Day 0 - Sequences and Series Pacing

Design:The topics in this pacing guide are based on the CollegeBoard’s “Big Idea 4: Series” for AP Calculus BC. Within that header come Enduring Understandings (EU: Students will understand that…; think overall goal for a unit), Learning Objectives (LO: Students will be able to….), and Essential Knowledge (EK: Students will know that…; daily goals). These are shown in the columns below, when applicable. The topics are organized (generally) by the order of the WCSD AP Calculus book: “Calculus AP Edition: Briggs, Cochran, Gillett © 2014”. However, except for the assigned bookwork, the lessons are intended to be used independently of the textbook.The lessons are built around an approximate 45-minute class period and are intended to reach an AP Calculus BC audience.

Why:Personally, I always finish my Sequences and Series Unit and realize I’ve forgotten at least one topic. The goal of the lessons below are two-fold: (1) Include everything that actually needs to be taughtin the Sequences and Seriesunit for success in AP Calculus BC and (2) Give teachers who do not teach BC Calculus (or even Calculus at all) a look at how the content they teach in lower-level high school math classes connects to the highest level of Calculus in the WCSD.

Access to All: It’s important to realize that the majority of Sequences and Series does not involve Calculus. In fact, the few sections that do utilize Calculus only involve basic derivatives and integrals. Unit 1 requires almost no knowledge of Calculus (except for the Integral Test). The overall idea of Unit 2 involves basic Calculus, but once you have the patterns down, it’s approachable to just about anyone.

Unit 1: Sequences, and Tests for Series Convergence

Unit 1 covers Enduring Understanding EU 4.1: The sum of an infinite number of real numbers MAY converge. The two Essential Knowledge strands are LO 4.1A: Determine whether an infinite series converges or diverges, and LO 4.1B:Determine or estimate the sum of an infinite series.

Day / Topic / Essential Knowledge Strand
1 / Introduction to Sequences / Prerequisite from Pre-Calculus
2 / Recursive Sequences / Prerequisite from Pre-Calculus
3 / Writing General Terms for Sequences / Prerequisite from Pre-Calculus
4 / Summation Notation / Prerequisite from Pre-Calculus
5 / ConvergentSequences / New to Calculus, could be introduced in Pre-Calculus
6 / Limits of Sequences Growth Rates / New to Calculus, could be introduced in Pre-Calculus
7 / Geometric Sequences + Practice Quiz / Prerequisite from Pre-Calculus
8 / Introduction to Infinite Series / EK 4.1A1, EK 4.1A2
9 / Geometric Series / EK 4.1A3, EK4.1B1
10 / Telescoping Series / EK 4.1A3
11 / Divergence(th term) Test & Harmonic Series / EK 4.1A2, EK 4.1A3, EK 4.1A6
12 / Integral Test / EK 4.1A6
13 / P-Series Test + Practice Quiz / EK 4.1A3
14 / Ratio Test / EK 4.1A6
15 / Root Test / NOT TESTED but falls under EK 4.1A6
16 / (Direct) Comparison Test / EK 4.1A6
17 / Limit Comparison Test + Choosing a Test / EK 4.1A6 (new for 2016-2017)
18 / Alternating Series Test + Alternating Error Bound / EK 4.1A6
19 / Absolute & Conditional Convergence / EK 4.1A4, EK 4.1A5, EK 4.1B3 (new for 2016-2017)
20 / General Review / All of EU 4.1
21 / Unit 1 Exam: Days 1 – 20

Unit 2: Power Series

Unit 2 covers Enduring Understanding EU 4.2:A function can be represented by an associated power series over the interval of convergence for the power series. The three Essential Knowledge strands are LO 4.2A:Construct and use Taylor polynomials, LO 4.2B:Write a power series representing a given function, and LO 4.2C: Determine the radius and interval of convergence of a power series.

Day / Topic / Essential Knowledge Strand
22 / Approximating Functions with Polynomials / EK 4.2A2
23 / Taylor Polynomials and / EK 4.2A1, EK 4.2A2, EK 4.2A3
24 / Maclaurin PolynomialsAnalyzing Terms of Series / EK 4.2A1, EK 4.2A2
25 / /Geometric Polynomials / EK 4.2B3
26 / Radius of Convergence / EK 4.2A3, EK 4.2C2
27 / Interval of Convergence / EK 4.2A3, EK 4.2C1
28 / Compositions & Combinations of Power Series / EK 4.2B5, EK 4.2C4
29 / Differentiation & Integration of Power Series / EK 4.2B5, EK 4.2C3
30 / Error Bounds for Series / EK 4.2A4
31 / AP FRQ Series Practice
32 / Unit 2 Exam: Days 22 – 31