DITF - Relative Magnitudes of Functions and Their Rates of Change Lesson

Relative Magnitudes of Functions and Their Rates of Change

Based on a comparison of their derivatives, the greater the value of a for exponential functions, the faster a^x increases. Similarly, for logarithmic functions, the greater the value of a, the slower log_a x increases. text annotation indicator As the value of x increases and approaches infinity, both of these functions approach infinity, but their rates of change differ as do their magnitudes. By examining the graphs of exponential, logarithmic, and polynomial functions, the relative magnitudes of these functions can be compared. Relative magnitude in this context refers to a comparison and ordering of the relative size of the functions.

Comparison of Exponential, Polynomial, and Logarithmic Growth

Over a short period of time the relative magnitude of a function and its derivative may vary greatly. It is impossible to determine whether the exponential, polynomial, or logarithmic function is always greater. However, over the long run exponential functions grow faster than polynomial functions, which grow faster than logarithmic functions. The table below summarizes the long term behavior of exponential, polynomial, and logarithmic functions as the independent variable x approaches infinity.

Type of Growth	Growth Rate Description	Long Term Behavior	Concavity	Function Behavior at x = 0	Function Magnitude Ranking
Exponential $y=a^x$ (for a > 1)	If a > 1, growth is proportional to the size of a.	If a > 1, it grows slowly and then grows to infinity more and more quickly.	Up	1	1
Polynomial $y=x^n$	Continuous growth occurs when n is even.	Grows at an ever increasing rate when n is even, ultimately increases when n is odd, and is eventually surpassed by an exponential function.	Up (for n an even integer) Both up and down (for n an odd integer)	0	2
Logarithmic $y=\log_ax$	Growth is inversely proportional to the size of a.	Rate of growth slows but stays positive and eventually approaches infinity.	Down	Undefined	3

Type of Growth

Growth Rate Description

Long Term

Behavior

Concavity

Function Behavior at x = 0

Function Magnitude Ranking

Exponential

equation image indicator $y=a^x$

(for a > 1)

If a > 1, growth is proportional to the size of a.

If a > 1, it grows slowly and then grows to infinity more and more quickly.

Polynomial

equation image indicator $y=x^n$

Continuous growth occurs when n is even.

Grows at an ever increasing rate when n is even, ultimately increases when n is odd, and is eventually surpassed by an exponential function.

Up (for n an even integer)

Both up and down (for n an odd integer)

Logarithmic

equation image indicator $y=\log_ax$

Growth is inversely proportional to the size of a.

Rate of growth slows but stays positive and eventually approaches infinity.

Down

Undefined

Relative Magnitudes of Functions and Their Rates of Change Practice

Relative Magnitudes of Functions and Their Rates of Change: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of relative magnitudes of functions and their rates of change.

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