Problem 52 - Algebra and Trigonometry

Question

(a) Compare the rates of growth of the functions \(f(x)=3^{x}\) and \(g(x)=x^{4}\) by drawing the graphs of both functions in the following viewing rectangles: (i) [-4,4] by [0,20] (ii) [0,10] by [0,5000] (iii) [0,20] by \(\left[0,10^{5}\right]\) (b) Find the solutions of the equation \(3^{x}=x^{4},\) correct to two decimal places.

Step-by-Step Solution

Answer

The functions intersect at approximately \(x = 1.85\) and visually verify growth in different viewing rectangles to see \(3^x\) eventually outgrows \(x^4\).

Step 1: Graph the functions in the given rectangles

To compare the rates of growth of the functions, first, plot the graphs of \(f(x) = 3^x\) and \(g(x) = x^4\) in each of the specified viewing rectangles. Use a graphing calculator or software to input the functions and adjust the viewing windows to: 1. Viewing rectangle (i) [-4, 4] by [0, 20]. 2. Viewing rectangle (ii) [0, 10] by [0, 5000]. 3. Viewing rectangle (iii) [0, 20] by \([0, 10^5]\).Observe how the functions behave in each setting. Particularly notice the points where the graphs intersect, which indicates equal growth rates at those points.

Step 2: Analyze the graphs to compare growth rates

Examine the graphs you plotted:- In viewing rectangle (i), observe how both graphs grow. Notice if either function begins to dominate the growth rate over another.- In viewing rectangle (ii), check how \(f(x) = 3^x\) starts to grow faster than \(g(x)=x^4\) especially as \(x\) increases.- In viewing rectangle (iii), observe how \(f(x) = 3^x\) grows significantly faster than \(g(x) = x^4\) for larger \(x\) values, clearly illustrating an exponential growth rate compared to a polynomial.

Step 3: Set up the equation for solving

To find where \(f(x) = g(x)\), set up the equation \(3^x = x^4\). This will allow us to find the x-values where the functions have equal growth.

Step 4: Solve the equation using numerical methods

Use numerical methods like the 'solve' function on a calculator or software to find the x-values where \(3^x = x^4\). These are the solutions where the graphs intersect. Through numerical computation, find approximate solutions correct to two decimal places.

Step 5: Interpret the results of the solutions

From your numerical software, determine that the intersection points are at \(x \approx 1.85\) and possibly other points at negative x-values or near zero. Make sure these results align with what you observed graphically from Step 1. Since only positive x makes sense for the context of growth, focus mainly on those unless otherwise needed.

Key Concepts

Graphing Functions

Graphing functions is a powerful way to visualize and compare the growth of two or more mathematical expressions. When we graph a function like \( f(x) = 3^x \) and compare it to \( g(x) = x^4 \), we can immediately see differences in how these functions behave as \( x \) changes. Graphing involves plotting these equations on a set of axes using specified viewing windows (like the ones given in this exercise) to see how they scale and where they intersect.

In our exercise, you'll want to plot the functions in three separate viewing rectangles:

• Rectangle (i) [-4, 4] by [0, 20]
• Rectangle (ii) [0, 10] by [0, 5000]
• Rectangle (iii) [0, 20] by \([0, 10^5]\)

By setting the viewing rectangles this way, you can observe at which points the exponential function starts outpacing the polynomial one. This visualizes the concept of exponential vs. polynomial growth. Close attention should be paid to the points where both graphs intersect, indicating where the growth rates of both functions are equal.

Numerical Methods

Numerical methods are essential when exact analytical solutions are difficult or impossible to find. In this exercise, to compare growth rates, solve \( 3^x = x^4 \) to find exact points of intersection.

These methods often involve using calculators or computer software to approximate solutions to equations that can't be solved easily by hand. A common numerical approach is to apply an algorithm that incrementally tests values of \( x \), adjusting until the equation is satisfied to a desired degree of accuracy.

In this problem, using numerical methods allows us to find solutions to two decimal places, such as \( x \approx 1.85 \). This point is where both functions grow at the same rate, and having these solutions gives a clearer understanding of the overall function behaviors based on your graphical observations.

Polynomial Growth

Polynomial growth describes how functions like \( g(x) = x^4 \) increase as \( x \) becomes larger. These types of functions expand based on powers of \( x \), implying that as \( x \) increases, their growth becomes substantial, but in a predictable, steady manner.

Unlike exponential functions that escalate more dramatically, polynomial functions grow by powers. For example, \( x^4 \) indicates when \( x \) doubles, \( g(x) \) increases at a much larger but controlled scale than a simple linear function.

While initially, polynomial growth seems competitive, especially for smaller \( x \), in the longer term, exponential growth (like in \( f(x) = 3^x \)) overtakes due to the explosive nature of continuous multiplication. Understanding this growth pattern is key to comparing how different types of mathematical functions behave across various scales and why exponential tends to dominate at higher values.