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Tangent Lines & Derivatives definitions

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  • Tangent Line
    A straight line that touches a curve at exactly one point, representing the instantaneous direction of the curve at that point.
  • Secant Line
    A straight line that intersects a curve at two distinct points, representing the average rate of change between those points.
  • Point of Tangency
    The specific location on a curve where a tangent line touches, corresponding to a single x-value of interest.
  • Slope
    A measure of steepness of a line, calculated as the ratio of vertical change to horizontal change between two points.
  • Limit
    A mathematical concept describing the value a function approaches as the input approaches a specific point.
  • Difference of Squares
    An algebraic expression of the form a^2 - b^2, which factors into (a + b)(a - b).
  • Derivative
    A function that gives the slope of the tangent line to a curve at any point, representing instantaneous rate of change.
  • Instantaneous Rate of Change
    The rate at which a function changes at a single point, found using the slope of the tangent line.
  • Average Rate of Change
    The change in a function's value over an interval, calculated using the slope of a secant line.
  • Point-Slope Form
    An equation format for a line using a known point and the slope, typically written as y - y₁ = m(x - x₁).
  • Prime Notation
    A shorthand symbol (') used to denote the derivative of a function, such as f'(x).
  • Limit Definition of Derivative
    A formula expressing the derivative as the limit of the difference quotient as the interval approaches zero.
  • Difference Quotient
    An expression representing the average rate of change of a function over an interval, used in the definition of the derivative.
  • Function Notation
    A way to represent functions, typically written as f(x), indicating the output for a given input x.
  • Factoring
    An algebraic process of rewriting an expression as a product of its simpler components, often used to simplify limits.