Exponential Functions

• Population Growth: India: Second most populous country with about 1.25 billion in 2013.

  • Growth rate: 1.2% per year.

  • Prediction: Will exceed China's population by 2031.

Exponential Functions

• Definition: Exponential growth occurs when populations grow rapidly, described mathematically as follows: General formula: f(x) = ab^x

  • Where:

    • a is any nonzero number.

    • b is a positive real number with restrictions:

      • b > 0

      • Ensure all outputs are real numbers.

Characteristics of Exponential Functions

• Base Restrictions:

  • Base b must be positive and not equal to 1 to prevent undefined outputs.

  • If b > 1: function increases rapidly.

  • If 0 < b < 1: function decreases.

• Constant Function Example:

  • f(x) = 1, results in constant output regardless of input.

Evaluating Exponential Functions

• Example: Evaluate f(x) = 2^x where x = 3:

  • f(3) = 2^3 = 8

• Order of Operations:

  • Follow PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).

• Example: Evaluating f(x) = 30(2)^x:

  • Evaluate f(3) = 30(2)^3:

  • = 30 times 8 = 240.

Real-World Application: Population Growth Models

• India Population Model:

  • P(t) = 1.25(1.012)^t: where t is years past 2013.

  • To find population in 2031 (t = 18):

  • P(18) = 1.25(1.012)^{18}

  • Approximately 1.549 billion.

• China Population Model:

  • P(t) = 1.39(1.006)^t: To predict 2031 population:

Compound Interest Formula

• Understanding Compound Interest:

  • A(t) = P(1 + r/n)^{nt}

  • Where:

    • A(t) = accumulated amount.

    • P = principal amount.

    • r = annual interest rate (as a decimal).

    • n = number of compounding periods per year.

    • t = number of years.

• Example of Compound Interest:

  • Invest 3000 at 3% interest compounded quarterly for 10 years:

  • Calculate using formula leading to:

  • A(10) approximately 4045.05 (round to two decimals).

Understanding the Number e

• Definition:

  • e arises from the limit: limit as n approaches infinity (1 + 1/n)^n

  • Approximately 2.718282.

  • Important in calculations involving continuous growth.

• Uses of e:

  • Frequently appears in real-world exponential models and calculations.

• Calculator Usage:

  • Use [e^x] function to compute powers of e.

  • Example: Calculate e^3.14:

  • Result: e^3.14 approximately 23.10387 (round to five decimals).