Unit 4 Contectual Application of Differentiation

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ENDURING UNDERSTANDING

CHA-3 Derivatives allow us to solve real-world problems involving rates of change.

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 Topic Name Essential Knowledge 4.1 Interpreting the Meaning of the Derivative in Context LEARNING OBJECTIVE 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${f}'\left( x \right)$ is the unit for f divided by the unit for x.

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At Just the Right Time  A good problem

Units

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 4.2 Straight Line Motion: Connecting Position, Velocity, and Acceleration LEARNING OBJECTIVE 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.

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The Ubiquitous Particle Motion Problem  – a PowerPoint Presentation and its Handout

Motion Problems: Same Thing Different Context (11-16-2012)   Matching Motion (9-16-2016)

Motion Matching  A quick quiz

Speed (11-19-2012)

Speed Activity An exploration on Speed

A Note on Speed (4-21-2018) An analytic approach

Brian Leonard’s Particle Motion Game Velocity Game  and answers Velocity game Answers

Type 2 Questions: Linear Motion

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 4.3 Rates of Change in Applied Contexts Other than Motion LEARNING OBJECTIVE 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.

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 4.4 Introduction to Related Rates LEARNING OBJECTIVE 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.

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Related Rates Problems 1

Good Question 9  Baseball and Related Rates

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 4.5 Solving Related Rate Problems LEARNING OBJECTIVE 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.

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Related Rates Problems 1

Good Question 9  Baseball and Related Rates

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

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Local Linearity The graphical manifestation of the derivative

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ENDURING UNDERSTANDING

LIM-4 L’Hospital’s Rule allows us to determine the limits of some indeterminate forms.

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 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms LEARNING OBJECTIVE LIM-4.A Determine limits of functions that result in indeterminate forms. LIM-4.A.1 When the ratio of two functions tends to $\frac{0}{0}$  or $\frac{\infty }{\infty }$  in the limit, such forms are said to  be indeterminate. LIM-4.A.2 Limits of the indeterminate forms $\frac{0}{0}$  or $\frac{\infty }{\infty }$   may be evaluated using L’Hospital’s Rule.

EXCLUSION STATEMENT: There are many other indeterminate forms, such as $\infty -\infty$ , 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.

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Determining the Indeterminate 2 Same name, different post. Examining an implicit relation

Locally Linear L’Hôpital  Demonstrating L’Hôpital’s Rule (a/k/a L’Hospital’s Rule)

L’Hôpital’s Rules the Graph

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