Calculus: Derivatives and Exponential Functions

Derivative

• Definition: A derivative measures the rate of change of a function with respect to one of its variables.
• Notation: f'(x) or (d/dx)f(x)
• Interpretation: The derivative of a function f at a point x represents the rate of change of the function at that point.
• Geometric Interpretation: The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

Application of Exponential Functions

• Exponential functions: f(x) = a^x, where 'a' is a positive real number.
• Derivative of exponential functions: f'(x) = a^x * ln(a)
• Applications:
  • Population growth and decay
  • Compound interest
  • Chemical reactions

Application of Trigonometry

• Trigonometric functions: sine, cosine, and tangent.
• Derivatives of trigonometric functions:
  • (d/dx)sin(x) = cos(x)
  • (d/dx)cos(x) = -sin(x)
  • (d/dx)tan(x) = sec^2(x)
• Applications:
  • Modeling periodic phenomena (e.g., sound waves, light waves)
  • Analyzing circular motion

Maxima and Minima

• Local maximum: A point at which the function has a maximum value in a small region around that point.
• Local minimum: A point at which the function has a minimum value in a small region around that point.
• Critical points: Points at which the derivative of the function is zero or undefined.
• First Derivative Test:
  • If f'(x) changes sign from positive to negative, then x is a local maximum.
  • If f'(x) changes sign from negative to positive, then x is a local minimum.

Concave Up and Down

• Concave up: A function is concave up if its derivative is increasing.
• Concave down: A function is concave down if its derivative is decreasing.
• Inflection points: Points at which the concavity of a function changes.
• Second Derivative Test:
  • If f''(x) > 0, then the function is concave up.
  • If f''(x) < 0, then the function is concave down.

Derivative

• Measures the rate of change of a function with respect to one of its variables
• Notated as f'(x) or (d/dx)f(x)
• Represents the rate of change of the function at a point
• Geometrically, it's the slope of the tangent line to the graph of the function at a point

Exponential Functions

• Defined as f(x) = a^x, where 'a' is a positive real number
• Derivative: f'(x) = a^x * ln(a)
• Applies to:
  • Population growth and decay
  • Compound interest
  • Chemical reactions

Trigonometric Functions

• Include sine, cosine, and tangent
• Derivatives:
  • (d/dx)sin(x) = cos(x)
  • (d/dx)cos(x) = -sin(x)
  • (d/dx)tan(x) = sec^2(x)
• Apply to:
  • Modeling periodic phenomena (e.g., sound waves, light waves)
  • Analyzing circular motion

Maxima and Minima

• Local maximum: A point with the maximum value in a small region
• Local minimum: A point with the minimum value in a small region
• Critical points: Where the derivative is zero or undefined
• First Derivative Test:
  • Sign change from positive to negative: local maximum
  • Sign change from negative to positive: local minimum

Concave Up and Down

• Concave up: Increasing derivative
• Concave down: Decreasing derivative
• Inflection points: Where concavity changes
• Second Derivative Test:
  • f''(x) > 0: Concave up
  • f''(x) < 0: Concave down