Problem 20 - Essential Calculus Early Transcendentals

Question

Compare the rates of growth of the functions \(f(x)=x^{5}\) and \(g(x)=5^{x}\) by graphing both functions in several view- ing rectangles. Find all points of intersection of the graphs correct to one decimal place.

Step-by-Step Solution

Answer

The graphs intersect around \(x \approx 2.4\) and \(x \approx 0.1\). Exponential growth surpasses polynomial for large \(x\).

Step 1: Understand the Functions

We have two functions, \(f(x) = x^5\) and \(g(x) = 5^x\). The first is a polynomial function, and the second is an exponential function. We are interested in comparing their rates of growth and finding where they intersect.

Step 2: Graph the Functions on Common Intervals

Start by plotting both functions \(f(x) = x^5\) and \(g(x) = 5^x\) on a graphing tool or calculator. Choose a range that includes negative and positive values for \(x\), for example, from \(-2\) to \(2\) initially. This helps understand their basic behavior visually.

Step 3: Broaden the Viewing Window if Necessary

If the intersection points aren’t clear between \(-2\) and \(2\), try a broader range, such as \(-5\) to \(5\). This might help capture where the growth behaviors change to clearly show intersections or differentiate their growth rates.

Step 4: Zoom In on Points of Intersection

Once a broad window reveals intersections or potential intersections, refine your window around these areas to get exact coordinates. For precision correct to one decimal place, use smaller increments such as 0.1 or 0.01 in these areas.

Step 5: Use Analytical or Calculative Tools for Precision

For finding precise intersection points, you may use a graphing calculator that can numerically solve \(x^5 = 5^x\). Input this into your calculator to get exact intersection points for required precision.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number exponents. A simple example is the function \( f(x) = x^5 \). Here, the term "polynomial" refers to the sum of monomials—a single term or multiple terms with variables raised to a non-negative integer exponent.

Polynomial functions can have different degrees, determined by the highest exponent of the variable. For our function \( f(x) = x^5 \), the degree is 5. Such functions can exhibit a range of behaviors depending on the degree, coefficients, and signs of the terms.

• When the degree is even, the ends of the graph will both rise or both fall.
• When the degree is odd, as with \( f(x) = x^5 \), one end rises and the other falls depending on the sign of the leading coefficient.

Understanding polynomial functions is crucial for analyzing their growth rate and how they may intersect with other types of functions, like exponential functions.

Exponential Functions

Exponential functions involve a constant raised to a variable exponent, such as \( g(x) = 5^x \). These functions exhibit a rapid rate of increase or decrease, depending on the base of the exponent. In our example, the base is 5, meaning the function grows quickly as \( x \) increases.

Unlike polynomial functions, exponential functions have distinct characteristics:

• They never touch the x-axis, although they get infinitely close, seen as approaching zero on the left side of the graph.
• Their initial growth can be slower compared to polynomial functions but eventually overtakes them as \( x \) grows large enough.

Exponential growth is a key feature that can significantly impact where and how these functions might intersect with others, especially over larger ranges of \( x \). Recognizing the properties of exponential functions helps when predicting behaviors and solving real-world problems involving rapid changes.

Intersection Points

Intersection points occur where two functions share the same value at a particular \( x \)-coordinate. For functions \( f(x) = x^5 \) and \( g(x) = 5^x \), these are the values of \( x \) where both expressions equal each other. Finding intersections involves solving the equation \( x^5 = 5^x \), which is not typically solvable via straightforward algebraic methods due to its complexity.

To accurately find these points:

• Visualize the problem by graphing both functions over a suitable range.
• Adjust the range and scale to clearly see where the graphs overlap.
• Utilize tools such as graphing calculators or software that can compute and show intersections precisely.

Intersection points indicate critical values where, for example, the growth rate of a polynomial function initially surpasses or equals an exponential function before being overtaken as \( x \) increases further. These intersections are valuable for understanding both mathematical and practical scenarios involving change and comparison of growth rates.