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Linear Systems and Matrix Inverses

• Example of a square matrix A which is not invertible.
• The equation A * x = b signifies that A is the coefficient matrix of the linear systems.
• Statement Analysis:
  • A: Any system of linear equations will have a unique solution. False
  • A unique solution requires an invertible coefficient matrix.
  • B: False
  • Dependent on the vector b.
  • C: False
  • Similar reason as B, influenced by b.
  • D: True
  • Can have infinitely many solutions or no solution, depending on b.
• Homogeneous system analysis:
  • E: A is a square matrix that is not invertible. False.
  • A homogeneous system always has at least the trivial solution (all variables = 0).
  • F: True
  • If the system is homogeneous.

Exponential Functions

• Defined as functions of the form f(x) = b^x where b is a positive constant.
• Characteristics:
  • Growth rate (derivative) is directly proportional to the function's value.
• Graph Behavior:
  • If base b > 1, function increases (exponential growth).
  • All functions of the form b^x pass through the point (0,1) since b^0 = 1.
  • The larger the base, the steeper the graph as x increases.
  • The horizontal asymptote is y = 0, meaning as x approaches negative infinity, f(x) approaches 0 but never reaches it.
• Base < 1:
  • The function decreases as x increases (reflected across the y-axis).
• Base = 1:
  • The function results in a constant function y = 1.
• Domain and Range:
  • Domain: All real numbers (-∞ to ∞).
  • Range: (0, ∞), 0 is not included as the function never equals 0.

Key Exponential Functions to Memorize

• Graph behavior for e^x and e^(-x) with their shapes, y-intercepts, and asymptotes.
• Application in complex functions, such as y = e^x * sin(x).

The Number e

• e ≈ 2.71828; it's an irrational number that never repeats or ends.
• Originally defined through compound interest (incremental interest compounding leads to e).
• Significant applications:
  • Describes normal distribution in statistics.
  • Models population growth in biology via logistic function.
  • Describes exponential decay in physics (e.g., radioactive decay).
• Unique property: For f(x) = e^x, the derivative is also e^x.

Logarithmic Functions

• Logarithms are inverses of exponential functions. If f(x) = b^x, then its log, log(b)(x), helps find the exponent needed to achieve x from the base b.
• Properties of Logarithms:
  • Log of a product: log(b)(x * y) = log(b)(x) + log(b)(y)
  • Log of a quotient: log(b)(x / y) = log(b)(x) - log(b)(y)
  • Log of a power: log(b)(x^n) = n * log(b)(x)
  • Log of 1: log(b)(1) = 0
• Graph Properties:
  • Log function has a domain of (0, ∞) and range of (-∞, ∞).
  • Vertical asymptote at x = 0.

Natural Logarithm

• Base e logarithm is termed as the natural logarithm, denoted as ln(x).
• Important properties:
  • ln(1) = 0 and ln(e) = 1.
  • Logarithm of fractions is derived through their exponential forms.

Solving Logarithmic and Exponential Equations

• To solve for x in equations involving logs or exponentials, use properties to combine or isolate logs:
  1. Combine logs into a single logarithm before applying the inverse.
  2. Ensure you apply operations to both sides of an equation correctly.