About the Course

Welcome to the journey of calculus!

What to know before taking Calculus

In some sense, the prerequisite for Calculus is to have an overall comfort with algebra, geometry, and trigonometry. After all, each new topic in math builds on previous topics, which is why mastery at each stage is so important.

However, for those of you who have taken courses in these subjects, but are looking to quickly brush up on the essentials before beginning calculus, this sheet is here to highlight a few of the more crucial skills that you should have going into the course.

Algebra

Manipulating expressions

• Know how to manipulate polynomial expressions.

  • Adding: $(x^2 + 2x + 3) + (3x^2 - 3x) = 4x^2 - x + 3$

• Multiplying and Factoring: $(x + 2)(3x - 5) \Leftrightarrow 3x^2 + x - 10$

• Know how to solve simple linear equations.

  • For example, $2x + 3 = 5x - 7$

• Know how to solve quadratic equations, such as $2x^2 + 3x - 5 = 0$

  • Completing the square

• Quadratic formula

• Know the properties of exponents.

  • $x^2 y^2 = (xy)^2$

• $(2^x)(2^y) = 2^{x + y}$

• Know how certain expressions are secretly exponentials in disguise

  • Reciprocals: For example, $\dfrac{1}{x} = x^{-1}$

• Roots: For example, $\sqrt{x} = x^{1/2}$

• Know what logarithms are, as well as their properties.

  • $y = 2^x$ says the same thing as $\log_2(y) = x$.

• $\log(x) + \log(y) = \log(xy)$.

• $\log(a^x) = x\log(a)$.

Functions

Calculus is all about functions, so it is helpful to be pretty fluent when it comes to thinking about functions, graphing functions, and using the appropriate terminology when talking about functions.

• Know what a function is

• Know how to represent a function with a graph.

• Know the graphs of various elementary functions.

  • Linear functions

• Quadratic functions

• Have at least a loose idea for what the graph of an $n^{\text{th}}$ degree polynomial might look like.

• Exponentials

• Logarithms

• Know how to manipulate functions.

  • Adding functions

• Multiplying functions

• It's also helpful to be familiar with function terminology

  • Domain

• Range

• Odd and even functions

• Periodic

• Zeros

• Intercepts

Geometry

• Know how to compute the area of simple shapes.

  • Rectangles

• Triangles

• Trapezoids

• Circles

• Know how to compute the volume of simple 3d shapes.

  • Cube

• Prism

• Sphere

• Know how to think about the cross section of a 3d shape

• Analytic geometry

  • Reasoning about points, lines, and shapes in the coordinate plane.

• Equations for parallel and perpendicular lines.

• Equation of a circle

Trigonometry

• Be comfortable with each of the basic trigonometry functions: $\sin(x)$, $\cos(x)$ and $\tan(x)$
  • Know what each one represents.
• Know the values of these functions when $x$ takes on one of the following values: $0$, $\dfrac{\pi}{6}$, $\dfrac{\pi}{4}$, $\dfrac{\pi}{3}$, $\dfrac{\pi}{2}$.
• Know what the graph of each of these functions looks like.

Know when you need to review

The thing about calculus is that it tends to pull examples from all sorts of areas in math. While the list above does cover the main things you should be comfortable with before starting a calculus class, you will inevitably come across some example or topic which references another bit of prerequisite knowledge.

In an ideal world, you would know everything about algebra, geometry and trigonometry 100% perfectly. But more realistically, there are a few things you did not learn perfectly the first time. It's totally fine if that happens, but it can sometimes be tricky to recognize when a calculus problem is hard because you don't know the fundamentals (e.g. algebra) or if it's hard because of the new material (i.e. the calculus itself).

Just make sure you are always willing to ask yourself "Do I have a strong feel for the concepts in this problem?" If the answer is no, don't be afraid to temporarily divert your attention away from the calculus material to review the necessary algebra, geometry or trigonometry. Trust me, in the long run, it is always worth taking a step back before moving forward.

Interview with Ben Eater

Although this is not directly about calculus preparation, we think this interview Sal did with one of our engineers, Ben Eater, serves as a telling example of why prerequisite knowledge is so important in the context of Calculus.

[[☃ video 1]]

Free-response questions

The purpose of this article is to learn more about the free-response section of the AP Calculus AB exam and how to write high-quality responses.

There are two parts to the free-response section of the exam:

• Part A (calculator) comprised of $2$ multi-part questions for $30$ minutes
• Part B (no calculator) comprised of $4$ multi-part questions for $60$ minutes

There may be parts of questions in Part A (calculator) that do not require a calculator, and most topics in the course can occur in either section.

These two free-response parts account for half the total exam grade. Each question is worth $9$ points, totaling $54$ possible points.

Characteristics of a good written response

Correct setup

Before writing or evaluating with the particulars of the question, always start by writing mathematically valid formulas, expressions, or models that match the situation.

This demonstrates conceptual understanding before diving into any computations.

Application of calculus principles

Written work must go beyond memorized formulas—it must show the calculus operations and concepts correctly applied.

Justification and reasoning

Explain or justify key steps whenever the question involves interpretation, such as why a derivative models a rate of change.

Proper substitution, evaluation, and precision

After setup, substitute values accurately and consistently with observable notation, such as parentheses. Be especially clear with time or rate values, using units whenever possible.

Make sure to compute the result to the required precision, and do not round until your final step to avoid a cascading rounding issue. Requirements are often $3$ decimal places or reasonable exact form.

Consistency with context

Final answers and interpretations must align with the problem’s situational meaning: no negative counts of people, for example.

Clear communication

Present work with correct notation and structure. Responses must be readable and logically organized.

Strategies

Free-response questions do not have to be completed in the order they appear on the exam.

• Start by reading the whole problem. It's important to read all parts of the question to understand the overall task and how parts may connect.

• Select the parts of questions that are easier, and do them first. Only work on the more difficult parts of questions after answering all the easier parts of the questions.

• The parts in each question are not generally dependent on each other. For example, an answer from one part of a question is usually not needed to answer another part.

• When parts are dependent, if an incorrect answer from one part is used correctly in another part of the question, then partial points are possible.

• When using a theorem that has conditions, we must write how those conditions have been met before applying the theorem.

• Label all graphs and functions clearly.

Scoring

How many points?

Each free-response question on the AP Calculus AB exam is worth $9$ points. In general there are $3$ to $4$ parts to each question. Most parts to a question are worth $2$ to $3$ points, but occasionally a part of a question is worth only $1$ point.

Rubrics

Graders score free-response questions using a scoring rubric for each question, which describes how each point can be earned, including partial credit. There are examples of scoring rubrics in the articles at the end of each unit.

We'll use these rubrics to help illustrate the key elements of a high-quality response. Here's is an example of a scoring rubric.

This part of the question is worth $2$ of the $9$ points.

• $1$ point for setting $f'(x)$ equal to $g'(x)$.

• $1$ point for the correct answer.

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Notice that just writing the correct answer will not achieve full credit, nor will just setting up the problem. We must clearly complete both steps to earn both points.