Mathematics 133 -- Intensive Calculus for Science

Discussion 2 -- Which Function Dominates?

September 5, 2000

Background

We have introduced the power functions f(x) = k xp last time and discussed the general shapes of their graphs. Today we want to experiment to determine what effect the coefficient k has on how fast the function grows, and also to compare power and exponential functions.

Discussion Questions

Using a graphing calculator investigate the following questions about which function is "dominant" in size (larger in value) for big positive values of x.

  1. Plot y = x3 and y = 5 x2 for x between 0 and 4. Which function is larger on this range? What if you increase the plotting range to x between 0 and 6? Where do the curves cross?
  2. Similarly, plot y = x3 and y = 10 x2 for x between 0 and 5. Which function is larger on this range? What if you increase the plotting range to x between 0 and 12? Where do the curves cross?
  3. Can you see a general pattern for the graphs y = x3 and y = A x2 with A > 0. At what point does the "dominant" function take over?
  4. Now consider y = 10x2 and y = 2x. Which is larger for relatively small x (say x between 0 and 8)? What happens if you plot a bigger range of x values? The graphs should eventually cross and a new "dominant" function should emerge!
  5. Try y = 100x2 and y = 2x. Which is larger for relatively small x (say x between 0 and 15)? What happens if you plot a bigger range of x values?

We can ask similar questions about which function is dominant for small positive values of x -- x close to zero.

  1. Which function dominates as x tends to zero (from the positive side) -- y = 5 x-2 or y = x-3? Do the graphs cross somewhere? Where?
  2. Same question for y = 10 x-2 and y = x-3.
  3. What is the general pattern for y = A x-2 and y = x-3?