Curriculum Documents by Quarter - Mathematics - AP Calculus AB

Unit of Study 1: 
Standards
Essential Questions

Learning Objectives

 

Students should develop a deeper understanding of very large and very small numbers and of various representations of them.

 

Students should be able to analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

 

Students should be able to judge the meaning, utility, and reasonableness of the re

  • Predict chemical formulas based on the number of valence electrons.
  • Name and write the chemical formulas for simple ionic and molecular compounds, including those that contain common polyatomic ions.
  • Calculate percent yield in a chemical reaction.

sults of symbol manipulations, including those carried out by technology.

 

Students should be able to develop and evaluate mathematical arguments and proofs. 

 

Students should be able to organize and consolidate their mathematical thinking through communication. 

 

  • Predict chemical formulas based on the number of valence electrons.
  • Name and write the chemical formulas for simple ionic and molecular compounds, including those that contain common polyatomic ions.
  • Calculate percent yield in a chemical reaction.

Students should be able to communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

 

Students should be able to use the language of mathematics to express mathematical ideas precisely.

 

Students should be able to use representations to model and interpret physical, social, and mathematical phenomena. 

 

What are different ways of representing numbers?

 

How are mathematical operations related?

 

How can change be analyzed?

 

Which strategies and mathematical operations are used to solve problems?

 

How do we monitor and reflect on the process of mathematical problem solving?

 

How can mathematical thinking be organized and presented so that it can be shared with others?

 

How can we develop a mathematical language that is useful in the real world?

 

What are the relationships between mathematical concepts?

 

How can mathematics be used in other disciplines, as well as the real world?

 

How can we use representations to model and interpret physical, social and mathematical phenomena?

 

Explain the limiting process. 

Estimate limits from graphs or tables of data, and calculate limits using algebra.

Identify asymptotes in terms of graphical behavior, and describe asymptotic behavior in terms of limits involving infinity.

Compare magnitudes of functions and their rates of change.

Explain continuity and its relationship to limits.

Identify graphically continuous functions.

Identify, define and apply the derivative as the slope of a curve at a point, as an instantaneous rate of change, and as the limit of a difference quotient.

Calculate derivatives of basic functions, and apply basic rules for the derivatives of sums, products, and quotients of functions.

Apply the chain rule and implicit differentiation to functions.

Define and explain rates of change with derivatives solve related rates problems.

Unit of Study 2:

 

Students should be able to analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

 

Students should be able to judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

 

Students should be able to develop and evaluate mathematical arguments and proofs. 

 

Students should be able to organize and consolidate their mathematical thinking through communication. 

 

Students should be able to communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

 

Students should be able to use the language of mathematics to express mathematical ideas precisely.

 

Students should be able to use representations to model and interpret physical, social, and mathematical phenomena. 

 

What are different ways of representing, organizing, and relating numbers to mathematical operations?

 

How do we understand and represent patterns, relationships, and change?

 

What are the characteristics, properties, and applications of multi-dimensional shapes?

 

How do we collect, organize, display, and analyze data?

 

Which strategies and mathematical operations are used to solve problems and arguments, and how can we develop and improve them?

 

How can mathematical thinking be effectively communicated to different audiences?

 

How are mathematical concepts related to one another, other disciplines, and to the real world?

 

Analyze of curves interms of monotonicity and concavity

Solve pptimization problems and calculate  both absolute (global) and relative (local) extrema.  

Model rates of change using deriviatives with specific application to related rates problems

Calculate the derivative of an inverse function using implicit differentiiation.  

Apply the derivative as a rate of change to varied applied contexts, including velocity, speed, and acceleration

Identify the increasing and decreasing behavior of f and the sign of f'

Identify the characteristics of the graphs of f, f', and f'' and their relationship to each other.  

Identify the relationship between concavity of f and the sign of f''

 
Identify points of inflection as places where the concavity changes

 

 

 

Unit of Study 3:

 

 

  • What are different ways of representing, organizing, and relating numbers to mathematical operations?
  • How do we understand and represent patterns, relationships, and change?
  • What are the characteristics, properties, and applications of multi-dimensional shapes? 
  • How do we collect, organize, display, and analyze data? 
  • Which strategies and mathematical operations are used to solve problems and arguments, and how can we develop and improve them?
  • How can mathematical thinking be effectively communicated to different audiences? 
  • How are mathematical concepts related to one another, other disciplines, and to the real world?

Vocabulary

Learning Objectives

Students can explain how antiderivatives follow directly from derivatives of basic functions.

Students can demonstrate how a definite integral is a limit of Riemann Sums.

Stduents can define basic properties of definite integrals (including additivity and linearity)

Students can use the Fundamental Thereom, to represent a particular antiderivatives and analytical and graphical analysis of functiosn so defined.

Students can find antideriviatives by substituting variables(including change of limits for definite integrals)

Students can use Riemann Sums (using left, right and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represnted algebraically, graphically, and by tables of values.  

Unit of Study 4:

 

 

 

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