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Chapter 4: Problem 14

Sketch a function that is continuous on \((-\infty, \infty)\) and has the following properties. Use a number line to summarize information about the function. $$f^{\prime}(-2)=f^{\prime}(2)=f^{\prime}(6)=0 ; f^{\prime}(x) \geq 0 \text { on }(-\infty, \infty)$$

Short Answer

Expert verified
Question: Sketch a continuous function that has the first derivative equal to zero at the points x = -2, x = 2, and x = 6 and is non-decreasing over its entire domain. Answer: To sketch a continuous function meeting the given conditions, follow these steps: 1. For the interval \((-\infty, -2)\), sketch the function with a non-decreasing behavior, starting from an arbitrary point. 2. At point \(x=-2\), add a local extremum, such as a local minimum. 3. For the interval \((-2, 2)\), sketch the function increasing from the local minimum at x=-2. 4. At point \(x=2\), add another local extremum, such as another local minimum. 5. For the interval \((2, 6)\), sketch the function increasing from the local minimum at x=2. 6. At the point \(x=6\), add another local extremum, such as a local minimum. 7. For the interval \((6, \infty)\), sketch the function increasing from the local minimum at x=6. This sketch represents a continuous function with first derivative equal to zero at the specified points (-2, 2, and 6) and non-decreasing behavior over its entire domain.

Step by step solution

01

1. Finding the intervals where the function increases or decreases

Since the first derivative \(f^{\prime}(x) \geq 0\) for all x on \((-\infty, \infty)\), we have that the function is always non-decreasing, that is, it is either increasing or staying constant.
02

2. Analyzing the zero points of the first derivative

We know that \(f^{\prime}(-2)=f^{\prime}(2)=f^{\prime}(6)=0\), which indicates that the function has local extrema (local minima, maxima, or inflection points with constant behavior) at these points.
03

3. Plotting the function on a number line

We will use the number line to indicate the behavior of the function depending on the intervals. 1. Interval \((-\infty, -2)\): on this interval, \(f^{\prime}(x) \geq 0\), so the function is non-decreasing. 2. At point \(x=-2\), \(f^{\prime}(-2)=0\), so there must be some local extremum. 3. Interval \((-2, 2)\): on this interval, \(f^{\prime}(x) \geq 0\), so the function is non-decreasing. 4. At point \(x=2\), \(f^{\prime}(2)=0\), so there must be some local extremum. 5. Interval \((2, 6)\): on this interval, \(f^{\prime}(x) \geq 0\), so the function is non-decreasing. 6. At point \(x=6\), \(f^{\prime}(6)=0\), so there must be some local extremum. 7. Interval \((6, \infty)\): on this interval, \(f^{\prime}(x) \geq 0\), so the function is non-decreasing.
04

4. Sketching the function using the number line and the given information

Now, we will sketch the function using the properties derived from the number line: 1. For the interval \((-\infty, -2)\), choose any arbitrary starting point, and sketch the function with a non-decreasing behavior. 2. At point \(x=-2\), add a local extremum, for example, a local minimum. 3. For the interval \((-2, 2)\), sketch the function increasing from the local minimum at x=-2. 4. At point \(x=2\), add another local extremum, for example, another local minimum. 5. For the interval \((2, 6)\), sketch the function increasing from the local minimum at x=2. 6. At the point \(x=6\), add another local extremum, for example, a local minimum. 7. For the interval \((6, \infty)\), sketch the function increasing from the local minimum at x=6. This sketch of the function provides us with a continuous function that has the first derivative equal to zero at the specified points and is non-decreasing over its entire domain.

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Key Concepts

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

Continuous Functions
Continuous functions are those in which small changes in the input result in small changes in the output. This means there are no jumps, breaks, or holes in the graph of the function. In our exercise, the function is continuous on the entire real line \((-fty, fty)\). This implies that whatever values the function takes at one part of its graph, it smoothly transitions to those of a nearby point.
  • For example, when you look at the graph, you can smoothly pass over any point on it without lifting your pen.
  • This continuous nature is crucial as it allows us to apply calculus methods, like derivatives, explicitly and accurately.
To concretely visualize continuity, consider how temperature measured over a day changes. It does so gradually without sudden spikes, similar to a continuous function. By understanding continuous functions, one can appreciate the seamless flow and prediction of changes across mathematical models.
Derivative Analysis
The derivative of a function is like the function's speedometer, showing how fast it is moving up or down at any point. In this exercise, we analyze the derivative \(f^{\prime}(x)\) to understand how the original function \(f(x)\) behaves.
  • The derivative tells us where the function is rising, falling, or staying level.
  • If \(f^{\prime}(x) > 0\), the function is increasing. If \(f^{\prime}(x) < 0\), the function is decreasing. With \(f^{\prime}(x) = 0\), the function is constant at that instant.
In our exercise, the derivative is always non-negative, \(f^{\prime}(x) \geq 0\). This analysis helps determine points of interest, such as where the function flattens out and changes direction, which are known as extrema.
Non-decreasing Function
A non-decreasing function is one where the function's value doesn't drop as you move from left to right on the x-axis. In the case of our exercise, since \(f^{\prime}(x) \geq 0\) across all \(x\), the function is non-decreasing. This means:
  • At every point, the function either increases or stays the same.
  • This creates a step-like or gently upward sloping graph.
Being non-decreasing, the function only maintains or raises your position as you move along the x-axis, like walking up a hill or on flat terrain without descending. Identifying non-decreasing behaviors is essential for comprehending patterns within data and ensuring positive growth tendencies in applications.
Extrema
Extrema in calculus refer to points at which a function reaches its lowest or highest values locally within a region, known as local minima or maxima. In our task, these points occur where the derivative equals zero, such as at \(x = -2\), \(x = 2\), and \(x = 6\).
  • Because \(f^{\prime}(x) = 0\) at these points, it suggests a change in direction, such as a peak or valley in the graph.
  • To visualize, consider hills and valleys in landscapes, representing maxima and minima respectively.
Understanding extrema helps us identify critical moments when the function behavior shifts, offering insights into key changes in scenarios modeled mathematically, like stock market price peaks and valleys.

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Most popular questions from this chapter

The complexity of a computer algorithm is the number of operations or steps the algorithm needs to complete its task assuming there are \(n\) pieces of input (for example, the number of steps needed to put \(n\) numbers in ascending order). Four algorithms for doing the same task have complexities of A: \(n^{3 / 2},\) B: \(n \log _{2} n,\) C: \(n\left(\log _{2} n\right)^{2},\) and \(D: \sqrt{n} \log _{2} n .\) Rank the algorithms in order of increasing efficiency for large values of \(n\) Graph the complexities as they vary with \(n\) and comment on your observations.

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation \(a(t)=v^{\prime}(t)=g,\) where \(g=-9.8 \mathrm{m} / \mathrm{s}^{2}\). a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A payload is released at an elevation of \(400 \mathrm{m}\) from a hot-air balloon that is rising at a rate of \(10 \mathrm{m} / \mathrm{s}\).

Let \(a\) and \(b\) be positive real numbers. Evaluate \(\lim _{x \rightarrow \infty}(a x-\sqrt{a^{2} x^{2}-b x})\) in terms of \(a\) and \(b\).

Determine the following indefinite integrals. Check your work by differentiation. $$\int(4 \cos 4 w-3 \sin 3 w) d w$$

Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function \(D(p)=500-10 p\) says that at a price of \(p=10,\) a quantity of \(D(10)=400\) units of the commodity can be sold. The elasticity \(E=\frac{d D}{d p} \frac{p}{D}\) of the demand gives the approximate percent change in the demand for every \(1 \%\) change in the price. a. Compute the elasticity of the demand function \(D(p)=500-10 p\) b. If the price is \(\$ 12\) and increases by \(4.5 \%,\) what is the approximate percent change in the demand? c. Show that for the linear demand function \(D(p)=a-b p\) where \(a\) and \(b\) are positive real numbers, the elasticity is a decreasing function, for \(p \geq 0\) and \(p \neq a / b\) d. Show that the demand function \(D(p)=a / p^{b}\), where \(a\) and \(b\) are positive real numbers, has a constant elasticity for all positive prices.
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