10.15 Representing Functions as Power Series

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.

Example 2:

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!