# Unit 4 Contectual Application of Differentiation

_______________________________________________________________________________________________

ENDURING UNDERSTANDING

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

_______________________________________________________________________________________________

 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.

Blog Posts

At Just the Right Time  A good problem

Units

_______________________________________________________________________________________________

 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.

Blog Posts

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

_______________________________________________________________________________________________

 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.

Blog Posts

_______________________________________________________________________________________________

 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.

Blog Posts

Related Rates Problems 1

Good Question 9  Baseball and Related Rates

_______________________________________________________________________________________________

 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.

Blog Posts

Related Rates Problems 1

Good Question 9  Baseball and Related Rates

_______________________________________________________________________________________________

 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.

Blog Posts

Local Linearity The graphical manifestation of the derivative

_______________________________________________________________________________________________

ENDURING UNDERSTANDING

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

_______________________________________________________________________________________________

 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.

Blog Posts

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

_______________________________________________________________________________________________