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Applications of Differentiation in Calculus

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Termini in questo insieme (21)

  • Increasing Function
    A function f is increasing on an interval I if for any a < b in I, f(a) < f(b).
  • Decreasing Function
    A function f is decreasing on an interval I if for any a < b in I, f(a) > f(b).
  • First Derivative Test for Relative Extrema
    If f' changes from negative to positive at c, f has a relative minimum at c. If f' changes from positive to negative at c, f has a relative maximum at c. If f' does not change sign at c, no relative extremum exists at c.
  • Critical Value
    A number c in the domain of f where f'(c) = 0 or f'(c) does not exist. The point (c, f(c)) is a critical point.
  • Concave Up
    A function f is concave up on interval I if f''(x) > 0 for all x in I.
  • Concave Down
    A function f is concave down on interval I if f''(x) < 0 for all x in I.
  • Second Derivative Test for Relative Extrema
    If f'(c) = 0 and f''(c) > 0, f has a relative minimum at c. If f''(c) < 0, f has a relative maximum at c. If f''(c) = 0, use the First Derivative Test.
  • Point of Inflection
    A point where f''(x) = 0 or f''(x) does not exist and the concavity of f changes.
  • Vertical Asymptote
    The line x = a is a vertical asymptote if f(x) approaches ±∞ as x approaches a.
  • Horizontal Asymptote
    The line y = b is a horizontal asymptote if f(x) approaches b as x approaches ±∞.
  • Extreme Value Theorem
    A continuous function on a closed interval [a, b] attains both an absolute maximum and an absolute minimum.
  • Optimization Steps
    1. Find f'(x). 2. Find critical values in [a, b]. 3. Evaluate f(x) at critical values and endpoints. 4. Largest value is absolute maximum; smallest is absolute minimum.
  • Marginal Cost
    The derivative of the total cost function, representing the approximate cost of producing one more unit.
  • Marginal Revenue
    The derivative of the total revenue function, representing the approximate revenue from selling one more unit.
  • Differential
    For y = f(x), the differential dy = f'(x) dx approximates the change in y for a small change dx in x.
  • Linearization
    The linear approximation of f(x) at x = a is L(x) = f(a) + f'(a)(x - a).
  • Elasticity of Demand
    E(p) = -p D'(p) / D(p), measuring responsiveness of quantity demanded to price changes.
  • Interpretation of Elasticity
    If E(p) < 1, demand is inelastic; if E(p) > 1, demand is elastic; if E(p) = 1, revenue is maximized (unit elasticity).
  • Implicit Differentiation
    Differentiate both sides of an equation with respect to x, treating y as a function of x.
  • Logarithmic Differentiation
    Take the natural logarithm of both sides before differentiating to simplify derivatives of complicated functions.
  • Related Rates Method
    1. Write an equation relating variables. 2. Differentiate with respect to time. 3. Substitute known values and solve for desired rate.