Calculus 1.4 Exponential Functions

Exponential Functions Overview

Definition

  • Exponential functions are defined by a constant base raised to a variable exponent.

Basic Operations with Exponents

  1. Product of Powers:

    • Formula: b^(x+y) = b^x * b^y

    • (Add exponents when multiplying)

  2. Quotient of Powers:

    • Formula: b^(x-y) = b^x / b^y

    • (Subtract exponents when dividing)

  3. Power of a Power:

    • Formula: (a^m)^n = a^(m*n)

    • (Multiply exponents)

  4. Power of a Product:

    • Formula: (ab)^n = a^n * b^n

    • (Raise each factor to the exponent)

Graphing Exponential Functions

  • Example: Sketching y = 3 - 2^x

    1. Start with y = 2^x:

      • Points: (0, 1), (1, 2), (2, 4), (3, 8)

      • As x decreases, y approaches 0.

    2. For y = -2^x, graph is a reflection over the x-axis.

    3. For y = 3 - 2^x, move up 3 units:

      • New point: (0, 2)

  • Domain and Range:

    • Domain: All real numbers

    • Range: (-∞, 3)

  • Observation: Exponential functions grow quickly and have a horizontal asymptote.

Comparing Exponential and Power Functions

  • Example: Compare f(x) = 2^x and g(x) = x^2:

    • Initially g(x) > f(x) for small x, but f(x) eventually grows faster (crossover around x = 4).

Half-Life Calculation

  • Example: Half-life of strontium-90 is 25 years.

    • Starting mass: 24 mg, after 25 years: 12 mg; after 50 years: 6 mg.

    • General formula: M(t) = 24 * (1/2)^(t/25).

    • After 40 years: M(40) ≈ 8 mg.

Natural Exponential Function

  • Defined as f(x) = e^x (where e ≈ 2.71828).

    • At x=0, f(0) = 1; slope of tangent line at (0, 1) is 1.

Transformations of Exponential Functions

  • Example: For y = 0.5e^(-x) - 1:

    1. Reflect y = e^(-x).

    2. Horizontal compression by 0.5.

    3. Shift down by 1; new asymptote at y = -1.

Finding When e^x > 1,000,000

  • Use graphing calculator:

    • Approx. x = 13.8 for e^x to exceed 1,000,000.