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    Calculus AB: Applications of the Derivative

    Math

    Study Guide
    • Study Guide
    • Topics
      • Preview of Applications of the Derivative
      • Rates of Change and Applications to Motion
      • Problems for "Rates of Change and Applications to Motion"
      • Related Rates Problems
      • Problems for "Related Rates"
      • Absolute and Local Extrema
      • Problems for "Absolute and Local Extrema"
      • The Mean Value Theorem
      • Problems for "The Mean Value Theorem"
      • Using the First Derivative to Analyze Functions
      • Problems from "Using the First Derivative to Analyze Functions"
      • Using the Second Derivative to Analyze Functions
      • Problems for "Using the Second Derivative to Analyze Functions"
      • Vertical and Horizontal Asymptotes
      • Problems for "Vertical and Horizontal Asymptotes"
      • Curve Sketching
      • Problems for "Curve Sketching"
      • Optimization
      • Problems for "Optimization"
    • Terms

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    Terms

    Terms
    • Absolute minimum

      The least single value that a function attains on a given interval. A function f has an absolute minimum at x = b if f (b)≤f (x) for all x in the domain of f.

    • Absolute maximum

      The greatest single value that a function attains on a given interval. A function f has an absolute maximum at x = b if f (b)≥f (x) for all x in the domain of f.

    • Acceleration

      The rate of change of velocity.

    • Average Rate of Change

      For a function f (x), this represents the total change in f over a given interval divided by the total change in x.

    • Concave Down

      A term applied to a section of the graph where the derivative is decreasing throughout.

    • Concave up

      A term applied to a section of the graph where the derivative is increasing throughout.

    • Critical points

      These are points in the domain of f where f'(x) = 0 or f'(x) is undefined.

    • Critical Point Theorem

      If f has a local extremum at x = b, then (b, f (b)) is a critical point.

    • Distance

      This is the scalar analog of the vector quantity position. The distance from a certain point is the absolute value of the position relative to that point. Thus, distances are always positive.

    • Horizontal asymptote

      A horizontal line that the graph of f (x) approaches but never touches as x approaches positive or negative infinity.

    • Instantaneous rate of change

      This is the rate of change of a function at a single point on its graph. The instantaneous rate of change of f at x = a is given by the derivative, f'(a).

    • Local extrema

      The collective term for local maxima and local minima.

    • Local maximum

      A function f has a local maximum at x = b if f (b) is the largest value that f attains "near b." More formally, a function f (x) has a local maximum at x = b if there is an open interval I in which f (b)≥f (x) for all x in I.

    • Local minimum

      A function f has a local minimum at x = b if f (b) is the smallest value that f attains "near b." More formally, a function f (x) has a local minimum at x = b if there is an open interval I in which f (b)≤f (x) for all x in I.

    • Mean value theorem

      If f is continuous on [a, b] and differentiable on (a, b), then there exists a point c on [a, b] for which

      f'(c) =    

    • Position

      This refers to the relative placement of an object with respect to another object, which is generally placed at the origin. This is a vector quantity, so it has both a magnitude and a direction.

    • Rectilinear motion

      A type of motion along a straight line, normally depicted by a number line.

    • Relative extrema

      Same as local extrema.

    • Relative maximum

      Same as local maximum.

    • Relative minimum

      Same as local minimum.

    • Rolle's theorem

      If f is continuous on [a,b] and differentiable on (a,b), and f (a) = f (b), then there is a c on (a, b) where f'(c) = 0. This is a special case of the mean value theorem.

    • Scalar quantities

      These are quantities that only express a magnitude. They are often compared against vector quantities, which have a magnitude and a direction.

    • Speed

      Speed is the scalar analog of velocity.

    • Vector quantities

      These are quantities that have both a magnitude (indicated by the number) and a direction (indicated by the presence or absence of a negative sign). They are often compared against scalar quantities, which only have a magnitude. For example, velocity is the vector analog of speed, because while speed depends only on the magnitude of the motion, velocity also depends on the direction. So, whether one is traveling with a velocity of 3 miles per hour in the positive direction or in the negative direction, one still has a speed of 3 miles per hour.

    • Velocity

      This is the rate of change of position.

    • Vertical asymptote

      This is a vertical line that the graph approaches, but never touches. Near a vertical asymptote, a function approaches positive or negative infinity. A vertical asymptote occurs at x = c when the following are all true
      1) f (c) is undefined
      2) f (x) = ∞ or - ∞
      3) f (x) = ∞ or - ∞

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