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?
 
 
Key Assessments--Overall
 

1. The achievement of the learning outcomes for each unit will be assessed by a Chapter Test which will be half multiple choice and half free response.  These tests will be graded on the AP scale.
2. There will be a Fall Midterm Exam.
3. There will be a Final exam each semester that will be designed after the AP exam.
4. Weekly review homework packets will be assigned in the Spring semester.

 
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 Knowledge and Skills (Performance Objectives)
 
By the end of this unit, the student will be able to:

• Use the Power Rule and Exponential Rules to find Derivatives.
• Find the Derivative of Composite Functions.
• Find Derivatives involving Trig, Trig Inverse, and Logarithmic Functions.

• Use the nDeriv function on the calculator to find numerical derivatives.

• Use the equation of a tangent line to approximate function values.
• Find the Derivative of a product or quotient of two functions.
• Find higher order derivatives.

 
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?

 
Unit Knowledge and Skills (Performance Objectives)
 
By the end of this unit, the student will be able to:

• Find the anti-derivative of a polynomial.
• Integrate functions involving Transcendental operations.
• Use Integration to solve rectilinear motion problems.
• Use the integration by substitution to integrate composite expressions.
• Use the Integration by Substitution to integrate integrands involving  

Sine and Cosine.

• Use the Integration by Substitution to integrate integrands involving  
• Secant and Tangent or Cosecant and Cotangent.
• Given a separable differential equation, find the general solution.
• Given a separable differential equation and an initial condition, find a particular solution.
• Given a differential equation, sketch its slope field.
• Given a slope field, sketch a particular solution curve.
• Given a slope field, determine the family of functions to which the solution curves belong.
• Given a slope field, determine the differential equation that it represents.

 
Unit III       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 Reimann 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 Knowledge and Skills (Performance Objectives)
 
By the end of this unit, the student will be able to:

• Find approximations of integrals using different rectangles.
• Use proper notation when dealing with integral approximation. 
• Differentiate integral expressions with the variable in the boundary
• Evaluate Definite Integrals
• Find the average value of a continuous function over a given interval
• Evaluate definite integrals using the Fundamental Theorem of Calculus.
• Evaluate definite integrals applying the Substitution Rule, when appropriate.
• Use proper notation when evaluating these integrals.
• Relate definite integrals to area under a curve.
• Understand the difference between displacement and total distance.
• Extend that idea to understanding the difference between the two concepts in other contexts.
• Analyze the interplay between rates and accumulation in context.

 
 
 
Unit IV       Applications of the Derivative I
 
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
• What concavity and inflection mean
• 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?

 
Unit Knowledge and Skills (Performance Objectives)
 
By the end of this unit, the student will be able to:

• Find critical values and extreme values for functions.
• Use the 1st and 2nd derivative tests to identify maxima vs. minima.
• Find Points of Inflection and Intervals of Concavity.
• Sketch the graph of a function using information from its first and/or second  
• derivatives.
• Sketch the graph of a first and/or second derivative from the graph of a function.
• Solve optimization problems.
• Eliminate the parameter of parametric equations.
• Interpret information in the graph of a derivative in terms of the graph of the “original” function.
• Use the graph of a function to answer questions concerning maximums, minimums, and intervals of increasing and decreasing
• Use the graph of a function to answer questions concerning points of inflection and intervals of concavity.
• Use the graph of a function to answer questions concerning the area under a curve.

 
Unit V        Applications of the Derivative II
 
Unit Enduring Understandings
 
By the end of this unit, the student will understand:

• to interpret rectilinear motion data presented in algebraic, graphical, and tabular formats.
• that position, velocity, and acceleration are interrelated by derivatives and integrals.
• that implicit differentiation is a variation on the Chain Rule.
• how to solve related rates problems through implicit differentiation.
• how the units in a related rates problem determine setup and calculations.

• what Logistic Growth is and how it differs from Exponential Growth

• what a differential equation is and how to solve one

 
 
 
 
Unit Essential Questions
 

1. How are position, velocity, speed, and acceleration related? 
2. How is rectilinear motion interpreted when introduced in function form vs. graphical form vs. numerical form?
3. How do I know to apply implicit differentiation to a problem?
4. How do I apply implicit differentiation?
5. How do I start a Related Rates word problem?
6. What is Logistic Growth and why is it in this unit?
7. How does Logistic Growth differ from Exponential Growth?
8. What do Logistic and exponential Growth have to do with separation of variables?

 
 
Unit Knowledge and Skills (Performance Objectives)
 
By the end of this unit, the student will be able to:

• Use the derivative to make conclusions about motion.
• Relate the position, velocity, and acceleration functions.

• Take derivatives of relations implicitly.
• Use implicit differentiation to find higher order derivatives.
• Solve related rates problems.
• Identify key information from a logistic growth equation
• Solve separable differential equations that arise from logistic or exponential growth

 
 
Unit VI       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 Knowledge and Skills (Performance Objectives)
 
By the end of this unit, the student will be able to:

• Find the area of the region between two curves.
• Find the volume of a solid rotated when a region is rotated about a given axis
• Find the volume of a solid rotated when a region is rotated about a given line
• Find the volume of a solid with given cross sections.
• Find the arc length of a function in Cartesian mode between to points.

 
 
 
Unit VII     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?

 
Unit Knowledge and Skills (Performance Objectives)
 
By the end of this unit, the student will be able to:

• Evaluate one-sided limits graphically, numerically, and algebraically.
• Evaluate two-sided limits in terms of one-sided limits.
• Prove continuity or discontinuity of a given function.
• Interpret Vertical Asymptotes in terms of one-sided limits.
• Determine if a function is differentiable or not.
• Demonstrate understanding of the connections and differences between differentiability and continuity.  
• Evaluate Limits algebraically.
• Evaluate Limits using L’Hopital’s Rule.
• Recognize and evaluate Limits which are derivatives.
• Evaluate Limits at infinity.
• Interpret Limits at infinity in terms of end behavior of the graph.