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Chapter 11: Problem 45

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 \(\lim _{x \rightarrow \infty} f(x)=\infty.\)

Short Answer

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

Step by step solution

01

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.
02

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 \( f(x) \).
03

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 \( g(x) \).
04

Counterexample Scenario

Let鈥檚 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.
05

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.

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Key Concepts

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

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.

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Most popular questions from this chapter

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 \(g(x)\) is increasing for \(x>0,\) then so is \(f(x).\)

The systems of differential equations model the interaction of two populations \(x\) and \(y .\) In each case, answer the following two questions: (a) What kinds of interaction (symbiosis, \(^{30}\) competition, predator-prey) do the equations describe? (b) What happens in the long run? (For one of the systems, your answer will depend on the initial populations.) Use a calculator or computer to draw slope fields. $$\begin{aligned} &\frac{1}{x} \frac{d x}{d t}=1-\frac{x}{2}-\frac{y}{2}\\\ &\frac{1}{y} \frac{d y}{d t}=1-x-y \end{aligned}$$

The radioactive isotope carbon- 14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon- 12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5730 years. Suppose \(C(t)\) is the amount of carbon- 14 present at time \(t.\) (a) Find the value of the constant \(k\) in the differential equation \(C^{\prime}=-k C.\) (b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained \(91 \%\) of the amount of carbon-14 contained in freshly made cloth of the same material. \(^{8}\) How old is the Shroud of Turin, according to these data?

An aquarium pool has volume \(2 \cdot 10^{6}\) liters. The pool initially contains pure fresh water. At \(t=0\) minutes, water containing 10 grams/liter of salt is poured into the pool at a rate of 60 liters/minute. The salt water instantly mixes with the fresh water, and the excess mixture is drained out of the pool at the same rate ( 60 liters/minute). (a) Write a differential equation for \(S(t),\) the mass of salt in the pool at time \(t .\) (b) Solve the differential equation to find \(S(t)\) (c) What happens to \(S(t)\) as \(t \rightarrow \infty ?\)

Are the statements in Problems \(59-62\) true or false? Give an explanation for your answer. The differential equation \(d y / d x-x y=x\) can be solved by separation of variables.
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