.

Understanding these limits is crucial for graphing the function and for predicting its long-term behavior.

Exponential Functions

Exponential functions are a critical category of mathematical functions that express quantities that grow or decay at a rate proportional to their current value. The general form of an exponential function is \( f(x) = a e^{bx} \), where \( e \) represents Euler's number, approximately equal to 2.71828. Exponential functions showcase some unique traits, such as their distinctive rapid growth or decay, which make them essential in modeling real-world phenomena like population growth and radioactive decay.

For the function \( f(x) = x^3 e^{x-1} \), the component \( e^{x-1} \) is an exponential function highlighting rapid increase as \( x \) becomes larger. This quality causes the exponential term to assert dominance over polynomial terms, particularly as \( x \rightarrow +\infty \). As a result, these functions frequently reveal crucial asymptotic behaviors, which are important for creating a sketch of the function based on its growth pattern.

When examining exponential functions, the base \( e \) ensures a constant relative rate of growth or decline. This is especially important in calculus, where derivatives and integrals involving \( e^x \) have simplified forms, making calculations more efficient. Recognizing how exponential functions interact with other functions, like polynomials, unlocks profound insights into the behavior of complex expressions.

Polynomials

Polynomials are algebraic expressions that consist of variables raised to non-negative integer powers with constant coefficients. The simplest form of a polynomial is \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \), where each coefficient \( a \) can be any number, and \( n \) is the highest degree. Understanding polynomials is fundamental to algebra and calculus, as they appear in many forms across various problems.

For \( f(x) = x^3 e^{x-1} \), the term \( x^3 \) represents a cubic polynomial. Such polynomials typically dictate certain local behaviors, such as critical points or inflection points, due to their powers. However, the presence of the exponential term modifies the polynomial's overall behavior at extreme values of \( x \).

Key characteristics of polynomials include:

• Determining roots or zeros, where the polynomial function crosses the x-axis.
• Identifying turning points, found by setting the derivative to zero and solving \( f'(x) = 0 \).
• Understanding end behavior which in simple polynomials depends only on the leading term's degree and sign. However, this changes when combined with exponential terms.

In the graphing and analysis of \( f(x) = x^3 e^{x-1} \), information from both polynomial and exponential insights must be combined, giving a more complete picture of the function's behavior and structure. Analyzing polynomials allows us to effectively predict how functions react to changes in input, both locally and globally.