__Enduring Understandings__1. At a basic level, the student will become conversant in the language and notation of math. The student will be able to translate mathematical concepts into “normal” language and vice versa.

2. The student will be able to reason logically through a problem.

3. The student will begin to see the underlying structure of the mathematical system so as to recognize the assumptions, flaws, and logical conclusions. The student may begin to see alternatives to the system. Ideally, the student will recognize this systemic approach in other fields and apply the same analysis to them.

4. The student will hopefully go beyond the utilitarian mode of learning (“When are we going to use this?” or “Is this going to be on the test?”) and begin to learn new things for the sake of learning them. Ultimately, the student might begin to appreciate the power and the intrinsic beauty people find in the material.

5. The student will begin to develop a meta-cognitive approach to learning. That is, the student will begin to think about his or her thinking process and how that affects his or her learning.

6. The student will develop a confidence and tenacity when approaching lengthy and intricate math problems. By breaking down a problem into its component parts, analyzing and resolving each part, and then reassembling the whole, the student will develop a sense that no problem is beyond his or her grasp.

7. The student will develop the ability to manage large amounts of information by learning to prioritize time, tasks, and goals.

__Essential Questions--Overall__1. What does this “math stuff” mean?

2. Where should this problem start and where does it end? How can I connect those?

3. How do all the problems relate to one another?

4. Why do some people find this interesting? Can it interest me?

5. What are learning preferences? What are my learning preferences? How might other approaches (that I do not prefer) help me learn?

6. How can I make this problem more manageable? What are the pieces of a particular problem and how do they fit together?

7. What information or techniques require details? When can I just know generalities? When should I skip a problem and move on?

__Unit Expectations__

__Unit I. Derivatives__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· The meaning of a derivative in context of graphing, motion, and rate of change

· How to calculate the derivative for various kinds of functions

· How a derivative helps to estimate function values

**Unit Essential Questions**

1. What is a derivative and how is this different in different contexts?

2. What are derivatives used for?

3. Why are there so many different rules for differentiation?

__Unit II Anti-Derivatives__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What an Anti-Derivative is and how it is calculated

· The graphic, algebraic and numerical representations of a differential equation are and how they are related

· What a differential equation is and how to solve one

· What a Slope Field is and how to use is to envision its solution curve

· What the solution function to a slope field might look like

**Unit Essential Questions**

1. What is an ant-derivative and how is it calculated?

2. What is a differential equation and how is it solved?

3. What is a slope field and what is its relationship to particular solution curves?

4. How does Euler’s Method relate to Separation of Variables and Slope Fields?

__Unit III Applications of the Derivative__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What a derivative tells about extremes

· The relationship between a function and its graph and how the derivative helps define that relationship

· What optimization is and how to find it

· The vector nature of position and its derivatives in 2D motion

· What concavity and inflection are

· How the graphs of a function, its derivative and its second derivative are related to one another

**Unit Essential Questions**

1. What does optimization mean?

2. What do derivative graphs tell about and unknown “original” function’s graph? How are the traits of one related to the traits of the others?

3. How are position, velocity, speed, and acceleration related?

__Unit IV Integrals__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What a Definite Integral is and how it is calculated

· The relationship between a definite integral and the are under a curve

· The interpretation of the definite integral in context of graphing and motion

· The difference between net change and total change in context of graphing and motion

· The idea of the definite integral as an accumulator.

· How trapezoids and Riemann Rectangles estimate the definite integral

· How the increasing or decreasing nature of a function relates to where an approximation is an under- or over-estimate.

**Unit Essential Questions**

1. What is a definite integral and how do I calculate it?

2. How is the definite integral set up in different contexts?

3. What are Riemann Rectangles and how are they used? How do I evaluate an integral when the equation for the integrand is known?

4. What is average value and how do I find it?

5. How do I tell the difference between the theorems (Average Value, Mean Value, Intermediate Value, etc) that all sound alike?

6. What does it mean for an integral to be “an accumulator?”

__Unit V Limits__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What a limit is and how it is calculated

· The difference between a one-sided limit and a two-sided limit

· The difference between an infinite limit and a limit at infinity, and how to calculate each

· The abstract nature of the underpinnings of the Calculus and how they relate to the concept of transfinite and indeterminate numbers.

· The concepts of continuity, differentiability, and smoothness

**Unit Essential Questions**

1. What are the different kinds of limits and how are they calculated?

2. What are the various kinds of discontinuity?

3. What do the various kinds of limits have to do with continuity and differentiability?

4. What is differentiability and how is it proven?

5. What is L’Hopital’s Rule and how and when do I use it?

6. What are improper integrals and how are they calculated?

7. How can an infinite region have a finite area?

__Unit VI Numerical Sequences and Series__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· The difference between sequences and series

· How to determine convergence or divergence of a sequence

· How to determine convergence or divergence of a series

· Which convergence test to use for a given series

· The difference between conditional and absolute convergence

**Unit Essential Questions**

1. What is the difference between a sequence and a series and how do I know which one they are talking about?

2. How do determine if a sequence is convergent or divergent?

3. Why are there so many tests for convergence of a series?

4. Which series test do I use when?

5. What is conditional convergence and how do I determine it?

__Unit VII Applications of the Integral__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· How to calculate volumes of solids formed by rotation or cross-section

· How to calculate volumes of solids formed by cross-sections arising from a base region

· How to ignore the distracter of visualizing three dimensional shapes in order to see the appropriate set-up of the problem

· How the Riemann Rectangles relate to the Volume of a solid

**Unit Essential Questions**

1. How do I calculate area for an irregular region?

2. How do I calculate volume of a solid formed by rotating irregular region?

3. How do I calculate volume of a solid formed by polygonal cross-sections rising from an irregular region?

4. How does the sample Riemann Rectangle determine which volume formula to use?

5. How do I see through the “white-noise” of a complicated 3D problem?

6. How do I calculate arc-length of an irregular region?

__Unit VIII Techniques of Integration__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· How the nature of an integrand determines what techniques of integration applies

· That their previous approach of segmenting their knowledge for short-term goals creates a major hurdle to overcome at this level

· What Logistic Growth is and how it differs from exponential growth

**Unit Essential Questions**

1. How does Integration by Parts work?

2. How do I decide what is

*u*and what is

*v*for Integration by Parts?

3. How do I attack integrals of rational function? What are the three integration formulas I should expect?

4. How to I find the coefficients for partial fractions?

5. Why are there three kinds of Partial Fractions? How are they different? And how do I know which one to use?

6. What is Logistic Growth and why is it in this unit?

__Unit IX Parametric and Polar Coordinates__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· The relationship between function, parametric, and polar modes of presenting functions and their graphs

· The nature of parametrics for describing two-dimensional motion

· How to calculate parametric and polar derivatives and integrals and how they interrelate

**Unit Essential Questions**

1. What are the advantages to parametric and polar representation over Cartesian representation? Why have multiple systems? Shouldn’t there be one “best” system?

2. What do the graphs of similar-looking equations look like in the different systems? How do the equations of similar-looking graphs compare in the different systems?

3. How do the systems interact and how does this affect my calculator use processes?

__Unit X Power Series__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· The meaning of a power series

· The relationship between a function and its Taylor polynomial

· How the function’s derivatives are related to the coefficients of the Taylor Polynomial

· What an error bound is and how to calculate it

· The meaning of Interval and Radius of Convergence

· Why a Taylor Polynomial might be more useful that the function it represents

**Unit Essential Questions**

1. What is a Power Series and why do we bother with them?

2. How do the coefficients tell me the Taylor coefficients and vice versa?

3. How do I know which of the four series I had to memorized is used for a new function? How do I use it to find the new series?

4. What are an Interval of Convergence and Radius of Convergence and how do they differ?

5. What is an Error Bound and how do I find it?

6. Which parts of the topic should I skip on the AP Test?