Problem 45 - Calculus

Question

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(\lim _{x \rightarrow \infty} g(x)=\infty,\) then \(\lim _{x \rightarrow \infty} f(x)=\infty.\)

Step-by-Step Solution

Answer

False. A counterexample proves the statement doesn't hold universally.

Step 1: Understand the Problem

We are given a differential equation \( \frac{d y}{d x} = g(x) \) where \( y = f(x) \) is a solution. We need to evaluate the statement: "If \( \lim_{x \rightarrow \infty} g(x) = \infty \), then \( \lim_{x \rightarrow \infty} f(x) = \infty \)." We are determining if this statement is true or false and verifying it with an explanation or a counterexample.

Step 2: Definition of Limits and Differentiation

Recall that if \( \lim_{x \rightarrow \infty} g(x) = \infty \), it means that as \( x \) approaches infinity, \( g(x) \) grows unbounded. Similarly, we are looking to see if this implies the same for \( f(x) \), given that \( \frac{d y}{d x} \) is \( g(x) \), implying \( f'(x) = g(x) \). The derivative \( f'(x) \) indicates the rate of change of \( f(x) \).

Step 3: Explore the Relationship

Consider the relationship between \( g(x) \) and \( f(x) \). Since \( f'(x) = g(x) \), \( f(x) \) is the antiderivative of \( g(x) \). If \( g(x) \to \infty \), \( f(x) \) could increase without bound, but it's not guaranteed because the growth of \( f(x) \) depends on the exact form of \( g(x) \).

Step 4: Counterexample Scenario

Let’s analyze a scenario for a counterexample. Consider \( g(x) = x \sin(x) \). As \( x \to \infty \), the limit does not exist, and \( f(x) \) does not have a defined limit either. If \( g(x) = \sin(x) \), it oscillates between -1 and 1, implying \( \lim_{x \to \infty} g(x) \) does not exist, but the integral may remain bounded or undefined in a particular form.

Step 5: Conclusion

From our counterexample, we observe that even if \( \lim_{x \rightarrow \infty} g(x) = \infty \), it is not necessary that \( \lim_{x \rightarrow \infty} f(x) = \infty \). \( f(x) \) may not approach infinity due to oscillating behavior or integral properties. Thus the original statement is false.

Key Concepts

Limits

In calculus, a limit describes the value that a function approaches as the input approaches some value. When we talk about a limit approaching infinity, we mean observing the behavior of functions as the input grows larger without bound.
This is a key idea, especially for understanding how functions behave at their extremes.

• If \( \lim_{x \to \infty} g(x) = \infty \), it implies \( g(x) \) keeps increasing endlessly as \( x \) becomes very large.
• This concept is crucial in our problem because we need to determine whether the same behavior can be expected from \( f(x) \).

In this context, limits help us theorize about the end behavior of functions but not necessarily dictate outcomes without further information.

Antiderivative

An antiderivative or an indefinite integral of a function is a function whose derivative is the original function. It helps in understanding the accumulation of quantity from a rate of change.
Given that \( f'(x) = g(x) \), it follows that \( f(x) \) is the antiderivative of \( g(x) \).

• This relationship means \( f(x) \) accumulates all changes depicted by \( g(x) \).
• However, \( f(x) \) behaves based on the specific form of \( g(x) \) and could majorly vary in its outcomes.

In our problem, this leads to the realization that just because \( g(x) \) increases without bound doesn't necessarily conclude \( f(x) \) will. It very much depends on how \( g(x) \) behaves precisely.

Unbounded Growth

Unbounded growth refers to the concept where a function continuously increases tends towards infinity without reaching a maximum. In terms of \( g(x) \), it can mean increasing monotonically without limits.

• Many real-world phenomena can be modeled with unbounded functions, yet their integral derivatives, like \( f(x) \), may not share the same unbounded property.
• A function like \( g(x) = x \sin(x) \) proves that unbounded growth of the derivative does not ensure unbounded behaviour in its antiderivative \( f(x) \).

Counterexamples illustrate cases where, despite \( g(x) \) tending toward infinity, \( f(x) \) might remain bounded or act erratically due to oscillations in \( g(x) \). Thus, this highlights how nuanced growth predictions are.