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Chapter 11: Problem 55

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
The statement is false; a counterexample shows inconsistent results despite \(\lim_{x \rightarrow \infty} g(x) = \infty\).

Step by step solution

01

Understand the Problem

We need to determine if the statement is true or false. The statement claims that if \(y = f(x)\) solves the differential equation \(\frac{d y}{d x} = g(x)\), and \(\lim_{x \rightarrow \infty} g(x) = \infty\), then \(\lim_{x \rightarrow \infty} f(x) = \infty\).
02

Analyze the Differential Equation

The given statement involves solving \(\frac{d y}{d x} = g(x)\). The solution for \(y\) is obtained by integrating \(g(x)\) with respect to \(x\), so \(y = f(x) = \int g(x) \, dx + C\), where \(C\) is a constant of integration.
03

Evaluate the Limit Behavior

Given that \(\lim_{x \rightarrow \infty} g(x) = \infty\), it suggests that \(g(x)\) increases without bound. However, \(f(x) = \int g(x) \, dx + C\) integrates \(g(x)\) and may not necessarily diverge to infinity, based on the specific function \(g(x)\).
04

Construct a Counterexample

Consider \(g(x) = x^2\). In this case, \(\lim_{x \rightarrow \infty} g(x) = \infty\). If \(y = f(x) = \int x^2 \, dx = \frac{x^3}{3} + C\), then \(\lim_{x \rightarrow \infty} f(x) = \infty\). This agrees with the statement. However, if we choose a different function such as \(g(x) = 2x\sin^2(x) + 1\), the behavior of \(f(x)\) may not necessarily be \(\infty\) despite \(\lim_{x \rightarrow \infty} g(x) = \infty\).
05

Falsehood Explanation

The statement is false because there can be behavior that leads to an inconsistent result even if \(\lim_{x \rightarrow \infty} g(x) = \infty\). Specifically, consider functions with oscillatory behavior causing intervals where contributions to \(f(x)\) do not add to infinity, such as intermittent negative and positive slopes.

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Key Concepts

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

Limits
Limits are fundamental in understanding the behavior of functions as variables approach a specific point. They help us analyze what happens as the input grows extremely large or small.

In our exercise, we used limits to determine if the growth of the function's derivative, given as \(g(x)\), implies similar growth in the original function \(f(x)\). This involved assessing if the limit \(\lim_{x \rightarrow \infty} g(x) = \infty\) necessarily results in \(\lim_{x \rightarrow \infty} f(x) = \infty\).

Steps to understanding limits:
  • Consider what happens to the function as \(x\) approaches a certain value or infinity.
  • Evaluate if the function heads towards a specific number, or grows without bound, like heading towards infinity or negative infinity.
  • Recognize that limits help us predict function behavior asymptotically or at points where they might be undefined in a straightforward evaluation.
Integration
Integration is the process of finding the antiderivative. It allows us to calculate the accumulation of quantities, like area under a curve.

In the context of our exercise, integration transforms the derivative \(g(x)\) into the function \(f(x)\). Specifically, \(f(x) = \int g(x) \, dx + C\), where \(C\) is a constant of integration.

Key points about integration:
  • Integration can demonstrate how changes accumulate over a period.
  • The presence of a constant \(C\) shows that multiple functions can share the same derivative but differ by a constant value.
  • Limits and behavior at infinity can be different after integration, as seen in our exercise.
Asymptotic Behavior
Asymptotic behavior refers to how functions behave as they approach bounds, like infinity.

In our task, we examined whether \(g(x)\) and \(f(x)\) (an integral of \(g(x)\)) both tended towards infinity. We found that even if \(g(x)\) grows without limit, \(f(x)\) might not, due to varying rates or characteristics of growth.

Understanding asymptotic behavior involves:
  • Examining dominance between functions - which function dictates growth at bounds.
  • Assessment of changing slopes and boundaries, since variables may behave differently.
  • Recognizing oscillations or oscillatory functions, which can dynamically affect overall trends. This can lead to averaging out or dampening effects over the growth of integrated or original functions.

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Most popular questions from this chapter

For each of the differential equations find the values of \(c\) that make the general solution: (a) overdamped, (b) underdamped,(c) critically damped. $$s^{\prime \prime}+4 s^{\prime}+c s=0$$

Analyze the phase plane of the differential equations for \(x, y \geq 0 .\) Show the nullclines and equilibrium points, and sketch the direction of the trajectories in each region. $$\begin{aligned}&\frac{d x}{d t}=x\left(1-\frac{x}{2}-y\right)\\\&\frac{d y}{d t}=y\left(1-\frac{y}{3}-x\right)\end{aligned}$$

Each of the differential equations (i)-(iv) represents the position of a 1 gram mass oscillating on the end of a damped spring. Pick the differential equation representing the system which answers the question. (i) \(\quad s^{\prime \prime}+s^{\prime}+4 s=0\) (ii) \(s^{\prime \prime}+2 s^{\prime}+5 s=0\) (iii) \(s^{\prime \prime}+3 s^{\prime}+3 s=0\) (iv) \(\quad s^{\prime \prime}+0.5 s^{\prime}+2 s=0\) Which spring exerts the smallest restoring force for a given displacement?

Give an example of: A differential equation all of whose solutions are increasing and concave up.

Let \(L,\) a constant, be the number of people who would like to see a newly released movie, and let \(N(t)\) be the number of people who have seen it during the first \(t\) days since its release. The rate that people first go see the movie, \(d N / d t\) (in people/day), is proportional to the number of people who would like to see it but haven't yet. Write and solve a differential equation describing \(d N / d t\) where \(t\) is the number of days since the movie's release. Your solution will involve \(L\) and a constant of proportionality, \(k.\)
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