Problem 51 - Algebra and Trigonometry

Question

(a) Compare the rates of growth of the functions \(f(x)=2^{x}\) and \(g(x)=x^{5}\) by drawing the graphs of both functions in the following viewing rectangles. (i) \([0,5]\) by \([0,20]\) (ii) \([0,25]\) by \(\left[0,10^{7}\right]\) (iii) \([0,50]\) by \(\left[0,10^{8}\right]\) (b) Find the solutions of the equation \(2^{x}=x^{5},\) correct to one decimal place.

Step-by-Step Solution

Answer

\(f(x)=2^x\) grows faster than \(g(x)=x^5\). Intersection points: \(x = 0.0\) and \(x = 4.7\).

Step 1: Understanding the problem

To compare the rates of growth of the functions \(f(x)=2^x\) and \(g(x)=x^5\), we need to draw their graphs over different intervals and then visually analyze their behavior. We will then solve the equation \(2^x = x^5\) to find where these functions intersect.

Step 2: Graphing the functions for (0,5) by (0,20)

Graph both functions on the interval \([0, 5]\) on the x-axis and the range \([0, 20]\) on the y-axis. Notice how the curve of \(g(x) = x^5\) grows quickly but not as quickly as \(f(x) = 2^x\) within this small interval.

Step 3: Graphing the functions for (0,25) by ([0,10^7])

Expand the viewing window to the interval \([0, 25]\) on the x-axis and range \([0, 10^7]\) on the y-axis. Observe that \(f(x) = 2^x\) begins to significantly outpace \(g(x) = x^5\) as \(x\) increases.

Step 4: Graphing the functions for (0,50) by ([0,10^8])

Use the interval \([0, 50]\) by \([0, 10^8]\) to graph the functions. Notice that \(f(x) = 2^x\) rapidly grows more steeply than \(g(x) = x^5\), further reinforcing that it has a higher rate of growth.

Step 5: Solving the equation 2^x = x^5

Solve the equation \(2^x = x^5\) numerically to find the points where these functions intersect. Use numerical methods or graphing utilities to find approximate solutions as exact algebraic solutions aren't feasible. Solution points are approximately \(x = 4.7\) and \(x = 0.0\).

Step 6: Conclusion

From the graphs, \(f(x) = 2^x\) eventually surpasses \(g(x) = x^5\) as \(x\) increases, indicating \(f(x)\) has a faster growth rate. The intersection points approximate to \(x = 0.0\) and \(x = 4.7\).

Key Concepts

Graphing Functions

When graphing functions, the focus is on visualizing how a function behaves over a specified set of values. In our exercise, we are dealing with two functions: \(f(x) = 2^x\) and \(g(x) = x^5\). Graphing these functions helps us to visually compare their growth rates. Let's delve into how to approach this systematically.
Graphing involves selecting a range for both the x-axis (input values) and y-axis (output values). For example, using small increments on the x-axis, such as \([0, 5]\), allows us to see detailed changes in the functions' behavior. As you increase the interval size to \([0,50]\), the graph helps us identify long-term behavior.

• For a small interval \([0, 5]\) and range \([0, 20]\), you might notice \(g(x) = x^5\) initially grows fast, but \(f(x) = 2^x\) overtakes beyond certain points.
• Using larger intervals and ranges allows us to see how quickly \(f(x) = 2^x\) surpasses \(g(x) = x^5\), highlighting the exponential growth's nature.

Each graphing interval provides a visual snapshot of how a function behaves, hinting at underlying differences in growth rates.

Rate of Growth Comparison

When comparing the rate of growth between different functions, such as \(f(x) = 2^x\) and \(g(x) = x^5\), visual analysis through graphing is an effective initial approach. However, understanding these rates more profoundly involves grasping what each function represents.
Exponential functions like \(f(x) = 2^x\) grow proportionally to their current value, meaning growth becomes increasingly rapid as time increases. This characteristic results in a steep rise in the graph, especially visible over larger intervals like \([0, 50]\).
Polynomial functions, such as \(g(x) = x^5\), grow at a slower rate compared to exponential functions as they depend on powers of \(x\). Initially, they may appear to grow quickly in small intervals, but exponential functions eventually outpace them.

• In smaller intervals, like \([0, 5]\), both functions may look competitive with each other.
• As the interval expands to \([0, 50]\), \(f(x) = 2^x\) dominates significantly, illustrating the overarching power of exponential growth.

This comparison of graphs across multiple intervals reveals the stark contrast between the two growth types.

Numerical Solutions

Solving equations that involve complex and differing functions like \(2^x = x^5\) requires an analytical approach that often involves numerical solutions. Numerical methods, such as iteration or using graphing calculators, allow you to find approximate values where these functions intersect.
Why not algebraically? Because exponential and polynomial functions usually don’t intersect in straightforward points; they need approximation.
To solve \(2^x = x^5\), you can:

• Utilize graphing calculators to visually identify approximate points of intersection. This provides a rough estimate.
• Apply numerical methods like the Newton-Raphson method for better precision.

The numerical solution to this problem indicates two main intersection points: approximately at \(x = 0.0\) and \(x = 4.7\). These reflect valid solutions where the behavior of both functions aligns.

Intersection of Functions

Understanding where two functions intersect, such as for \(2^x = x^5\), helps identify points where both functions yield the same output. Intersection points are essential as they tell where one function transitions from being less than to greater than the other.
Intersections occur where \(f(x) = g(x)\), represented by their identical outputs on a graph. By examining graphs over different intervals, you can pinpoint these breaks or changes in domination.
For \(f(x) = 2^x\) and \(g(x) = x^5\), their intersections demonstrate key transition points:

• At \(x = 0.0\), both functions start at the same value.
• At approximately \(x = 4.7\), another intersection occurs before \(f(x)\) overtakes permanently.

Graphically identifying these intersections not only helps in solving equations but also provides insight into how each function behaves relative to the other across a domain.