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Chapter 11: Problem 52

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 \(g(0)=1\) and \(x(x)\) is increasing for \(x \geq 0,\) then \(f(x)\) is also increasing for \(x \geq 0.\)

Short Answer

Expert verified
True. Since \( g(x) \) is positive and increasing for \( x \geq 0 \), so is \( f(x) \).

Step by step solution

01

Understanding the Given Information

The problem states that the function \( y = f(x) \) is a solution to the differential equation \( \frac{d y}{d x} = g(x) \). We know that \( \frac{d y}{d x} \) represents the derivative of the function \( f(x) \), and \( g(x) \) is known to be increasing for \( x \geq 0 \) with \( g(0) = 1 \).
02

Analyzing the Derivative Behavior

Since \( \frac{d y}{d x} = g(x) \) and \( g(x) \) is increasing and positive starting from \( x = 0 \) (as \( g(0) = 1 \)), the derivative \( \frac{d y}{d x} \) is positive for \( x \geq 0 \). This indicates that the function \( f(x) \) has a positive slope and hence, is increasing for \( x \geq 0 \).
03

Drawing the Conclusion

Based on the behavior of the derivative \( \frac{d y}{d x} = g(x) \), which is positive for \( x \geq 0 \), it follows that \( f(x) \) must also be increasing in this interval. Therefore, the statement is true.

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Key Concepts

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

Increasing Functions
An increasing function is one where, as the input (or "x" value) gets larger, the output (or "y" value) also increases. This means the graph of the function consistently moves upward as you look from left to right.
In mathematics, we say that a function \( f(x) \) is increasing on an interval if for every pair of numbers \( x_1 \) and \( x_2 \) within that interval, whenever \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \).
This is a key concept because it helps us understand how functions behave over certain ranges.
  • A function with a positive derivative in a given interval is usually considered increasing on that interval.
  • Increasing doesn't mean it rises everywhere, just in the interval we examine.
Derivative
The derivative of a function is a fundamental concept in calculus that measures how a function changes as its input changes. In simpler terms, it tells us the rate of change or slope of the function at any given point.
For a function \( f(x) \), its derivative is often written as \( f'(x) \) or \( \frac{d f}{d x} \).
A positive derivative indicates that the function is increasing at that point, while a negative derivative shows it is decreasing.
  • A derivative equal to zero can mean a point of local maximum, minimum, or inflection.
  • Knowing the derivative helps us plot the function's graph by understanding its slope and direction at various points.
However, understanding a derivative isn't just about solving equations鈥攊t's about visualizing and predicting the behavior of functions.
Counterexample
In mathematical reasoning, a counterexample is an example that disproves a statement. When someone provides a logical assertion, a counterexample can show that the statement isn鈥檛 always true by providing a specific case where it fails.
  • Counterexamples are crucial in understanding limits and the boundaries of mathematical statements.
  • They reinforce the need to test hypotheses across varying scenarios.
In the context of increasing functions and derivatives, if a claim states that a function is increasing due to a property like derivatives, offering a counterexample could involve finding a function with the specified derivative behavior that doesn't increase over an interval.
Counterexamples help refine mathematical theories by highlighting exceptions and encouraging a deeper understanding.

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Most popular questions from this chapter

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) \(\quad s^{\prime \prime}+0.5 s^{\prime}+2 s=0\) In which system does the mass experience the frictional force of smallest magnitude for a given velocity?

Find the general solution to the given differential equation. $$z^{\prime \prime}+2 z=0$$

Morphine is often used as a pain-relieving drug. The half-life of morphine in the body is 2 hours. Suppose morphine is administered to a patient intravenously at a rate of \(2.5 \mathrm{mg}\) per hour, and the rate at which the morphine is eliminated is proportional to the amount present. (a) Use the half-life to show that, to three decimal places, the constant of proportionality for the rate at which morphine leaves the body (in mg/hour) is \(k=-0.347\) (b) Write a differential equation for the quantity, \(Q,\) of morphine in the blood after \(t\) hours. (c) Use the differential equation to find the equilibrium solution. (This is the long-term amount of morphine in the body, once the system has stabilized.)

Consider a conflict between two armies of \(x\) and \(y\) soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and \(t\) represents time since the start of the battle, then \(x\) and \(y\) obey the differential equations $$ \begin{array}{l} \frac{d x}{d t}=-a y \\ \frac{d y}{d t}=-b x \quad a, b>0 \end{array} $$ Near the end of World War II a fierce battle took place between US and Japanese troops over the island of Iwo Jima, off the coast of Japan. Applying Lanchester's analysis to this battle, with \(x\) representing the number of US troops and \(y\) the number of Japanese troops, it has been estimated \(^{35}\) that \(a=0.05\) and \(b=0.01\) (a) Using these values for \(a\) and \(b\) and ignoring reinforcements, write a differential equation involving \(d y / d x\) and sketch its slope field. (b) Assuming that the initial strength of the US forces was 54,000 and that of the Japanese was 21,500 draw the trajectory which describes the battle. What outcome is predicted? (That is, which side do the differential equations predict will win?) (c) Would knowing that the US in fact had 19,000 reinforcements, while the Japanese had none, alter the outcome predicted?

Find the general solution to the given differential equation. $$y^{\prime \prime}-3 y^{\prime}+2 y=0$$
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