Review 5ap Calculus
Add your email address to get news and announcements sent to you. BASIC CALCULUS REFRESHER Ismor Fischer, Ph.D. Of Statistics UW-Madison 1. This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in Mathematics, Statistics, Engineering, Pharmacy, etc. It is not comprehensive,.
In this article we take a look at how calculus can be used in the analysis of graphs. It’s important to know what limits and derivatives can tell us about graphs, not only on the AP Calculus exams, but also in real world applications. Wouldn’t it be nice to know about the ups and downs of the stock market?
What can Calculus Tell us about Graphs?
Calculus is a method for studying rates of change of functions. Because we often represent functions by their graphs, you could say that calculus is all about the analysis of graphs.
We will review the main topics that you’ll need to know for the AP Calculus exams. Analysis of graphs (or curve sketching) includes finding:
- Domain and range
- Intercepts
- Asymptotes
- Relative extrema and intervals of increase and decrease
- Points of inflection and intervals of concavity
Domain, Range, and Intercepts
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Actually, finding the domain, range, and intercepts requires no calculus at all. These are standard topics in algebra and pre-calculus. However, it doesn’t hurt to review the concepts.
Domain
The domain of a function f(x) is the set of all input values (x-values) for the function.
Unless further information is given, we look for the natural domain, which is the largest set of x-values that makes sense for the function.
Look at the form of the function. If there are fractions, then you must exclude any x-values that make the denominator equal to zero. If there are square roots, then exclude any x-values that make the expression under the radical negative.
Let’s find the domain of √x – 5 .
Here we set x – 5 ≥ 0, which implies that x ≥ 5. As an interval, the domain would be [5, ∞).
Range
The range of a function f(x) is the set of all output values (y-values) for the function.
Finding the range is quite a bit more challenging than finding domain in general. You can rely on certain properties of functions. For example, x2 ≥ 0, and √x ≥ 0.
Here’s a good refresher for domain and range.
Intercepts
The y-intercept of a function f(x) is the point where the graph crosses the y-axis. There can be only one. Simply plug in x = 0 to find it. But if 0 is not in the domain of the function, then it has no y-intercept.
For example, the function f(x) = x2 – 4 has a y-intercept at (0, -4), because f(0) = 02 – 4 = -4.
An x-intercept of a function f(x) is any point where the graph crosses the x-axis.
In contrast to the y-intercept, there can be many x-intercepts for a given graph. To find them, set f(x) = 0 and solve.
For example, there are two x-intercepts for f(x) = x2 – 4. Setting the function equal to 0 and solving, we get:
Therefore the points (2, 0) and (-2, 0) are the x-intercepts for the function.
Asymptotes
An important part of analysis of graphs is finding the asymptotes, if any exist. Limits can be used to find the various kinds of asymptote.
- A vertical asymptote for a function is a vertical line x = k showing where the function becomes unbounded.
- A horizontal asymptote for a function is a horizontal line that the graph of the function approaches as x approaches ∞ or -∞.
- An oblique asymptote for a function is a slanted line that the function approaches as x approaches ∞ or -∞.
For more information and examples, check out these article about horizontal asymptotes, vertical asymptotes, and oblique asymptotes.
Relative Extrema and Increase/Decrease
The term relative extrema refers to both relative minimum and relative maximum points on a graph. In other words, the “peaks and valleys.” In the intervals between the relative extrema, the function may increase or decrease. The key idea is that the first derivative measures increase and decrease.
- Find the first derivative, f '(x).
- Set f '(x) = 0 and solve for x to find the critical numbers for the function. These are the locations of possible relative extrema. The critical numbers also give you the endpoints of all the intervals of increase and/or decrease.
- Choose sample points within each interval.
- Plug each sample point into the first derivative, f '(x). If the results is positive, then f is increasing in that interval. If negative, then f is decreasing in that interval.
I like to organize my work in a table or on a number line.
Use the First Derivative Test to locate the relative minima and maxima. This is not too difficult; just look for critical numbers at which the direction changes. Equivalently, look for changes in the sign of the derivative.
If signs change from positive to negative (increase to decrease), then there is a relative maximum at that critical number. If signs change from negative to positive (decrease to increase), then there is a relative minimum there.
Example 1
Find the relative extrema and intervals of increase/decrease for f(x) = x3 – 27x + 10.
The first derivative is f '(x) = 3x2 – 27. Set this equal to 0 and solve:
The critical numbers are x = -3 and 3. This means there will be three intervals to keep track of: (-∞, -3), (-3, 3), and (3, ∞). Choose samples in each interval, for example, -4, 0, and 4. There will always be one more sample point than critical number.
Plug in each sample point into the first derivative and look for positive vs. negative values.| Interval | Sample | Derivative value | Conclusion |
|---|---|---|---|
| (-∞, -3) | -4 | 3(-4)2 - 27 = 21 > 0 | Increasing |
| (3, -3) | 0 | 3(0)2 - 27 = -27 < 0 | Decreasing |
| (3, ∞) | 4 | 3(4)2 - 27 = 21 > 0 | Increasing |
According to the table, there is a relative maximum at x = -3, and a relative minimum at x = 3. Finally, plugging those two points back into the original function, f(x) = x3 – 27x + 10, gives the y-coordinates for your answer:
Relative minimum point: (3, f(3)) = (3, -44).
Relative maximum point: (-3, f(-3)) = (-3, 64).
Increasing on: (-∞, -3) U (3, ∞).
Decreasing on: (-3, 3).
Below is a sketch of the graph showing the relative extrema. Notice how the curve increases until it reaches x = -3, decreases in the middle, and then increases when x > 3.
Inflection Points and Concavity
Concavity is a measure of how curved the graph is. Informally, we say a graph is “concave up if it looks like a cup, and concave down if it looks like a frown.” A given function may have both kinds of concavity in its graph. Any point at which concavity changes (from up to down or down to up) is called a point of inflection.
While it is sometimes difficult to determine the concavity of a graph visually, it is actually very easy to determine it using calculus. In fact, if you can do intervals of increase and decrease and relative extrema, then you already know the correct technique! Just apply all of the same steps to the second derivative rather than the first.
- Find the second derivative, f '(x).
- Set f '(x) = 0 and solve for x to find the possible points of inflection for the graph. These numbers also give you the endpoints of all the intervals of concavity.
- Choose sample points within each interval.
- Plug each sample point into the second derivative, f '(x). If the results is positive, then f is concave up in that interval. If negative, then f is concave down in that interval.
Next look to see if there are any changes of concavity. What we have called possible points of inflection will be actual points of inflection only if the concavity is different on either side of the point.
Example 2
Find the point(s) of inflection and intervals of concavity for f(x) = x3 – 27x + 10.
We already have the first derivative, which is f '(x) = 3x2 – 27. But now we need the second.
f '(x) = 6x.
Set equal to 0 and solve: 6x = 0. Thus x = 0 is the only possible point of inflection.
The table below records our analysis. Note, because of the change in concavity, there is an inflection point at x
= 0.| Interval | Sample | Second Derivative | Conclusion |
|---|---|---|---|
| (-∞, 0) | -1 | 6(-1) = -6 < 0 | Concave Down |
| (0, ∞) | 1 | 6(1) = 6 > 0 | Concave Up |
Inflection point: (0, f(0)) = (0, 10).
Concave up on: (0, ∞).
Concave down on: (-∞, 0).
Here is the graph with the inflection point highlighted. To the left of this point, the curve is concave down, while to the right it’s concave up.
Summary
There are many aspects to the analysis of graphs. Be sure you are familiar with each concept and know which tool helps to find it.
- Domain and range (algebra / pre-calculus)
- Intercepts (algebra / pre-calculus)
- Asymptotes (limits)
- Relative extrema and intervals of increase and decrease (first derivative)
- Points of inflection and intervals of concavity (second derivative)
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About Shaun Ault
Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!
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AP Calculus AB can be a challenging subject for many students. Sometimes classroom instruction isn’t enough to get the information you need. After all, an instructor can only provide so much information in such a limited amount of time.
Below, we give our top picks for AP Calculus AB review books that will help fill in what you’re missing in your course.
The Best Books to Prepare for the AP AB Calculus Test
Review 5ap Calculus Definition
Our #1 ChoiceTable of Contents
- Best AP AB Calculus Exam Prep Books
- Frequently Asked Questions
Best AP AB Calculus Exam Prep Books
1. AP Calculus Premium: With 12 Practice Tests (Barron’s Test Prep)
Barron’s AP Calculus Premium has all the information and review material you need to earn a passing score on the AP Calculus AB exam.
Publisher: Barron’s Test Prep
Year: 2019
Number of pages: 672 pages
Our Final Grade: A+
If you’re looking for a complete review for AP Calculus AB, then Barron’s test prep book may be the right choice for you. It contains a total of 12 practice tests, 6 in Calculus AB and 6 in Calculus BC. You’ll also get advice on ways to use your graphing calculators efficiently.
Pros
- 6 practice tests in AB Calculus
- Thorough and detailed review of all topics
- Teaches how to use a graphical calculator
Cons
- Practice tests may be a bit harder than the actual exam
2. Cracking the AP Calculus AB Exam 2020, Premium Edition: 6 Practice Tests + Complete Content Review (College Test Preparation)
Princeton Review offers a comprehensive study guide and additional prep resources to help you ace the AP Calculus AB Exam.
Publisher: Princeton Review
Year: 2019
Number of pages: 752 pages
Our Final Grade: A
By purchasing Princeton Review’s Premium Practice for AP Calculus AB, you’ll gain access to practice tests, detailed answer explanations, targeted test strategies, and access to online extras. Subjects are organized into units that are more manageable, which gives a more targeted focus on one topic at a time.
Pros
- 5 in-book practice tests and 1 online
- Contains a guide to essential calculus formulas
- Access to comprehensive online drills
Cons
- Contains some printing mistakes
3. 5 Steps to a 5: AP Calculus AB 2020 (5 Steps to a 5 AP Calculus AB/BC)
McGraw-Hill Education’s 5 Steps to a 5 AP Calculus AB prep book is a step-by-step multi-platform study guide.
Publisher: McGraw-Hill Education
Year: 2019
Number of pages: 432 pages
Our Final Grade: A
McGraw-Hill Education’s 5 Steps to a 5 AP Calculus AB prep book is different from competitors in that it’s available in multiple platforms, including print, online, and mobile. In addition, it contains step-by-step explanations for almost 800 AP Calculus AB problems and online flashcards, games, and more.
Pros
- 4 practice exams, 2 in the book and 2 online
- Straightforward guide
- Access to online resources
Cons
- Online resources may be unintuitive in some cases
4. 320 AP Calculus AB Problems
CreateSpace’s 320 AP Calculus AB Problems review is a great resource for students who want to solely focus on practice questions and problems instead of the fundamentals of calculus.
Publisher: CreateSpace Independent Publishing Platform
Year: 2016
Number of pages: 214 pages
Our Final Grade: B
CreativeSpace’s review book provides a comprehensive collection of 320 AP Calculus AB problemswith 160 additional questions with answers. What’s useful about this prep book is that they arrange the problems by topic and difficulty, which helps you more easily find the problems you struggled with, which will in turn help you improve your skills and knowledge in calculus.
Pros
- Great for practice
- Problems arranged by topic and difficulty level
- Covers all AP Calculus AB topics for the exam
Cons
- Doesn’t provide an in-depth topic review
5. AP Calculus AB Prep Plus 2020 & 2021
Kaplan’s AP Calculus AB Prep Plus is a versatile go-to resource for a variety of study styles, whether you need targeted prep or a more comprehensive review.
Publisher: Kaplan Publishing
Year: 2020
Number of pages: 888 pages
Our Final Grade: B-
Featuring 1,000 practice problems and questions, 8 practice tests, answer explanations, and pre-chapter quizzes to aid you in gauging your knowledge. The book also contains an extensive, specific review of content that appears most frequently on the AP exam. What’s more, this resource allows you to study according to your very own style, which makes for more flexible learning.
Pros
- 8 full-length practice exams
- Online quizzes and workshops
- Flexible study plans
Cons
- Review could be a bit longer
6. AP® Calculus AB & BC Crash Course, 2nd Ed.
REA’s AP Calculus AB and BC Crash Course is the ideal resource for students who want to get in some last-minute cramming.
Publisher: Research and Education Association
Year: 2016
Number of pages: 224 pages
Our Final Grade: C+
If you’re looking for a quick refresher or you’ve waited until the last minute to study, then REA’s AP Calculus AB & BC Crash Course might be the right choice for you. The review book contains AP-style practice questions and detailed answer explanations to help you pinpoint your strengths and weaknesses.
Pros
- Quick review format for last-minute cramming
- Online practice exam with feedback
- Helpful test strategies
Cons
- Comes with only 1 practice test
- The updated (3rd edition) version is upcoming but not out yet
Frequently Asked Questions
What Differentiates AP Calculus AB From AP Calculus BC?
AP Calculus BC is an extension of AP Calculus AB. The two courses don’t differ by difficulty, but by the scope of the course material. For example, AP Calculus AB covers techniques and applications of the derivative, the definite integral, and the Fundamental Theorem of Calculus.
On the other hand, AP Calculus BC includes all the topics of AP Calculus AB. The difference is that it includes other topics, such as parametric, polar, vector functions, and series.
Should I Take AP Calculus AB or AP Calculus BC?
Because AP Calculus BC covers more topics, it can be more challenging than AP Calculus AB. But if you’re up to the challenge and you’re quite proficient in math and analysis, you might consider enrolling in AP Calculus BC. Plus, BC Calculus typically gives you more math credit in college.
However, with AB Calculus, you’re less likely to experience burnout and, you may be able to free up your schedule for an additional AP class. Yet, AP Calculus AB is still a great option and will provide you with core calculus concepts that can help prepare you for college math.
What College Course Is the Equivalent of AP Calculus AB?
AP Calculus AB is the equivalent to a college calculus course that focuses on topics in differential and integral calculus.
What Is the AP Calculus AB Exam Format?
The AP Calculus AB exam contains two sections: multiple choice and free response. The multiple-choice section is divided into parts A and B. Part A has a total of 30 questions, while Part B has 15 questions. You can’t use a calculator on Part A, but you can on Part B.
The free-response section of the AP Calculus AB exam also has a Part A and Part B. Part A has two problems, while Part B has four problems. On Part A, you can use a graphing calculator, but you can’t on Part B.
What Kinds of Questions Will Be on the AP Calculus AB Exam?
The AP Calculus AB exam will contain questions and problems on a wide range of topics. The fundamental topics include:
- Functions, Graphs, and Limits (i.e., analysis of graphs, limits, etc.)
- Differential Calculus (i.e., derivatives)
- Integral Calculus (i.e. Fundamental Theorem of Calculus)
How Much of the AP Calculus AB Exam Requires a Calculator?
About half of the AP Calculus AB exam requires a calculator. Part B of the multiple-choice section and Part A of the free-response section require a calculator.
How Vital Are Test Prep Books for AP Calculus AB?
Because learning the concepts of AP Calculus AB requires lots of practice and repetition, a prep book on the subject can be extremely beneficial to you when studying for the exam.
A test prep book will not only help you get used to potential questions and problems that will be on the AP exam, but it will also give you an idea about what areas you still need to work on. The key to learning any subject is knowing what areas need improvement so that you can make adjustments to your study plan.
How Many Prep Books Do I Need to Prepare for the AP Calculus AB Exam?
The number of prep books you use to prepare for the AP Calculus AB exam will depend on how much information you’re already getting through your AP course. So if your instructor is providing you with lots of supplement material in addition to classroom lectures and textbook reading, you may only need just one prep book to serve as a refresher.
But if your instructor is only providing you with the bare bones, leaving you to fend for yourself, you might benefit from multiple prep books. So one that provides a comprehensive overview and another that gives you a quick review of the course material should be sufficient in helping you prepare for the exam.
How Many Practice Tests Should an AP Calculus AB Prep Book Include?
The AP Calculus AB prep book of your choosing should have at least 3 practice tests. One test should be used as a pre-test to determine what you know about the subject so far.
You should take the second one midway through the prep book’s content to see how well you’re retaining the information. Finally, take the last practice test to see if you’ve mastered the course material in its entirety. The score you receive on the third practice test should guide you in your final review of the content.
When Should I Start Using a Test Prep Book to Study for the AP Calculus AB Exam?
It’s important to grab a test prep book for the AP Calculus AB exam and start studying it as soon as possible. The sooner you gain access to the study material, the sooner you can work on your weakest areas so that you’re ready when exam time rolls around.
Conclusion
AP Calculus AB can be a difficult, yet rewarding course if you put enough time and effort into it. As long as you have the drive to study and prepare for the exam, you’ll likely do well. No matter your study style or aptitude in calculus, supplemental study and review material can be essential for your success on the exam. Hopefully, with our guidance, you can find the best AP Calculus AB prep book for you.