Relative Rates of Growth

1. Relative Rates of Growth Section 8.2

2. Comparing Rates of Growth The exponential function grows so rapidly and the natural logarithm function grows so slowly that they set standards by which we can judge the growth of other functions...

3. As an illustration of how rapidly grows, imagine graphing the function on a board with the axes labeled in centimeters… At x = 1 cm, the graph is cm high. At x = 6 cm, the graph is m high. At x = 10 cm, the graph is m high. At x = 24 cm, the graph is more than half way to the moon. At x = 43 cm, the graph is light-years high (well past Proxima Centauri, the nearest star to the Sun).

4. Faster, Slower, Same-rate Growth as x Let f (x) and g(x) be positive for x sufficiently large. 1. f grows faster than g (and ggrows slower than f) as if or, equivalently, if 2. f and ggrow at the same rate as if (L finite and not zero)

5. Faster, Slower, Same-rate Growth as x According to these definitions, does not grow faster than as . The two functions grow at the same rate because which is a finite nonzero limit. The reason for this apparent disregard of common sense is that we want “f grows faster than g” to mean that for large x-values, g is negligible in comparison to f.

6. Transitivity of Growing Rates If f grows at the same rate as g as and g grows at the same rate as h as , then f grows at the same rate as h as .

7. Guided Practice Determine whether the given function grows faster than, at the same rate as, or slower than the exponential function as x approaches infinity. Our new rule: (because the base is less than one!) Grows slower than as

8. Guided Practice Determine whether the given function grows faster than, at the same rate as, or slower than the exponential function as x approaches infinity. Grows slower than as

9. Guided Practice Determine whether the given function grows faster than, at the same rate as, or slower than the squaring function as x approaches infinity. Grows faster than as

10. Guided Practice Determine whether the given function grows faster than, at the same rate as, or slower than the squaring function as x approaches infinity. Grows at the same rate as as

11. Guided Practice Determine whether the given function grows faster than, at the same rate as, or slower than the squaring function as x approaches infinity. Grows faster than as

12. Guided Practice Determine whether the given function grows faster than, at the same rate as, or slower than the natural logarithm function as x approaches infinity. Grows at the same rate as as

13. Guided Practice Show that the three functions grow at the same rate as x approaches infinity.

14. Compare the first and second functions: Rational Function Theorem! Compare the first and third functions: By transitivity, the second and third functions grow at the same rate, so all three functions grow at the same rate!