Problem 45 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises 45-48, find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function. $$y=e^{x}-2 x$$

Step-by-Step Solution

Answer

The right end behavior model for the function \(y=e^{x}-2x\) is \(e^{x}\), and the left end behavior model is \(-2x\).

Step 1: Right End Behavior Model

Consider the function as \(x\) approaches positive infinity. As \(x\) increases without bound, the term \(e^{x}\) will dominate the term \(-2x\). So, at large \(x\) values, the function behaves like \(e^{x}\). Hence, \(e^{x}\) is a simple basic function that describes the right end behavior of the function.

Step 2: Left End Behavior Model

Consider the function as \(x\) approaches negative infinity. As \(x\) decreases without bound, the term \(e^{x}\) will tend towards 0, and the \(-2x\) term will dominate. Hence at large negative \(x\) values, the function behaves like \(-2x\). So, \(-2x\) is a simple basic function that describes the left end behavior of the function.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent, and they are often denoted as \( e^x \). Here, \( e \) is the exponential constant, approximately 2.718. These functions have unique characteristics compared to other types of functions, like linear or polynomial. They grow rapidly with increasing \( x \), which makes them particularly helpful in modeling real-world situations like population growth or radioactive decay.

In the given exercise, the function \( y = e^x - 2x \) contains an exponential term. As \( x \) becomes significantly large in the positive direction, \( e^x \) grows very quickly, which profoundly impacts the overall behavior of the function. This is because exponential terms can outpace both linear and polynomial growth. Understanding exponential functions is key to analyzing functions' behaviors at the extreme values of \( x \).

Key characteristics of exponential functions:

• Exponential growth is faster than linear or polynomial growth.
• They have a horizontal asymptote on one side; for instance, \( y = e^x \) approaches zero as \( x \) approaches negative infinity.
• Exponential functions are often used in real-world applications where rapid growth or decay is involved.

Dominant Terms in Limits

In calculus, understanding which term dominates a function as the variable approaches a certain value is crucial. This is especially important when both exponential and polynomial terms are present in the function.

Considering the function \( y = e^x - 2x \), to determine the behavior at extreme values of \( x \), identify which term becomes more significant in influence. For the right end behavior (as \( x \) approaches positive infinity), the term \( e^x \) grows significantly faster than the linear term \(-2x\). Thus, \( e^x \) is the dominant term, and the function mimics \( e^x \) behavior.

For the left end behavior (as \( x \) approaches negative infinity), \( e^x \) diminishes towards zero, making \(-2x\) the dominant term. Therefore, in this situation, the function behaves similar to \(-2x\).

Steps to identify dominant terms in limits:

• Compare the growth rates of the terms involved as \( x \) approaches the specified value.
• Assess whether exponential growth outpaces polynomial or linear terms when \( x \) is large.
• Determine which term retains significance when others become negligible.

Negative Infinity Behavior

Negative infinity behavior describes how a function behaves as the variable \( x \) becomes very large in the negative sense. This is essential to predict the function's trend when \( x \) is extremely low.

For the function \( y = e^x - 2x \), as \( x \) approaches negative infinity, the exponential term \( e^x \) approaches zero because the base \( e \) (about 2.718) raised to a large negative power results in a very small value. In contrast, the linear term \(-2x\) grows in magnitude since \( x \) is multiplying by a constant \(-2\). Therefore, \(-2x\) is the dominant term in this context, dictating that the function's behavior mimics that of \(-2x\) when approaching negative infinity.

Analyzing negative infinity behavior involves:

• Identifying which terms reduce towards zero and which terms increase in value.
• Recognizing that exponential functions become negligible as \( x \) trends negatively hard.
• Concluding the overall composition of the function by the persistence of non-negligible terms.