Calculus Overview

Calculus

• Definition: Branch of mathematics that studies continuous change, dealing with derivatives and integrals.

• Key Concepts:

  • Limits: Fundamental concept for defining derivatives and integrals.

    • Notation: (\lim_{x \to a} f(x))

  • Derivatives: Measure of how a function changes as its input changes.

    • Notation: (f'(x)) or (\frac{dy}{dx})
    • Rules:
      • Power Rule: (\frac{d}{dx}(x^n) = nx^{n-1})
      • Product Rule: (\frac{d}{dx}(uv) = u'v + uv')
      • Quotient Rule: (\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2})
      • Chain Rule: (\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x))

  • Integrals: Measure of the area under a curve.

    • Indefinite Integral: Represents a family of functions.
      • Notation: (\int f(x)dx)
      • Fundamental Theorem of Calculus: Connects differentiation and integration.
    • Definite Integral: Represents the area under the curve between two bounds.
      • Notation: (\int_{a}^{b} f(x)dx)

• Applications:

  • Optimization problems (finding maximum and minimum values)
  • Area and volume calculations
  • Motion analysis (velocity and acceleration)

Formulas

• Basic Algebra:

  • Quadratic Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})

• Geometry:

  • Area of a Circle: (A = \pi r^2)
  • Circumference of a Circle: (C = 2\pi r)
  • Area of a Triangle: (A = \frac{1}{2}bh)

• Trigonometry:

  • Sine: (\sin(\theta) = \frac{opposite}{hypotenuse})
  • Cosine: (\cos(\theta) = \frac{adjacent}{hypotenuse})
  • Tangent: (\tan(\theta) = \frac{opposite}{adjacent})
  • Pythagorean Identity: (\sin^2(\theta) + \cos^2(\theta) = 1)

• Calculus:

  • Derivative of a constant: ( \frac{d}{dx}(c) = 0)
  • Integral of a constant: (\int c , dx = cx + C)

• Statistics:

  • Mean: (\bar{x} = \frac{\sum x_i}{n})
  • Variance: (\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n})
  • Standard Deviation: (\sigma = \sqrt{\sigma^2})

• Exponential and Logarithmic Functions:

  • Exponential Growth: (y = ae^{bt})
  • Natural Logarithm: (\ln(x)) is the inverse of (e^x)

• Miscellaneous:

  • Distance Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
  • Slope of a Line: (m = \frac{y_2 - y_1}{x_2 - x_1})

Calculus

• Branch of mathematics focusing on continuous change, involving derivatives and integrals.

• Limits:

  • Essential concept in calculus for defining derivatives and integrals.
  • Notation: (\lim_{x \to a} f(x)).

• Derivatives:

  • Represents the rate of change of a function as its input changes.
  • Common notations include (f'(x)) or (\frac{dy}{dx}).
  • Key Rules:
    • Power Rule: (\frac{d}{dx}(x^n) = nx^{n-1}).
    • Product Rule: (\frac{d}{dx}(uv) = u'v + uv').
    • Quotient Rule: (\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}).
    • Chain Rule: (\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)).

• Integrals:

  • Calculate the area under a curve.
  • Indefinite Integral: Represents a family of functions, noted as (\int f(x)dx).
  • Fundamental Theorem of Calculus: Links differentiation with integration.
  • Definite Integral: Measures area under a curve between two limits, expressed as (\int_{a}^{b} f(x)dx).

• Applications:

  • Solving optimization problems to find maximum and minimum values.
  • Calculating areas and volumes of geometric shapes.
  • Analyzing motion through velocity and acceleration.

Formulas

• Basic Algebra:

  • Quadratic Formula used to solve (ax^2 + bx + c = 0): (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).

• Geometry:

  • Area of a Circle: (A = \pi r^2).
  • Circumference of a Circle: (C = 2\pi r).
  • Area of a Triangle: (A = \frac{1}{2}bh).

• Trigonometry:

  • Sine: (\sin(\theta) = \frac{opposite}{hypotenuse}).
  • Cosine: (\cos(\theta) = \frac{adjacent}{hypotenuse}).
  • Tangent: (\tan(\theta) = \frac{opposite}{adjacent}).
  • Pythagorean Identity: (\sin^2(\theta) + \cos^2(\theta) = 1).

• Calculus:

  • Derivative of a constant: (\frac{d}{dx}(c) = 0).
  • Integral of a constant: (\int c , dx = cx + C).

• Statistics:

  • Mean: (\bar{x} = \frac{\sum x_i}{n}).
  • Variance: (\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}).
  • Standard Deviation: (\sigma = \sqrt{\sigma^2}).

• Exponential and Logarithmic Functions:

  • Exponential Growth Model: (y = ae^{bt}).
  • Natural Logarithm: (\ln(x)) serves as the inverse function of (e^x).

• Miscellaneous:

  • Distance Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).
  • Slope of a Line: (m = \frac{y_2 - y_1}{x_2 - x_1}).