Math 108 - Lecture 19: Exponential Growth and Decay

Math 108 Lecture Notes

Lecture 19: Exponential Growth and Decay

Definition of Exponential Functions
  • Definition 3.1.2: Let ( b ) be a real number such that ( b > 0 ) and ( b \neq 1 ).
    • The function defined by ( f(t) = b^t ) is called an exponential function with base ( b ).
Importance of the Condition ( b \neq 1 )
  • The condition ( b \neq 1 ) is essential because:
    • If ( b = 1 ), the function becomes constant: ( f(t) = 1^t = 1 ), which does not represent growth or decay.
Value at Zero
  • For exponential functions, calculate the value at zero:
    • ( f(0) = b^0 = 1 ) (for any value of ( b ) where ( b > 0 )).
General Form of Exponential Functions
  • In applications, we consider functions of the form:
    • ( f(t) = a b^t ) where ( a, b ) are positive constants and ( b \neq 1 ).
    • ( a ): This is a scaling factor that affects the initial value of the function.
    • ( b ): This is referred to as the growth factor.
      • If ( b = (1 + r) ), where ( r ) is the growth rate.
      • Growth Rates:
      • If ( b > 1 ), the function represents exponential growth.
      • If ( 0 < b < 1 ), the function represents exponential decay.
Effect of Parameter ( a ) on the Graph
  • The parameter ( a ) expands or compresses the graph vertically:
    • Larger values of ( a ) result in a higher initial point (y-intercept) on the graph, while smaller values compress the graph downwards towards the x-axis.

Activity 3.1.2: Exploring Exponential Functions in Desmos

  • Task: Define the function ( g(t) = ab^t ) in Desmos.
    • Create sliders for both ( a ) and ( b ).
    • Set minimum value for each slider to 0.1 and maximum to 10.
    • Note that to qualify as an exponential function, ( b ) must not equal 1, despite the slider allowing this value.
Questions for Consideration

  1. Domain of ( g(t) = ab^t ):

    • The domain consists of all real numbers, ( (-\infty, \infty) ), because ( t ) can take any real value.

  2. Range of ( g(t) = ab^t ):

    • The range is ( (0, \infty) ) since ( g(t) ) is always positive for ( a > 0 ) and ( b > 0 ).

  3. Y-Intercept of ( g(t) = ab^t ):

    • The y-intercept occurs when ( t = 0 ): ( g(0) = ab^0 = a ). Hence, the y-intercept is ( a ).

  4. Effect of Changing ( b ):

    • As the value of ( b ) changes:
      • If ( b > 1 ), the graph rises steeply, indicating exponential growth.
      • If ( 0 < b < 1 ), the graph falls, indicating exponential decay.
      • The steepness and direction of the graph depend directly on the value of ( b ).

  5. Correspondence Between ( b ) and Growth Rate:

    • For positive growth rates, ( b ) must be greater than 1: ( b > 1 \rightarrow r > 0 ).
    • For negative growth rates, ( 0 < b < 1 \rightarrow r < 0 ).

  6. Comparison of Exponential Functions: Graphs of ( p(t) = ab^t ) and ( q(t) = cd^t ):

    • In Figure 3.1.3, the graphs of the functions ( p ) and ( q ) show:
      • All coefficients ( a, b, c, \, d ) are positive,
      • Both ( b ) and ( d ) cannot equal 1 to maintain their identity as exponential functions.
      • Comparisons involving the size of ( a, b, c, d ):
      • Direct relationships may provide insights into which graph grows faster or intersects.
      • The analysis is constrained to the fact that they must be positive and adhere to the exponential properties defined above.
Reference Graph: Figure 3.1.3
  • Description: Graphs of the exponential functions ( p ) and ( q ).
    • Visual representation helps illustrate differences in behaviors under various configurations of ( a, b, c, ) and ( d ).