We can use power series to approximate functions, which are called Taylor Series.
Once we know the formula for a power series, we can do several things to find other series, as we shall demonstrate from the following examples.
Using the Maclaurin Series above, find the sixth-order Maclaurin polynomial for the following functions.
Congratulations, you’re done with this unit, and as a result, you’ve also reached the end of AP Calculus BC! Now, you have all the tools you need to ace that AP test this May!
Now, the question is, what’s next after AP Calculus BC? If you are mainly interested in the humanities or want to go into research sciences, then AP Statistics is a great option. A lot of statistics is based off of calculus, even if the relationship isn’t explicitly stated.
Many science courses are also deeply rooted in calculus as well. AP Chemistry utilizes calculus indirectly with the concepts of reaction rates and rate laws. However, AP Physics C uses calculus freely and is a good challenge if you are interested in physics or want a little bit of a challenge!
If you want to go into higher STEM courses, then you may consider higher mathematical courses. However, there are some concepts in single-variable calculus that aren’t covered in AP Calculus that are useful for higher math courses such as hyperbolic functions, the epsilon-delta definition of a limit, and trigonometric substitutions. After this, there are multiple courses which follow, which include multivariable calculus, differential equations, real and complex analysis, linear and abstract algebra, , and geometry and topology.
Multivariable calculus applies the concepts of single-variable into multiple dimensions and also expands the concept of vector calculus, which eventually sums up and generalizes most of the important theorems in calculus, especially the Fundamental Theorem of Calculus.
Differential equations is all about finding ways to solve many different types of differential equations, especially those which are highly applicable to other disciplines. It covers both ordinary differential equations, differential equations with respect to one variable, and partial differential equations, differential equations with respect to multiple variables.
Analysis is the study of change, especially relating to functions and other curves, and this serves as a theoretical extension to calculus, with real analysis covering real-valued functions and complex analysis covering complex numbers and complex-valued functions.
Algebra is the study of structure (not to be confused with high school algebra). Linear algebra is usually the first algebra class one will take and will involve the study of matrices and vectors, and introduces the topic of a vector space (which includes more than just vectors). Abstract algebra then includes more abstract structures, such as groups, rings, and fields. Number theory, which deals with the properties of numbers, is also a subset of algebra as well.
Geometry and topology are the study of shape and are two related fields. Geometry is the study of structures and objects in different dimensions, and topology is the study of structures and their properties under different transformations.
No matter what path you choose, remember that calculus, and more generally mathematics is all around us, and keep on studying and remember to make your exams all Fiveable!
Calculus II. Paul's Online Math Notes, http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx
Stewart, James. Calculus — Early Transcendentals, 8th Edition. Cengage Publishing, 2016.
“AP Calculus AB and BC Course and Exam Description, Effective Fall 2019,” The College Board, 2019.
✍️ 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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