Course Syllabus: Syllabus
The following links are the blank and filled-in notes and assignments for AP Calculus AB/BC.
All notes and assignments follow the textbook: Calculus of a Single Variable Tenth Edition, AP* Edition, (2015), Larson.
Here is a digital version of the textbook: https://www.nxtbook.com/nxtbooks/ngsp/calculus_singlevariable/index.php#/1
The following links are the blank and filled-in notes and assignments for AP Calculus AB/BC.
All notes and assignments follow the textbook: Calculus of a Single Variable Tenth Edition, AP* Edition, (2015), Larson.
Here is a digital version of the textbook: https://www.nxtbook.com/nxtbooks/ngsp/calculus_singlevariable/index.php#/1
Chapter 1 Limits and Their Properties
This first chapter involves the fundamental calculus elements of limits. While limits are not typically found on the AP test, they are essential in developing and understanding the major concepts of calculus: derivatives & integrals. These notes cover the properties of limits including: how to evaluate limits numerically, algebraically, and graphically. An important characteristic of functions, continuity, is also discussed in greater detail than in previous math classes. This chapter also contains two major theorems: The Intermediate Value Theorem (IVT) and the Squeeze Theorem. While neither are prominent on the AP test, the IVT has applications with derivative tests found in the third chapter. Lastly, the idea of infinity is discussed in greater detail.
This first chapter involves the fundamental calculus elements of limits. While limits are not typically found on the AP test, they are essential in developing and understanding the major concepts of calculus: derivatives & integrals. These notes cover the properties of limits including: how to evaluate limits numerically, algebraically, and graphically. An important characteristic of functions, continuity, is also discussed in greater detail than in previous math classes. This chapter also contains two major theorems: The Intermediate Value Theorem (IVT) and the Squeeze Theorem. While neither are prominent on the AP test, the IVT has applications with derivative tests found in the third chapter. Lastly, the idea of infinity is discussed in greater detail.
1.2 Finding Limits Graphically and Numerically |
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1.3 Evaluating Limits Analyically |
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1.4 Continuity and One-Sided Limits |
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1.4 Intermediate Value Theorem&Squeeze Theorem |
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1.5/3.5 Infinite Limits and Limits at Infinity |
Chapter 2 Derivatives
The second chapter concerns derivatives. The beginning lesson establishes the meaning of a derivative and how it is developed from limits. The limit definition of a derivative is almost always found as a multiple choice question on the AP test. Mechanically finding the derivative of a multitude of functions follows once the definition of a derivative is understood. Derivative rules include: power rule, product rule, higher order derivatives (2nd derivative, 3rd derivative, etc.), quotient rule, chain rule, and implicit differentiation. Together the derivative rules cover how to find derivatives for all types of mathematical operations: addition, subtraction, multiplication, division, and composition. Incorporated into this chapter are some applications of derivatives: position/velocity/acceleration, tangent lines, and related rates. Related rates can be a challenging section, so I devote several days to it in class. Related rates are also a favorite of the AP writers. Expect to see one in the multiple choice and one in the free response every couple of years. Additionally, I have included the lyrics to a song about the quotient rule that I sing to my students. Encores are requested each year without fail, but more importantly help them remember how to find the derivative of a quotient.
The second chapter concerns derivatives. The beginning lesson establishes the meaning of a derivative and how it is developed from limits. The limit definition of a derivative is almost always found as a multiple choice question on the AP test. Mechanically finding the derivative of a multitude of functions follows once the definition of a derivative is understood. Derivative rules include: power rule, product rule, higher order derivatives (2nd derivative, 3rd derivative, etc.), quotient rule, chain rule, and implicit differentiation. Together the derivative rules cover how to find derivatives for all types of mathematical operations: addition, subtraction, multiplication, division, and composition. Incorporated into this chapter are some applications of derivatives: position/velocity/acceleration, tangent lines, and related rates. Related rates can be a challenging section, so I devote several days to it in class. Related rates are also a favorite of the AP writers. Expect to see one in the multiple choice and one in the free response every couple of years. Additionally, I have included the lyrics to a song about the quotient rule that I sing to my students. Encores are requested each year without fail, but more importantly help them remember how to find the derivative of a quotient.
2.1 Definition of a Derivative |
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2.2 Basic Differentiation Rules |
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2.1 and 2.2 Tangent Lines and Horizontal Tangents |
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2.2 Rates of Change |
Homework |
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2.3 Higher Order Derivatives & Product Rule |
Homework |
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2.3 Quotient Rule & Trig Derivatives |
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2.4 Chain Rule |
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2.4 Chain Rule Part II & Absolute Value |
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Practice AP Questions |
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2.5 Implicit Differentiation |
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2.6 Related Rates Day 1 |
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2.6 Related Rates Day 2 |
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2.5-2.6 Practice AP Questions |
Chapter 3 Applications of Derivatives
This third chapter gives applications of derivatives ranging from the shape of a graph to finding the maximum or minimum values that would optimize a given scenario. The major theorem for this chapter is the Mean Value Theorem for Derivatives; make sure the initial conditions are met before you can apply the theorem. The 1st Derivative Test also plays a major role in this chapter and beyond.
This third chapter gives applications of derivatives ranging from the shape of a graph to finding the maximum or minimum values that would optimize a given scenario. The major theorem for this chapter is the Mean Value Theorem for Derivatives; make sure the initial conditions are met before you can apply the theorem. The 1st Derivative Test also plays a major role in this chapter and beyond.
3.1 Extrema |
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3.2 Rolle's Theorem and Mean Value Theorem |
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3.3 Increasing, Decreasing, & the 1st Derivative Test |
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3.4 Concavity and the 2nd Derivative Test |
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3.6 Curve Sketching |
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3.7 Optimization |
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AP Questions 3.1-3.6 |
Chapter 4 Antiderivatives and Integration
Chapter 4 introduces the next big concept: integration. Integration is the process of finding the antiderivative of a function. There are many applications of integrals that are studied in this section. Three big theorems are found in this chapter: 1st Fundamental Theorem of Calculus, 2nd Fundamental Theorem of Calculus, and the Mean Value Theorem for Integrals. The 1st Fundamental Theorem of Calculus is an extremely important theorem that allows us to find the area under a curve over an interval. Riemann Sums are also part of chapter 4 and are included to find the area under a curve before the introduction of the 1st Fundamental Theorem of Calculus.
Chapter 4 introduces the next big concept: integration. Integration is the process of finding the antiderivative of a function. There are many applications of integrals that are studied in this section. Three big theorems are found in this chapter: 1st Fundamental Theorem of Calculus, 2nd Fundamental Theorem of Calculus, and the Mean Value Theorem for Integrals. The 1st Fundamental Theorem of Calculus is an extremely important theorem that allows us to find the area under a curve over an interval. Riemann Sums are also part of chapter 4 and are included to find the area under a curve before the introduction of the 1st Fundamental Theorem of Calculus.
4.1 Antidifferentiation and Indefinite Integrals |
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4.1 Trig Antiderivatives |
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4.1 Particular Solutions for Integrals |
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4.1 Particular Solutions for Integrals Part II |
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4.3 Riemann Sums |
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4.3 Midpoint and Trapezoidal Sums |
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4.3 Properties of Definite Integrals |
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AP Questions 4.1-4.3 |
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4.4 1st Fundamental Theorem of Calculus |
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4.4 Mean Value Thm. and 2nd Fundamental Thm. of Calculus |
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4.5 U-Substitution |
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4.5 Change of Variable |
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Practice Integration Problems #1 |
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Practice Integration Problems #2 |
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AP Free Response Questions Chapter 4 |
Chapter 5 Transcendental Functions
Transcendental functions include logarithms, exponential, and inverse function. The properties of each type of function are discussed in this chapter in addition to finding the derivative and integral for each type of function. Inverse functions include polynomial and rational function as well as inverse trigonometric functions.
Transcendental functions include logarithms, exponential, and inverse function. The properties of each type of function are discussed in this chapter in addition to finding the derivative and integral for each type of function. Inverse functions include polynomial and rational function as well as inverse trigonometric functions.
5.1 The Natural Log |
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5.2 Natural Log Integration |
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5.2 Trig Integration |
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5.3 Inverse Functions |
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5.3 Derivative of Inverse Functions |
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5.4 e^x Properties and Derivative |
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5.4 e^x Integration |
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5.5 Bases Other than e |
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5.5 Exponential Functions with Function Bases |
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5.6 Inverse Trig Functions |
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5.6 Inverse Trig Differentiation |
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5.7 Inverse Trig Integration |
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AP Questions Chapter 5 |
Chapter 6 Differential Equations
Chapter 6 delves into the study of differential equations. Differential equations can be solved multiple ways including analytically, graphically, and by approximating the solution numerically. Each method is examined through the use of separation of variables, slope fields, and Euler's Method, respectively. Additional topics of study include exponential growth and decay and logistic growth for populations with limiting factors.
Chapter 6 delves into the study of differential equations. Differential equations can be solved multiple ways including analytically, graphically, and by approximating the solution numerically. Each method is examined through the use of separation of variables, slope fields, and Euler's Method, respectively. Additional topics of study include exponential growth and decay and logistic growth for populations with limiting factors.
6.3 Separation of Variables |
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6.1 Slope Fields |
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6.1 Euler's Method |
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6.2 Growth and Decay |
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6.3 Logistic Growth |
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AP Questions Chapter 6 |
Chapter 7 Area and Volume
Chapter 7 introduces more applications of the integral; namely finding the area between two curves and finding the volume of a three-dimensional figure. Volume of a three-dimensional figure is found by using cross-sections and rotation. Additionally, volume by shells is included although it is not found on the AP test. Also not found on the AP test but found in this unit is the surface area of a curved plane. Including surface area provides the opportunity to inspect interesting characteristics of Gabriel's Horn. Unique to the BC curriculum is length of a curve.
Chapter 7 introduces more applications of the integral; namely finding the area between two curves and finding the volume of a three-dimensional figure. Volume of a three-dimensional figure is found by using cross-sections and rotation. Additionally, volume by shells is included although it is not found on the AP test. Also not found on the AP test but found in this unit is the surface area of a curved plane. Including surface area provides the opportunity to inspect interesting characteristics of Gabriel's Horn. Unique to the BC curriculum is length of a curve.
7.1 Area Between 2 Curves |
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7.1 Area Between 2 Curves Part II |
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7.2 Volume by Cross Sections |
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7.2 Volume by Disks |
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7.2 Volume by Washers |
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7.3 Volume by Shells |
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7.4 Arc Length and Surface Area |
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AP Questions Chapter 7 |
Review of AP Calculus AB Material
We now pause at the conclusion of the Calculus 1 material to review the concepts that are found on the AP Calculus AB Examination. Not every topic will be hit, but the majority of them will be covered. The review will consist of 3 days separated by concepts. The first day will cover limits, derivatives, and applications of derivatives. The second day focuses on integrals and the fundamental theorems of calculus. The final day concludes with transcendental functions and area/volume. My students will take a full-length AB test at the conclusion of the review to provide a capstone for the first calculus course and a checkpoint of their progression.
We now pause at the conclusion of the Calculus 1 material to review the concepts that are found on the AP Calculus AB Examination. Not every topic will be hit, but the majority of them will be covered. The review will consist of 3 days separated by concepts. The first day will cover limits, derivatives, and applications of derivatives. The second day focuses on integrals and the fundamental theorems of calculus. The final day concludes with transcendental functions and area/volume. My students will take a full-length AB test at the conclusion of the review to provide a capstone for the first calculus course and a checkpoint of their progression.
AB Review Limits and Derivatives |
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AB Review Integrals, Area, and FTC |
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AB Review Transcendentals, Diff. Eqs, and Area/Volume |
Chapter 8 Advanced Integration Techniques
Chapter 8 takes us into the advanced integration techniques. This chapter is almost entirely exclusive to the BC classroom. The lone exception being L'Hopital's Rule for evaluating indeterminate limits. The advanced integration techniques include: integration by parts, partial fractions, integrals with infinite limits of integration, and integrals with vertical asymptotes. Partial fractions on the AP only includes non-repeating linear factors. We cover the other types in class regardless.
Chapter 8 takes us into the advanced integration techniques. This chapter is almost entirely exclusive to the BC classroom. The lone exception being L'Hopital's Rule for evaluating indeterminate limits. The advanced integration techniques include: integration by parts, partial fractions, integrals with infinite limits of integration, and integrals with vertical asymptotes. Partial fractions on the AP only includes non-repeating linear factors. We cover the other types in class regardless.
8.2 Integration by Parts |
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8.2 Integration by Parts II |
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8.3 Trig Functions with Powers |
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8.4 Trig Substitution |
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8.5 Partial Fractions (Linear Factors) |
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8.5 Partial Fractions (Quadratic Factors) |
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8.7 L'Hopital's Rule and Indeterminate Forms |
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8.8 Improper Integrals: Infinite Limits |
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8.8 Improper Integrals: Infinite Discontinuities |
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AP Questions Chapter 8 |
Chapter 9 Infinite Series
Chapter 9 opens with information about sequences. The remainder of the chapter details series: the summation of the infinite terms of a sequence. The first dealings with series involves convergence tests to determine whether a series will converge or diverge. These tests will be important for checking the boundaries of an interval of convergence for a power series. Power series, Taylor series, and Maclaurin series follow the convergence tests. These series, in conjunction with power series manipulation and Lagrange Error Bound, allow mathematicians to approximate definite integrals of non-integrable functions with an acceptable amount of error.
Chapter 9 opens with information about sequences. The remainder of the chapter details series: the summation of the infinite terms of a sequence. The first dealings with series involves convergence tests to determine whether a series will converge or diverge. These tests will be important for checking the boundaries of an interval of convergence for a power series. Power series, Taylor series, and Maclaurin series follow the convergence tests. These series, in conjunction with power series manipulation and Lagrange Error Bound, allow mathematicians to approximate definite integrals of non-integrable functions with an acceptable amount of error.
9.1 Sequences |
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9.2 Series and the nth Term Test for Divergence |
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9.2 Geometric Series |
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9.3 Integral Test |
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9.3 p-Series Test |
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9.4 Direct Comparison Test |
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9.4 Limit Comparison Test |
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9.5 Alternating Series |
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9.5 Alternating Series Remainder and Absolute/Conditional Convergence |
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9.6 Root Test |
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9.6 Ratio Test |
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9.6 All Tests |
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AP Questions 9.1-9.6 |
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9.10 Taylor and Maclaurin Series |
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9.10 Taylor and Maclaurin Series Day II |
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9.10 Taylor and Maclaurin Manipulation |
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9.7 Approximations and Lagrange Error Bound |
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9.8 Interval and Radius of Convergence |
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9.8 Interval and Radius of Convergence 2 |
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9.9 Geometric Power Series |
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AP Questions Taylor/Maclaurin Day 1 |
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AP Questions Taylor/Maclaurin Day 2 |
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AP Questions Taylor/Maclaurin Day 3 |
Chapter 10 Parametrics, Vectors, and Polar
The final chapter looks at different types of functions where calculus can be applied: parametric equations, vectors, and polar equations. Keep calculus concepts of derivatives, integrals, and area are reviewed in a new light with these functions.
The final chapter looks at different types of functions where calculus can be applied: parametric equations, vectors, and polar equations. Keep calculus concepts of derivatives, integrals, and area are reviewed in a new light with these functions.
10.2 Parametrics and Eliminating the Parameter |
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10.3 Parametric Equations and Calculus |
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10.3 Vectors Day 1 |
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10.3 Vectors Day 2 |
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10.4 Polar Coordinates and Graphs |
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10.5 Polar Area Day 1 |
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10.5 Polar Area Day 2 |
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AP Questions Para/Vector/Polar |