Mathematical Analysis (Honors)
Unit 8: Limits and Series
In this unit, students extend understanding of functions to explore sequences and series. First, students determine whether a sequence converges or diverges. They find the partial sum of an arithmetic or geometric sequence, as well as the sum of a converging geometric sequence. Students develop an understanding that an infinite series is found by taking the limit of partial sums.
Students apply their understanding of functions and use a problem-solving approach in order to calculate limits algebraically as well as sketch functions without the use of a graphing calculator and identify key elements of the function including intercepts, end behavior, turning points, and intervals for which the function is increasing/decreasing/or remains constant using limits. They use limits to describe continuity of functions and be able to discuss one-sided limits and determine if a limit fails to exist at a value. Students calculate average rate of change of a functions and use limits to find the instantaneous rate of change of a function at a point and instantaneous rates of change as defined by limits and discuss the use of limits to develop the concept of the derivative of a function. Students use functions to model real-world situations and make predictions about future behavior.
Students apply their understanding of functions and use a problem-solving approach in order to calculate limits algebraically as well as sketch functions without the use of a graphing calculator and identify key elements of the function including intercepts, end behavior, turning points, and intervals for which the function is increasing/decreasing/or remains constant using limits. They use limits to describe continuity of functions and be able to discuss one-sided limits and determine if a limit fails to exist at a value. Students calculate average rate of change of a functions and use limits to find the instantaneous rate of change of a function at a point and instantaneous rates of change as defined by limits and discuss the use of limits to develop the concept of the derivative of a function. Students use functions to model real-world situations and make predictions about future behavior.
Students will:
Analyze limits of functions.
MA.LS.A.1
Calculate limits algebraically and estimate limits from graphs and tables of values.
MA.LS.A.2
Use and apply the properties of limits to find the limit of various functions.
MA.LS.A.3
Determine the limit of a function as the domain approaches infinity.
Analyze and apply limits of functions to describe function behavior.
PC.AF.C.8
Use one-sided limits to describe continuity of functions.
PC.AF.C.9
Determine if a function is continuous at a point or over an interval.
PC.AF.C.10
Apply limits using the definition of derivative with respect to tangent lines of functions.
Use appropriate notation when working with series.
MA.LS.B.4
Use factorial and summation notation appropriately.
Analyze and apply series to solve real-world and mathematical problems.
PC.S.A.1
Determine whether a sequence converges or diverges.
PC.S.A.2
Find the partial sum of an arithmetic or geometric series.
PC.S.A.3
Define an infinite series as the limit of the partial sums.
PC.S.A.4
Find the sum of a convergent infinite geometric series.
Analyze limits of functions.
MA.LS.A.1
Calculate limits algebraically and estimate limits from graphs and tables of values.
- Check for Understanding: Finding Limits Numerically | Limit from Graph
- Review/Rewind:: Intro to Limits | Limits intuition | Estimating Limits Numerically | Limit from graph
MA.LS.A.2
Use and apply the properties of limits to find the limit of various functions.
- Check for Understanding: Finding Limits Algebraically | Limits using Advanced Algebra
- Review/Rewind: Finding Limits Algebraically | Properties of Limits | Finding Limits with Advanced Algebra
MA.LS.A.3
Determine the limit of a function as the domain approaches infinity.
- Check for Understanding: Limits at Infinity
- Review/Rewind: Limits and Infinity | Limits at Positive and Negative Infinity | More Limits at Infinity
Analyze and apply limits of functions to describe function behavior.
PC.AF.C.8
Use one-sided limits to describe continuity of functions.
- Review/Rewind: Limits and Continuity
PC.AF.C.9
Determine if a function is continuous at a point or over an interval.
PC.AF.C.10
Apply limits using the definition of derivative with respect to tangent lines of functions.
- Check for Understanding: Continuity
- Review/Rewind: Limit and Function at Discontinuity | Making a Function Continuous
Use appropriate notation when working with series.
MA.LS.B.4
Use factorial and summation notation appropriately.
- Check for Understanding: Sigma Notation
- Review/Rewind: Sigma Notation | Factorials
Analyze and apply series to solve real-world and mathematical problems.
PC.S.A.1
Determine whether a sequence converges or diverges.
- Check for Understanding: Infinite Series
- Review/Rewind: Infinite Geometric Series | Convergent or Divergent Series
PC.S.A.2
Find the partial sum of an arithmetic or geometric series.
- Background Info.
- Check for Understanding: Arithmetic Series | Finite Geometric Series
- Review/Rewind: Arithmetic Series | Geometric Series | Geometric Series with Sigma Notation
PC.S.A.3
Define an infinite series as the limit of the partial sums.
- Review/Rewind: Infinite Geometric Series as a Limit
PC.S.A.4
Find the sum of a convergent infinite geometric series.
- Check for Understanding: Calculating Infinite Series | Word Problem: Bouncing Ball
- Review/Rewind: Infinite Geometric Series | Convergent or Divergent Series
What are some signs of student mastery?
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Tools & Technology
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