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Chapter 4: Problem 31

Which of the seven basic models (linear, exponential, logarithmic, quadratic, logistic, cubic, and sine) could have relative maxima or minima?

Short Answer

Expert verified
Quadratic, logistic, cubic, and sine models can have relative maxima or minima.

Step by step solution

01

List Basic Models

Identify the seven basic models listed in the exercise: linear, exponential, logarithmic, quadratic, logistic, cubic, and sine.
02

Understand Relative Extrema

Recall that relative maxima or minima are points where a function reaches local highest or lowest values nearby. These are a result of the function's changing direction.
03

Analyze Each Model for Extrema

Evaluate the potential for each model to have relative maxima or minima: - **Linear functions** have a constant slope and no curvature, so they cannot have relative extrema. - **Exponential functions** are monotonic (constantly increasing or decreasing), so they don't have relative extrema. - **Logarithmic functions** are also monotonic, generally increasing or decreasing. - **Quadratic functions** are parabolas, which can have one relative maximum or minimum, depending on the parabola's orientation. - **Logistic functions** are S-shaped and can have relative maximum/minimum, typically near the inflection point. - **Cubic functions** can have up to two relative extrema due to their S-shaped curve. - **Sine functions** are periodic and have recurring maxima and minima due to their oscillating nature.
04

Identify Models with Relative Extrema

From the analysis: - Quadratic functions have one relative extrema. - Logistic functions have relative extrema near inflection points. - Cubic functions can have two relative extrema. - Sine functions have multiple relative extrema due to their periodic nature.

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Key Concepts

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

Relative Maxima
In mathematical models, relative maxima refer to the points where a function reaches a local peak compared to its immediate surroundings. These points are significant as they indicate the highest value in a particular area of the function. Relative maxima are common in quadratic, cubic, and sine functions due to their distinctive shapes and behavior. For instance:
  • Quadratic functions: These functions form a parabola, with the vertex being a relative maximum if the parabola opens downwards.
  • Cubic functions: Depending on the curve's shape, they can have up to two relative maxima due to their S-shape.
  • Sine functions: As periodic functions, they have recurring relative maxima, correlating with their oscillating waves.
Recognizing relative maxima is crucial in determining the behavior and trajectory of various functions in real-world scenarios.
Relative Minima
Relative minima are points where a function attains a local low point within a specified neighborhood. These points signal where the function drops to its lowest value locally, providing valuable information on the function's behavior. Relative minima appear prominently in functions due to their nature:
  • Quadratic functions: In an upward opening parabola, the vertex is a relative minimum.
  • Cubic functions: Their S-curve offers potential for relative minima as the curve dips.
  • Sine functions: Due to their periodic nature, sine functions exhibit continuous relative minima, reflecting their wave pattern.
Understanding relative minima is vital in identifying points of interest where functions change direction from decreasing to increasing.
Quadratic Functions
Quadratic functions often appear as parabolic curves in a graph, defined by the equation \( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants. These functions are revered for their ability to model naturally occurring phenomena such as projectile motion. Depending on the sign of \(a\):
  • If \(a > 0\), the parabola opens upwards, and the vertex serves as a relative minimum.
  • If \(a < 0\), the parabola opens downwards, and the vertex is a relative maximum.
The vertex can be found using the formula \( x = -\frac{b}{2a} \), and it plays a central role in analyzing the function's behavior, providing information about the peak or dip of the curve.
Logistic Functions
Logistic functions are characterized by their S-shaped curve, commonly seen in processes of growth which eventually plateau, such as populations reaching carrying capacity. The standard form is \( f(x) = \frac{L}{1 + e^{-k(x-x_0)}} \), where \(L\) represents the curve's maximum value. Logistic functions can have relative maxima or minima near their inflection points鈥攚here the curve transitions from increasing at a decreasing rate, to increasing at an increasing rate. This behavior provides pivotal insights into growth dynamics over time, making logistic functions useful in biological, ecological, and sociological studies.
Cubic Functions
Cubic functions are defined by the general form \( f(x) = ax^3 + bx^2 + cx + d \), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(aeq 0\). These functions are known for their distinctive S-shaped curve, which can feature up to two relative maxima and minima.
  • Cubic functions can represent more complex behaviors and are used to model variables where changes aren't linear or parabolic.
  • The function's inflection point occurs where the curve changes concavity, a key detail in charting the cubic function's growth pattern.
The real-world applications of cubic functions include analyzing economic trends, physics simulations, and optimizing engineering design processes.
Sine Functions
Sine functions are a fundamental element in trigonometry, known for their periodic and oscillating nature. They are expressed in the form \( f(x) = A \sin(Bx + C) + D \), where \(A\), \(B\), \(C\), and \(D\) adjust the amplitude, frequency, phase shift, and vertical shift, respectively. This waveform is crucial in analyzing phenomena that repeat cyclically, such as sound waves and tides.
  • Sine functions have a symmetric pattern, which includes repeated relative maxima and minima across their cycle.
  • The maximum occurs at \(f(x) = A + D\) while the minimum is at \(-A + D\).
Understanding sine functions allows one to predict and model periodic behaviors accurately in various fields, including engineering, physics, and music theory.

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Most popular questions from this chapter

Rose Sales A street vendor constructs the table below on the basis of sales data. Sales of Roses, Given the Price per Dozen \begin{tabular}{|c|c|} \hline Price (dollars) & Sales (dozen roses) \\ \hline 20 & 160 \\ \hline 25 & 150 \\ \hline 30 & 125 \\ \hline 32 & 85 \\ \hline \end{tabular} a. Find a model for quantity sold. b. Construct a model for consumer expenditure (revenue for the vendor). c. What price should the street vendor charge to maximize consumer expenditure? d. If each dozen roses costs the vendor \(\$ 10,\) what price should he charge to maximize his profit?

For Activities 7 through \(18,\) write the first and second derivatives of the function. \(g(x)=e^{3 x}-\ln 3 x\)

A Cobb-Douglas function for the production of mattresses is $$ M=48.1 L^{0.6} K^{0.4} \text { mattresses } $$ where \(L\) is measured in thousands of worker hours and \(K\) is the capital investment in thousands of dollars. a. Write an equation showing labor as a function of capital. b. Write the related-rates equation for the equation in part \(a,\) using time as the independent variable and assuming that mattress production remains constant. c. If there are currently 8000 worker hours, and if the capital investment is \(\$ 47,000\) and is increasing by \(\$ 500\) per year, how quickly must the number of worker hours be changing for mattress production to remain constant?

Senior Population (Projected) The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled as $$ \begin{aligned} p(x)=&-0.00022 x^{3}+0.014 x^{2} \\ &-0.0033 x+12.236 \text { percent } \end{aligned} $$ where \(x\) is the number of years since \(2000,\) data from \(0 \leq x \leq 50\) (Source: Based on data from U.S. Census Bureau, National Population Projections, \(2008 .)\) a. Determine the year between 2000 and 2050 in which the percentage is predicted to be increasing most rapidly, the percentage at that time, and the rate of change of the percentage at that time. b. Repeat part \(a\) for the most rapid decrease.

Boyle's Law for gases states that when the mass of a gas remains constant, the pressure \(p\) and the volume \(v\) of the gas are related by the equation \(p v=c,\) where \(c\) is a constant whose value depends on the gas. Assume that at a certain instant, the volume of a gas is 75 cubic inches and its pressure is 30 pounds per square inch. Because of compression of volume, the pressure of the gas is increasing by 2 pounds per square inch every minute. At what rate is the volume changing at this instant?
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