Exponential Growth and Decay and Inverse Trigonometric Functions
MAT1241: Calculus 1
University of Eswatini.
Exponential Growth and Decay
1. Exponential Growth
● Definition:
○ Exponential growth occurs when a quantity increases at a constant percentage rate
over equal time intervals.
○ The general form of an exponentially growing function is , where:
■ (a) is the initial value (usually the value at (x = 0)).
■ (b) is the growth factor (greater than 1).
● Example 1: Population Growth
○ Suppose a population of bacteria doubles every hour. If there are initially 100
bacteria, the population after (t) hours is given by .
○ Calculate the population after 3 hours:
● Example 2: Compound Interest
○ The compound interest formula is , where:
■ (A) is the final amount.
■ (P) is the principal amount.
■ (r) is the annual interest rate (expressed as a decimal).
■ (n) is the number of times interest is compounded per year.
■ (t) is the time in years.
○ Solve for (A) when (P = 1000), (r = 0.05), (n = 4), and (t = 5):
2. Exponential Decay
● Definition:
○ Exponential decay occurs when a quantity decreases at a constant percentage rate
over equal time intervals.
○ The general form of an exponentially decaying function is , where:
■ (a) is the initial value.
■ (b) is the decay factor (between 0 and 1).
MAT1241: Calculus 1
University of Eswatini.
Exponential Growth and Decay
1. Exponential Growth
● Definition:
○ Exponential growth occurs when a quantity increases at a constant percentage rate
over equal time intervals.
○ The general form of an exponentially growing function is , where:
■ (a) is the initial value (usually the value at (x = 0)).
■ (b) is the growth factor (greater than 1).
● Example 1: Population Growth
○ Suppose a population of bacteria doubles every hour. If there are initially 100
bacteria, the population after (t) hours is given by .
○ Calculate the population after 3 hours:
● Example 2: Compound Interest
○ The compound interest formula is , where:
■ (A) is the final amount.
■ (P) is the principal amount.
■ (r) is the annual interest rate (expressed as a decimal).
■ (n) is the number of times interest is compounded per year.
■ (t) is the time in years.
○ Solve for (A) when (P = 1000), (r = 0.05), (n = 4), and (t = 5):
2. Exponential Decay
● Definition:
○ Exponential decay occurs when a quantity decreases at a constant percentage rate
over equal time intervals.
○ The general form of an exponentially decaying function is , where:
■ (a) is the initial value.
■ (b) is the decay factor (between 0 and 1).
