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)
Well after going through countless limits, derivatives, and integrals, you’ve made it here! Welcome to Unit 10, the last unit of BC Calculus! Before we move on, let’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 — series. This unit emphasizes the study of series analysis. If you haven’t ever heard of analysis in a mathematical before, this is just the study of functions and how they behave over time. Sound familiar? It’s just a fancy term for calculus used in university, so don’t get bogged down by this term!
There are two main parts to this unit — Convergence and Divergence being one and the Series Approximation to Functions being the other. The Convergence and Divergence part is concerned with the question, “How does this series behave over a long period of time as we add more and more terms”? On the other hand, the Series Approximation to Functions is concerned with the question, “How can I simply represent a function that will make it easier for me to perform complex operations on them?”
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.
Convergence and Divergence Tests
nth Term Test/Limit Test
Comparison Tests (Limit and Direct)
Alternating Series Test
Taylor and Maclaurin Series
Lagrange Error Bounds
✍️ Free Response Questions (FRQ)
Calculus Free Response Questions
👑 Unit 1: Limits & Continuity
1.5Determining Limits Using Algebraic Properties of Limits
1.6Determining Limits Using Algebraic Manipulation
1.10Exploring Types of Discontinuities
1.11Defining Continuity at a Point
1.12Confirming Continuity over an Interval
🤓 Unit 2: Differentiation: Definition & Fundamental Properties
2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
🤙🏽 Unit 3: Differentiation: Composite, Implicit & Inverse Functions
3.0Unit 3 Overview: Differentiation: Composite, Implicit, and Inverse Functions
3.1The Chain Rule
3.3Differentiating Inverse Functions
3.4Differentiating Inverse Trigonometric Functions
👀 Unit 4: Contextual Applications of the Differentiation
4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
4.4Intro to Related Rates
4.6Approximating Values of a Function Using Local Linearity and Linearization
✨ Unit 5: Analytical Applications of Differentiation
5.0Unit 5 Overview: Analytical Applications of Differentiation
5.2Extreme Value Theorem, Global vs Local Extrema, and Critical Points
5.3Determining Intervals on Which a Function is Increasing or Decreasing
5.4Using the First Derivative Test to Determine Relative (Local) Extrema
5.7Using the Second Derivative Test to Determine Extrema
🔥 Unit 6: Integration and Accumulation of Change
6.11Integrating Using Integration by Parts
💎 Unit 7: Differential Equations
7.0Unit 7 Overview: Differential Equations
7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
🐶 Unit 8: Applications of Integration
8.1Finding the Average Value of a Function on an Interval
8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
8.4Finding the Area Between Curves Expressed as Functions of x
8.5Finding the Area Between Curves Expressed as Functions of y
8.6Finding the Area Between Curves That Intersect at More Than Two Points
8.7Volumes with Cross Sections: Squares and Rectangles
8.8Volumes with Cross Sections: Triangles and Semicircles
8.9Volume with Disc Method: Revolving Around the x- or y-Axis
8.10Volume with Disc Method: Revolving Around Other Axes
8.11Volume with Washer Method: Revolving Around the x- or y-Axis
🦖 Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)
9.0Unit 9 Overview: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.1Defining and Differentiating Parametric Equations
♾ Unit 10: Infinite Sequences and Series (BC Only)
10.0Unit 10 Overview: Infinite Series and Sequences
10.1Defining Convergent and Divergent Infinite Series
10.6Comparison Tests for Convergence
10.7Alternating Series Test for Convergence
10.11Finding Taylor Polynomial Approximations of Functions
10.14Finding Taylor or Maclaurin Series for a Function
🧐 Multiple Choice Questions (MCQ)
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