Jordan High School Summer Review for AP Calculus

Hi there,

Welcome to Advanced Placement Calculus! I’m Dr. Jason Elsinger, and I’m excited to work with you this year.

To help you start strong, I’ve prepared a summer review with practice problems and instructional videos covering the key algebra and precalculus topics you’ll need. These materials aren’t busy work—they’ll help you honestly assess your readiness and strengthen any areas that need work.

What to do:

• Use the diagnostic practice set below to check what you remember and what you need to review. You can also review the learning outcomes for the course. You are expected to know the unit 0 learning outcomes.

• Watch the video lessons to review prerequisite topics [see below].

• Visit the additional websites for further practice problems with solutions and recordings if you’d like more review [see below].

Important notes:

• The focus is on understanding math, not just getting the right answer. You’ll learn by showing your work and explaining your reasoning.

• Assessments will require you to complete problems without using a calculator/Desmos.

• All assessments will be free-response with no multiple-choice questions to develop strong problem-solving and communication skills.

• If, after working through the review, you feel you’re not ready for AP Calculus, please reach out so we can discuss the best placement for you.

I encourage you to start now and build a solid foundation for the year ahead. I’m looking forward to seeing you in class this fall.

Best regards,

Dr. Jason Elsinger

Mathematics Instructor

C. E. Jordan High School

Summer Review Problems (Start Here)

Below is the List of Learning Outcomes for Honors Precalculus that will be available on Canvas. I also provide an example assessment for the review topics R1 - R7. This assessment displays the format, to be completed without a calculator/Desmos. In my courses, I refer to exams as a "Celebration of Knowledge."

To help students begin the year prepared for AP Calculus, I strongly encourage everyone to complete the summer review problems before the first day of class. The packet is designed to refresh the prerequisite algebra, geometry, and function skills that are essential for success in the course.

As an incentive, students who complete the entire packet in earnest and submit it on the first day of class will receive the following benefit:

If your final course average is a 59, 69, 79, or 89, I will round your grade up to the next letter grade—no questions asked.

This incentive is intended to reward students who make a genuine effort to prepare for the course before the school year begins. The expectation is that the packet reflects your own honest work and is completed thoughtfully rather than simply copied or rushed. I emphasize that Desmos or other calculators are not allowed in completing this work. 

If, after working through the review, you still find that much of the material is unfamiliar or cannot be completed without significant assistance, I encourage you to work through the resources below, which I have tried to prioritize. 

Reading Resources

Dr. Elsinger's Handout on Functions

This packet is meant to be a Crash Course of the common types of functions studied in precalculus (it is currently a work in progress). Some of the material in this packet is not prerequisite material. To prepare for precalculus, read through the following sections carefully: 1, 2, 3, 4, 5.3, 5.4, 8.2, 8.3, 8.4.

Precalculus with Limits Text by Ron Larson 

The appendices in this text is an excellent resource for review of foundational mathematical concepts. Read through pg. 891 (A1) - pg. 947 (A5).

1. Exponent rules and radicals

1. Properties of Exponents — The Organic Chemistry Tutor
   Covers product, quotient, power, zero, and negative-exponent rules. This is the best general starting lesson for students who missed basic exponent questions.

2. Simplifying Exponents with Fractions, Variables, and Negative Exponents
   Better match for AP Calculus preparation because the examples are more involved than a standard introductory exponent lesson.

3. Fractional Exponents — The Organic Chemistry Tutor
   Covers conversion between radicals and rational exponents, including negative rational exponents.

2. Rational expressions and equations

1. Simplifying Rational Expressions — The Organic Chemistry Tutor
   This is particularly well established: search results show more than two million views and approximately 32,000 likes. It emphasizes factoring before cancellation—exactly the habit calculus students need.

2. Rational Expressions: Adding, Subtracting, Multiplying, and Dividing
   A comprehensive lesson on operations with rational expressions. This is more valuable for calculus readiness than a video limited to graphing rational functions.

3. Solving Rational Equations with Different Denominators
   Covers clearing denominators, solving the resulting equations, and checking for excluded values.

4. Simplifying Complex Rational Expressions
   Especially useful preparation for simplifying difference quotients and derivative expressions.

3. Trigonometry

Trig should probably be divided into separate remediation categories. One “trig review” video is unlikely to be enough.

Unit-circle meaning and exact values

1. The Unit Circle: Basic Introduction — The Organic Chemistry Tutor
   Explains sine and cosine as coordinates and works in both degrees and radians. This is the best conceptual starting point.

2. How to Remember the Unit Circle — NancyPi
   Concise, visual, and extremely student-friendly. It is good for memorization after students understand the underlying construction.

3. How to Remember the Unit Circle Fast — The Organic Chemistry Tutor
   Another strong memorization and reference-angle lesson.

4. Finding Exact Values of Trigonometric Functions
   Connects the unit circle to 30-60-90 and 45-45-90 triangles.

I would assign both a conceptual video and a memorization video. NancyPi’s video alone is efficient but may encourage memorizing a chart without understanding why it works.

Deeper unit-circle instruction

5. Trigonometric Functions and the Unit Circle — Professor Leonard
   A full lecture connecting right-triangle trigonometry, the unit circle, and the definitions of the trig functions.

Professor Leonard is slower and more thorough than The Organic Chemistry Tutor. He is excellent for students who genuinely did not learn the material, but probably excessive for students who only need review.

Trigonometric identities

6. Introduction to Using Trigonometric Identities — Professor Leonard
   Develops reciprocal, quotient, and Pythagorean identities conceptually.

7. Pythagorean Identities for Trigonometric Functions — Professor Leonard
   A thorough treatment of the identities most often used in calculus.

8. How to Prove Trigonometric Identities — Professor Leonard
   Search results show roughly 79,000 views for this lesson. It emphasizes algebraic reasoning instead of treating identity verification as guesswork.

For AP Calculus AB readiness, students primarily need:

sin2x+cos2x=1,1+tan2x=sec2x,1+cot2x=csc2x,

along with reciprocal and quotient identities. Sum-to-product and half-angle identities are much less important.

Solving trigonometric equations

9. Solving Trigonometric Equations Using Identities, Multiple Angles, and Factoring
   A comprehensive Organic Chemistry Tutor lesson with a broad variety of equations.

10. Solving Trigonometric Equations by Factoring
    More focused and appropriate if your diagnostic only tests equations reducible by factoring.

11. How to Solve Trigonometric Equations — Professor Leonard
    A 55-minute full lesson with approximately 87,000 visible views. Better for relearning than quick review.

Inverse trigonometric functions

12. An In-Depth Look at Inverse Trigonometric Functions — Professor Leonard
    Covers restricted domains, ranges, exact values, and interpretation. 

Complete Playlists for Additional (Optional) Review

• Professor Leonard: Complete Precalculus and Trigonometry
  Best for students who need to relearn entire topics. The series is explicitly intended to prepare students for Calculus I.

• Professor Leonard: Trigonometry Playlist
  Approximately 32 lessons covering angles, radians, the unit circle, graphs, identities, inverse trig, and equations.

• The Organic Chemistry Tutor: Precalculus/Trigonometry
  Better for shorter, problem-oriented remediation.

Other Additional (Optional) Recources

1) Algebra Rules Cheat Sheet (not allowed on assessments)

2) Paul’s Online Notes: Algebra 

An excellent recourse for notes, examples, and problems with solutions for all topics in algebra. You can also check out Paul's other courses!

3) Precalculus at Fordham University (Fall 2019)

Entire series: https://www.youtube.com/playlist?list=PLYoxM3oLTvxK-uD8QszitlemtP7X2g7LN

Course website: https://jhevonorg.wordpress.com/math-1000-r03-precalculus-fall-2019/