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Unit 4: Contextual Applications of Differentiation

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  1. 4.1 Interpreting the Meaning of the Derivative in Context

    1. CHA-3.A—Interpret the meaning of a derivative in context

      • CHA-3.A Interpret the meaning of a derivative in context. • CHA-3.A.1 The derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable. • CHA-3.A.2 The derivative can be used to express information about rates of change in applied contexts. • CHA-3.A.3 The unit for f′(x) is the unit for f divided by the unit for x. • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

  2. 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration

    1. CHA-3.B—Calculate rates of change in applied contexts

      • CHA-3.B Calculate rates of change in applied contexts. • CHA-3.B.1 The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration. • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

  3. 4.3 Rates of Change in Applied Contexts Other Than Motion

    1. CHA-3.C—Interpret rates of change in applied contexts

      • CHA-3.C Interpret rates of change in applied contexts. • CHA-3.C.1 The derivative can be used to solve problems involving rates of change in applied contexts. • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

  4. 4.4 Introduction to Related Rates

    1. CHA-3.D—Calculate related rates in applied contexts

      • CHA-3.D Calculate related rates in applied contexts. • CHA-3.D.1 The chain rule is the basis for differentiating variables in a related rates problem with respect to the same independent variable. • CHA-3.D.2 Other differentiation rules, such as the product rule and the quotient rule, may also be necessary to differentiate all variables with respect to the same independent variable. • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

  5. 4.5 Solving Related Rates Problems

    1. CHA-3.E—Interpret related rates in applied contexts

      • CHA-3.E Interpret related rates in applied contexts. • CHA-3.E.1 The derivative can be used to solve related rates problems; that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known. • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

  6. 4.6 Approximating Values of a Function Using Local Linearity and Linearization

    1. CHA-3.F—Approximate a value on a curve using the equation of a tangent line

      • CHA-3.F Approximate a value on a curve using the equation of a tangent line. • CHA-3.F.1 The tangent line is the graph of a locally linear approximation of the function near the point of tangency. • CHA-3.F.2 For a tangent line approximation, the function’s behavior near the point of tangency may determine whether a tangent line value is an underestimate or an overestimate of the corresponding function value. • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

  7. 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms

    1. LIM-4.A—Determine limits of functions that result in indeterminate forms

      • LIM-4.A Determine limits of functions that result in indeterminate forms. • LIM-4.A.1 When the ratio of two functions tends to 0/0 or ∞/∞ in the limit, such forms are said to be indeterminate. - Exclusion statement: There are many other indeterminate forms, such as ∞ − ∞, for example, but these will not be assessed on either the AP Calculus AB or BC Exam. However, teachers may include these topics, if time permits. • LIM-4.A.2 Limits of the indeterminate forms 0/0 or ∞/∞ may be evaluated using L’Hospital’s Rule. • Enduring understanding LIM-4: L’Hospital’s Rule allows us to determine the limits of some indeterminate forms.