Problem 42 - Precalculus Mathematics for Calculus

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 \(3^{x}=x^{4},\) rounded to two decimal places.

Step-by-Step Solution

Answer

The functions intersect near \( x \approx 2.49 \) and \( x \approx -0.76 \).

Step 1: Plot the Functions in Viewing Rectangle (i)

Start by plotting the functions \( f(x) = 3^x \) and \( g(x) = x^4 \) within the range of \([-4, 4]\) for the x-axis and \([0, 20]\) for the y-axis. You should observe that \( f(x) \) grows exponentially and reaches noticeable values for positive x, while \( g(x) \) is much slower in growth within this range.

Step 2: Plot the Functions in Viewing Rectangle (ii)

Next, plot the functions within \([0, 10]\) on the x-axis and \([0, 5000]\) on the y-axis. You'll notice \( f(x) = 3^x \) overtakes \( g(x) = x^4 \) as x increases, due to the exponential nature of \( f(x) \). In this rectangle, \( f(x) \) grows significantly faster than \( g(x) \).

Step 3: Plot the Functions in Viewing Rectangle (iii)

Lastly, plot the functions with x-axis \([0, 20]\) and y-axis \([0, 10^5]\). Here, the rapid increase of \( f(x) \) becomes even more evident as it quickly surpasses \( g(x) \) with increasing x values. You can see \( f(x) \) continues to grow exponentially, outpacing \( g(x)\)'s polynomial rate.

Step 4: Identify Intersections Numerically

To find where these two functions intersect, solve the equation \( 3^x = x^4 \) numerically. This requires solving and verifying potential solutions within the ranges previously plotted.

Step 5: Calculate Approximate Solutions

Using numerical methods, find the values of x where \( 3^x = x^4 \). For these values, an approximate numerical solution shows that they intersect at \( x \approx 2.49 \) and \( x \approx -0.76 \), rounded to two decimal places.

Key Concepts

Exponential Growth

Exponential growth is a concept where functions increase at a rate proportional to their current value. This creates a curve that rises steeper and steeper as it progresses. Consider the function \( f(x) = 3^x \). As \( x \) becomes larger, \( f(x) \) grows much faster compared to linear or polynomial functions. Here are some characteristics of exponential growth:

• The base, which is greater than 1, causes the function to accelerate its growth as \( x \) increases.
• The rate of growth is not constant. As the values of \( x \) increase, the function's values rapidly increase.
• In a graph, it appears as a curve that starts slow but becomes steeper and steeper.

When you plot \( f(x) = 3^x \) in different ranges, you will notice its rapid increase, overtaking polynomial growth quickly. This unique feature of exponential growth makes it useful for modeling populations, investments, and more.

Polynomial Growth

Polynomial growth is observed when a function is expressed as a polynomial, like \( g(x) = x^4 \). Unlike exponential functions, polynomial functions grow at a slower, predictable rate. The degree of the polynomial (in this case, 4) significantly influences how the function behaves as \( x \) increases.Key points about polynomial growth:

• The growth rate depends on the highest power of \( x \). Higher powers result in slower initial growth followed by faster increases as \( x \) becomes large.
• In the early stages, polynomial growth will seem slower compared to exponential growth, especially over smaller ranges.
• In a graph, polynomial growth is depicted as a smooth curve, initially flatter and gradually becoming steeper.

When comparing \( g(x) = x^4 \) with \( f(x) = 3^x \), you'll see \( g(x) \) grow steadily, but it is quickly surpassed by \( f(x) \) as \( x \) becomes larger. This occurs because exponential growth accelerates more dramatically than polynomial growth.

Graphical Analysis

Graphical analysis involves visually comparing the growth of functions by plotting them on a graph. This helps to understand how different functions behave and where they might intersect. When examining \( f(x) = 3^x \) and \( g(x) = x^4 \), plotting them in different viewing rectangles illustrates their growth dynamics.Here's how to approach graphical analysis:

• Use appropriate scales for your axes to capture the key features of the functions without oversimplifying or missing critical points.
• Observe how \( f(x) = 3^x \) quickly surpasses \( g(x) = x^4 \) due to its exponential nature. In smaller ranges, \( f(x) \) may rise gradually, but eventually, it accelerates rapidly.
• Identify intersections: The points where the graphs meet represent solutions to the equation \( 3^x = x^4 \). You can estimate these points by observing the graph or by calculating them more precisely.

By solving \( 3^x = x^4 \) numerically, intersections are found at approximately \( x = 2.49 \) and \( x = -0.76 \). Graphical analysis, supported by mathematical calculations, provides a powerful method to visualize and compare the functions' growth.