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Chapter 3: Problem 134

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

Short Answer

Expert verified
After close analysis, the order from the one that increases most slowly to the one that increases most rapidly is: \(y=\ln x\), \(y=\sqrt{x}\), \(y=x\), \(y=x^{2}\), \(y=e^{x}\), \(y=x^{x}\).

Step by step solution

01

Understand and identify each function type

Identify the type of each function presented: \(y=x\) is a linear function, \(y=\sqrt{x}\) is a square root function, \(y=e^{x}\) is an exponential function, \(y = \ln x\) is a natural logarithmic function, \(y=x^{x}\) is an exponential power function, \(y=x^{2}\) is a quadratic function.
02

Graph the functions

Graph each function in a suitable graphing software tool or use a graphic calculator. Ensure all graphs are visible in the same viewing window.
03

Analyze the rate of increase

Observe the graphs. As \(x\) increases, analyze how each function's \(y\) values change. The function that stays consistently closer to the x-axis increases at a slower rate compared to others. The function that moves away from the x-axis the quickest has the highest rate of increase.
04

Rank the functions

Based on their rates of increase observed from their graphs, rank these functions from the one that increases most slowly to the one that increases most rapidly.

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Key Concepts

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

Linear Function
A linear function is perhaps the simplest type of function to understand. It is represented in the form \( y = x \), where there is a constant rate of change in \( y \) with respect to \( x \). This means that if you plot this on a graph, you will get a straight line. The slope of this line is constant, meaning the function increases or decreases evenly as \( x \) changes.

Key characteristics of a linear function include:
  • Straight line graph
  • Constant slope
  • No curve or bending
Linear functions are useful for representing scenarios where there is a steady rate of change, like calculating speed or converting units.
Exponential Function
Exponential functions can dramatically increase and decrease. The function \( y = e^x \) represents exponential growth, where \( e \) is Euler's number, approximately 2.718. As \( x \) increases, \( y \) rises rapidly, making the graph rise sharply. This is a typical feature of exponential functions: they start slow and then rapidly shoot upwards.

Some features of exponential functions are:
  • Curved graphs that rise quickly
  • Growth rate increases with time
  • Used in modeling population growth, compound interest, etc.
These functions are excellent for understanding processes that experience rapid growth or decay.
Quadratic Function
Quadratic functions take on the form \( y = x^2 \). When graphed, they produce a U-shaped curve known as a parabola. The vertex of this parabola represents the function's lowest point if it opens upwards, which is the case for \( y = x^2 \). Quadratic functions exhibit a unique symmetry about the vertex, making them distinct from linear and exponential functions.

Characteristics of quadratic functions include:
  • Parabolic shape
  • Symmetrical graph
  • Rate of increase/decrease changes; initially slow, then faster
Quadratic functions are often used in physics to describe projectile motion and other phenomena involving acceleration.
Natural Logarithmic Function
The natural logarithmic function \( y = \ln x \) uses the logarithm base \( e \). It is the inverse of the exponential function, \( y = e^x \). Unlike other functions that shoot up quickly or drop sharply, the natural logarithmic function grows slowly and steadily. Its graph is a curve that increases at a diminishing rate.

Key points about natural logarithmic functions:
  • Slower growth rate compared to exponential functions
  • Graph approaches infinity but never touches the x-axis
  • Used in complex calculations like solving equations involving exponential growth
These functions are especially useful in mathematics for dealing with exponential-related problems.
Square Root Function
The square root function is depicted by \( y = \sqrt{x} \). Its graph only exists for non-negative \( x \) values since you can't take the square root of a negative number (in real numbers). It starts at the origin \( (0,0) \) and curves upwards more gently than a linear function. As \( x \) increases, \( y \) still increases but at a decreasing rate.

Main features to note about square root functions:
  • Graph is a gentle upward curve
  • Only defined for \( x \geq 0 \)
  • Slow rate of increase
Square root functions are frequently used in statistics and various fields to simplify squared data or express proportions.

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Most popular questions from this chapter

Suppose that a population that is growing exponentially increases from 800,000 people in 2010 to 1,000,000 people in \(2013 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\)

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's power regression option to obtain a model of the form \(y=a x^{b}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

The exponential growth models describe the population of the indicated country, \(A\), in millions, \(t\) years after 2006 $$\begin{array{l}\mathrm{Camada}\quadA=33.1e^{0.009\mathrm{t}}\\\\\mathrm{U}_{\mathrm{ganda}}\quad A=28.2 e^{0.034 t}\end{array}$$ In Exercises \(81-84,\) use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.

Exercises \(153-155\) will help you prepare for the material covered in the next section. U.S. soldiers fight Russian troops who have invaded New York City. Incoming missiles from Russian submarines and warships ravage the Manhattan skyline. It's just another scenario for the multi-billion-dollar video games Call of Duty, which have sold more than 100 million games since the franchise's birth in 2003 The table shows the annual retail sales for Call of Duty video games from 2004 through 2010 . Create a scatter plot for the data. Based on the shape of the scatter plot, would a logarithmic function, an exponential function, or a linear function be the best choice for modeling the data? $$\begin{array}{cc} \hline \text { Year } & \begin{array}{c} \text { Retail Sales } \\ \text { (millions of dollars) } \end{array} \\ \hline 2004 & 56 \\ 2005 & 101 \\ 2006 & 196 \\ 2007 & 352 \\ 2008 & 436 \\ 2009 & 778 \\ 2010 & 980 \end{array}$$
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