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Chapter 7: Problem 7

(Graphing program recommended.) If \(x\) is positive, for what values of \(x\) is \(3 \cdot 2^{x}<3 \cdot x^{2}\) ? For what values of \(x\) is \(3 \cdot 2^{x}>3 \cdot x^{2} ?\)

Short Answer

Expert verified
For \( 2 < x < 4 \), \( 2^x < x^2 \). For \( 0 < x < 2 \) and \( x > 4 \), \( 2^x > x^2 \).

Step by step solution

01

Divide both sides by 3

Start with the original inequality: \[ 3 \times 2^x < 3 \times x^2 \] Divide both sides by 3 to simplify: \[ 2^x < x^2 \]
02

Find the points of intersection

Next, solve for the intersection points of the functions \( 2^x \) and \( x^2 \). Setting \( 2^x = x^2 \), we see that \( x = 2 \) is one solution since \( 2^2 = 2^2 = 4 \). Another value is \( x = 4 \) since \( 2^4 = 16 \) and \( 4^2 = 16 \). Therefore, the points of intersection are \( x = 2 \) and \( x = 4 \).
03

Test values around the intersection points

To determine the inequality intervals, test values of \( x \) around the intersection points. Choose a value in the intervals: - For \( 0 < x < 2 \), let \( x = 1 \): \( 2^1 = 2 \) and \( 1^2 = 1 \), so \( 2 > 1 \). Thus, \( 2^x > x^2 \) for \( 0 < x < 2 \). - For \( 2 < x < 4 \), let \( x = 3 \): \( 2^3 = 8 \) and \( 3^2 = 9 \), so \( 8 < 9 \). Thus, \( 2^x < x^2 \) for \( 2 < x < 4 \). - For \( x > 4 \), let \( x = 5 \): \( 2^5 = 32 \) and \( 5^2 = 25 \), so \( 32 > 25 \). Thus, \( 2^x > x^2 \) for \( x > 4 \).
04

Summarize the intervals

We can summarize the intervals based on our testing: - \( 2^x < x^2 \) for \( 2 < x < 4 \). - \( 2^x > x^2 \) for \( 0 < x < 2 \) and \( x > 4 \).

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Key Concepts

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

Inequality Solving
Inequality solving involves finding the range of values for which an inequality holds true. In the given problem, we need to determine for which values of x the inequality \( 3 \times 2^x < 3 \times x^2 \) is satisfied. We began by simplifying the inequality by dividing both sides by 3, giving us \( 2^x < x^2 \). This step is crucial as it simplifies the problem to comparing two simpler functions, \( 2^x \) and \( x^2 \). By finding where these two functions intersect (where they are equal), we can test intervals around these points to see where the original inequality is satisfied.
Exponential Functions
Exponential functions, like \( 2^x \), grow extremely fast as x increases. In an exponential function \( f(x) = a^x \), 'a' is a constant called the base, and 'x' is the exponent. For the problem, \( 2^x \) represents an exponential growth, meaning as x gets larger, \( 2^x \) increases rapidly.
Understanding the growth behavior of exponential functions is essential because exponential and quadratic functions can have different rates of increase or decrease. This helps in finding where one function overtakes another and also assists in checking specific intervals of x.
Quadratic Functions
Quadratic functions have the form \( f(x) = ax^2 + bx + c \). In our exercise, the function \( x^2 \) is a simple quadratic function with no linear or constant terms. Quadratic functions form parabolas when graphed, and they grow much slower compared to exponential functions as x becomes very large.
To solve the inequality \( 2^x < x^2 \), understanding the behavior of quadratic functions allows us to compare their rates of growth. For small values of x, quadratic functions may be smaller than exponential functions, but beyond certain points, they dominate due to their squared term.
Graphical Analysis
Graphical analysis involves plotting functions to visually inspect their points of intersection and the regions where one function is greater than another. In this exercise, plotting \( 2^x \) and \( x^2 \) reveals their intersection at x = 2 and x = 4.
Using a graphing program, you can visually confirm these points and observe how the functions behave around these intersections. By testing values around these points鈥攍ike x = 1, x = 3, and x = 5鈥攜ou can determine the intervals on which \( 2^x < x^2 \) or \( 2^x > x^2 \). Visual inspection alongside analytical testing ensures the accuracy of your solution.

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Most popular questions from this chapter

Given \(h(x)=0.5 x^{2}\) a. Calculate \(h(2)\) and compare this value with \(h(6)\). b. Calculate \(h(5)\) and compare this value with \(h(15)\). c. What happens to the value of \(h(x)\) if \(x\) triples in value? d. What happens to the value of \(h(x)\) if \(x\) is divided by \(3 ?\)

Given the functions \(f(x)=x^{4}\) and \(g(x)=(4)^{x}\) a. Find \(f(3 x)\) and \(3 f(x)\). Summarize the difference between these functions and \(f(x)\). b. Find \(g(3 x)\) and \(3 g(x)\). Summarize the difference between these functions and \(g(x)\).

A box has volume \(V=\) length \(\cdot\) width \(\cdot\) height. a. Find the volume of a cereal box with dimensions of length \(=19.5 \mathrm{~cm},\) width \(=5 \mathrm{~cm},\) and height \(=27 \mathrm{~cm} .\) (Be sure to specify the unit.) b. If the length and width are doubled, by what factor is the volume increased? c. What are two ways you could increase the volume by a factor of 4 and keep the height the same?

In "Love That Dirty Water" (Chicago Reader, April 5,1996 ), Scott Berinato interviewed Ernie Vanier, captain of the towboat Debris Control. The Captain said, "We've found a lot of bowling balls. You wouldn't think they'd float, but they do." When will a bowling ball float in water? The bowling rule book specifies that a regulation ball must have a circumference of exactly 27 inches. Recall that the circumference of a circle with radius \(r\) is \(2 \pi r\). The volume of a sphere with radius \(r\) is \(\frac{4}{3} \pi r^{3}\). a. What is a regulation bowling ball's radius in inches? b. What is the volume of a regulation bowling ball in cubic inches? (Retain at least two decimal places in your answer.) c. What is the weight in pounds of a volume of water equivalent in size to a regulation bowling ball? (Water weighs \(0.03612 \mathrm{lb} / \mathrm{in}^{3}\) ). d. A bowling ball will float when its weight is less than or equal to the weight of an equivalent volume of water. What is the heaviest weight of a regulation bowling ball that will float in water? e. Typical men's bowling balls are 15 or 16 pounds. Women commonly use 12 -pound bowling balls. What will happen to the men's and to the women's bowling balls when dropped into the water? Will they sink or float?

(Graphing program optional.) Evaluate each of the following functions at \(0,0.5,\) and \(1 .\) Then, on the same grid, graph each over the interval [0,1] . Compare the graphs. a. \(y_{1}=x \quad y_{2}=x^{1 / 2} \quad y_{3}=x^{1 / 3} \quad y_{4}=x^{1 / 4}\) b. \(y_{5}=x^{2} \quad y_{6}=x^{3} \quad y_{7}=x^{4}\)
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