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AP Calculus AB/BC · Advanced Placement (AP) · Unit 4 of 10 · Gatekeeper unit

4Unit 4: Contextual Applications of Differentiation

Work through these 7 topics in order. Clear the gatekeeper and the rest of the course opens up.

Contents

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  1. 4.1
    4.1 Interpreting the derivative in context

    In a real situation, f′(a)f'(a) is the INSTANTANEOUS RATE of change of the quantity at the input aa — and it carries UNITS: (units of ff) per (unit of the input).

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  2. 4.2
    4.2 Straight-line motion: position, velocity, acceleration

    For motion on a line, velocity is the derivative of position and acceleration is the derivative of velocity.

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  3. 4.3
    4.3 Rates of change in applied contexts

    Any derivative is a rate of change of one quantity with respect to another — cost per item, people per year, liters per second.

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  4. 4.4
    4.4 Introduction to related rates

    When several changing quantities are tied together by an equation, their RATES are tied together too.

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  5. 4.5
    4.5 Solving related-rates problems

    Model the relationship, differentiate with respect to time, THEN substitute the values true at that instant, and solve for the unknown rate — reporting units.

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  6. 4.6
    4.6 Local linearity and linearization

    Zoomed in near a point, a smooth curve looks like its tangent line.

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  7. 4.7
    4.7 L'Hôpital's Rule for indeterminate forms

    When a limit gives the indeterminate form 00\tfrac{0}{0} or ∞∞\tfrac{\infty}{\infty}, compare how fast the numerator and denominator change — replace each by its derivative.

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