Problem 21 - Essential Calculus Early Transcendentals

Question

Compare the functions \(f(x)=x^{10}\) and \(g(x)=e^{x}\) by graph- ing both \(f\) and \(g\) in several viewing rectangles. When does the graph of \(g\) finally surpass the graph of \(f ?\)

Step-by-Step Solution

Answer

The graph \(g(x)\) surpasses \(f(x)\) around \(x = 10\).

Step 1: Understand the Functions

Compare the two functions, where \(f(x) = x^{10}\) represents a polynomial, and \(g(x) = e^x\) represents an exponential function. We aim to determine the point where \(g(x)\) surpasses \(f(x)\).

Step 2: Consider Graphing the Functions

Consider graphing each function using a graphing calculator or software like Desmos. Start with a viewing window from \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 100\). This provides a general view of how each function behaves.

Step 3: Analyze the Intersection Point

Within the graph, observe where the two curves intersect or where the exponential function \(g(x)\) starts to lie above \(f(x)\). Initially, \(f(x)\) grows faster for \(x < 1\), but as \(x\) increases, \(g(x)\) surpasses \(f(x)\).

Step 4: Adjust Viewing Rectangle

Zoom out to a larger range, such as \(-5 \leq x \leq 20\), to clearly see where \(g(x)\) overtakes \(f(x)\) as \(x\) increases. The point of interest where \(g(x)\) surpasses \(f(x)\) usually occurs around \(x = 10\).

Step 5: Confirm Intersection Algebraically

Verify the \(x\) value found graphically by solving \(x^{10} = e^x\) algebraically or checking values numerically. Trial and error or computational tools can confirm that \(g(x)\) exceeds \(f(x)\) for \(x > 10\).

Key Concepts

Polynomial Functions

Polynomial functions, like the function \(f(x) = x^{10}\), are algebraic expressions consisting of variables with whole number exponents. In this case, the highest power, or degree, of \(x\) is 10, which indicates it's a 10th-degree polynomial.

• As a polynomial function, \(f(x)\) follows a pattern where its graph is typically smooth and continuous.
• The leading term, \(x^{10}\), suggests that for very large values of \(x\), and especially positive \(x\), the function will grow quite rapidly.
• However, polynomial growth is slower compared to exponential growth over larger inputs.

Unlike linear functions, polynomial functions can have multiple turning points, showing more variability in their graphs.

Exponential Functions

Exponential functions, like \(g(x) = e^x\), are functions where the variable \(x\) is an exponent. These functions can exhibit rapid growth. Here, the base of the exponent is Euler’s number, \(e\), approximately equal to 2.718.

• Exponential functions increase more quickly than polynomial functions as \(x\) becomes large, a pivotal attribute in comparing \(g(x)\) to \(f(x)\).
• The graph of \(g(x)\) begins increasing more steeply and consistently for all real numbers \(x\).
• Unlike polynomial functions, they don't have turning points; they are steadily increasing (or decreasing, if it's a decaying exponential).

This steady growth makes exponential functions quite powerful in modeling real-world scenarios such as population growth or compound interest.

Graphical Analysis

Graphical analysis involves plotting functions on a coordinate plane to understand their behavior visually.

• For these functions, start with a standard viewing window from \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 100\) to see their general behavior.
• At first, the polynomial function \(f(x) = x^{10}\) appears to grow faster for values of \(x\) less than 1.
• By increasing the x-range to \( -5 \leq x \leq 20 \), the distinct point where the exponential function \(g(x) = e^x\) clearly surpasses \(f(x)\) becomes evident around \(x = 10\).

This graphical method is incredibly helpful when you need to quickly visualize where one function becomes larger than another.

Intersection Points

Intersection points are where two functions meet or cross on a graph. In this exercise, we focus on the point where \(g(x) = e^x\) surpasses \(f(x) = x^{10}\).

• At intersection points, the value of the functions \(f(x)\) and \(g(x)\) are equal.
• Graphically, this appears as the two curves touching or crossing each other, typically followed by one graph lying above the other.
• To find this point algebraically, one would solve the equation \(x^{10} = e^x\).

The problem identifies that \(g(x)\) overtakes \(f(x)\) at approximately \(x =10\), meaning for \(x > 10\), the values of \(g(x)\) are consistently greater than those of \(f(x)\). Calculating these numerically or using graphing software makes this task more straightforward.