AP Calc Crash Course for Spring 2021 โ™พ๏ธ

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๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

10.0 Unit 10 Overview: Infinite Series and Sequences

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#unit10

#series

#sequences

โฑ๏ธย ย 2 min read

written by

Peter Cao

peter cao


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Exam Weighting โš–

  • Only on the BC Test

  • 17-18% of the test

  • Roughly 7 to 8 multiple choice questions

  • Always one FRQ on this unit (usually FRQ 6)

Developing Understanding ๐Ÿง 

Well after going through countless limits, derivatives, and integrals, youโ€™ve made it here! Welcome to Unit 10, the last unit of BC Calculus! Before we move on, letโ€™s give yourself a pat on the back!

Anyways, this unit builds from past knowledge of limits, differentiation, and integration, but applies this to a new concept โ€” series. This unit emphasizes the study of series analysis. If you havenโ€™t ever heard of analysis in a mathematical before, this is just the study of functions and how they behave over time. Sound familiar? Itโ€™s just a fancy term for calculus used in university, so donโ€™t get bogged down by this term!

There are two main parts to this unit โ€” Convergence and Divergence being one and the Series Approximation to Functions being the other. The Convergence and Divergence part is concerned with the question, โ€œHow does this series behave over a long period of time as we add more and more termsโ€? On the other hand, the Series Approximation to Functions is concerned with the question, โ€œHow can I simply represent a function that will make it easier for me to perform complex operations on them?โ€

Mathematical Practices โž—

Four of the College Board's mathematical practices for AP Calculus are used in this unit, which will be outlined below.

1) Apply appropriate mathematical rules or procedures, with and without technology.

This means that you should know how to calculate the error bounds to analyze series, with the alternating error bound for alternating series and the Lagrange Error Bound for power series.

2) Explain how an approximated value relates to the actual value.

This means that you know how to interpret Lagrange and alternating series error bounds as the maximum error of an infinite series given a partial series.

3) Identify a re-expression of mathematical information presented in a given representation.

This means that you know how to rewrite any function as an infinite power series.

4) Apply an appropriate mathematical definition, theorem, or test.

There are many tests that can be used to determine series convergence and divergence. This means that it is up to you to find the one to use to fit that series.

Main Ideas for this Unit ๐Ÿ’ฌ

  • Convergence and Divergence Tests

    • nth Term Test/Limit Test

    • Geometric Series

    • Harmonic Series

    • p-Series

    • Comparison Tests (Limit and Direct)

    • Integral Test

    • Alternating Series Test

    • Ratio Test

  • Series Approximation

    • Taylor and Maclaurin Series

    • Lagrange Error Bounds

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