Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f, g,\) and \(f+g\) in the same viewing window. Which function contributes most to the magnitude of the sum when \(0 \leq x \leq 2 ?\) Which function contributes most to the magnitude of the sum when \(x>6 ?\) $$f(x)=3 x, \quad g(x)=-\frac{x^{3}}{10}$$

Step-by-Step Solution

Answer

Based on plotting and inspection, one can determine that the function \(f(x)=3x\) contributes more to the magnitude of the sum when \(0 \leq x \leq 2\), while \(g(x)=-x^3/10\) contributes more when \(x>6\).

Step 1: Graph the Functions

Plot the functions \(f(x)=3x\), \(g(x)=-x^3/10\), and the sum function \(f(x)+g(x)\) on the same graph. Confirm that all three functions are displayed correctly.

Step 2: Determine the Dominant Function Between 0 and 2

Inspect the graphs for the range \(0 \leq x \leq 2\). Considering the values and slopes of the functions within this range, one can tell which function is more dominant (which provides a greater contribution to the sum).

Step 3: Determine the Dominant Function for x>6

Inspect the graphs for the range \(x>6\). By comparing the values and slopes, it can be seen which function is more dominant (provides a greater contribution to the sum) in this range.