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CLEP Calculus
By Gregory Hill Research & Education Association
Copyright © 2013 Research & Education Association, Inc.
All rights reserved.
ISBN: 978-0-7386-8380-5
CHAPTER 1
PASSING THE CLEP CALCULUS EXAM
Congratulations! You're joining the millions of people who have discovered the value and educational advantage offered by the College Board's College-Level Examination Program, or CLEP. This test prep focuses on what you need to know to succeed on the CLEP Calculus exam, and will help you earn the college credit you deserve while reducing your tuition costs.
GETTING STARTED
There are many different ways to prepare for a CLEP exam. What's best for you depends on how much time you have to study and how comfortable you are with the subject matter. To score your highest, you need a system that can be customized to fit you: your schedule, your learning style, and your current level of knowledge.
This book, and the online tools that come with it, allow you to create a personalized study plan through three simple steps: assessment of your knowledge, targeted review of exam content, and reinforcement in the areas where you need the most help.
THE REA STUDY CENTER
The best way to personalize your study plan is to get feedback on what you know and what you don't know. At the online REA Study Center, you can access two types of assessment: a diagnostic exam and full-length practice exams. Each of these tools provides true-to-format questions and delivers a detailed score report that follows the topics set by the College Board.
Diagnostic Exam
Before you begin your review with the book, take the online diagnostic exam. Use your score report to help evaluate your overall understanding of the subject, so you can focus your study on the topics where you need the most review.
Full-Length Practice Exams
Our full-length practice tests give you the most complete picture of your strengths and weaknesses. After you've finished reviewing with the book, test what you've learned by taking the first of the two online practice exams. Review your score report, then go back and study any topics you missed. Take the second practice test to ensure you have mastered the material and are ready for test day.
If you're studying and don't have Internet access, you can take the printed tests in the book. These are the same practice tests offered at the REA Study Center, but without the added benefits of timed testing conditions and diagnostic score reports. Because the actual exam is Internet-based, we recommend you take at least one practice test online to simulate test-day conditions.
AN OVERVIEW OF THE EXAM
The CLEP Calculus exam consists of approximately 44 multiple-choice questions, each with five possible answer choices, to be answered in 90 minutes.
The exam covers the material one would find in a college-level Calculus course. The exam focuses on the test-taker's understanding of calculus and his/her experience with its methods and applications. The test-taker's knowledge of other areas of mathematics, including algebra, plane and solid geometry, trigonometry and analytic geometry is assumed. Use of a graphing calculator (non-CAS) is allowed ONLY in Section 2 of the exam.
The approximate breakdown of topics is as follows:
60% Limits and Differential Calculus
40% Integral Calculus
ALL ABOUT THE CLEP PROGRAM
What is the CLEP?
CLEP is the most widely accepted credit-by-examination program in North America. The CLEP program's 33 exams span five subject areas. The exams assess the material commonly required in an introductory-level college course. Examinees can earn from three to twelve credits at more than 2,900 colleges and universities in the U.S. and Canada. For a complete list of the CLEP subject examinations offered, visit the College Board website: www.collegeboard.org/clep.
Who takes CLEP exams?
CLEP exams are typically taken by people who have acquired knowledge outside the classroom and who wish to bypass certain college courses and earn college credit. The CLEP program is designed to reward examinees for learning — no matter where or how that knowledge was acquired.
Although most CLEP examinees are adults returning to college, many graduating high school seniors, enrolled college students, military personnel, veterans, and international students take CLEP exams to earn college credit or to demonstrate their ability to perform at the college level. There are no prerequisites, such as age or educational status, for taking CLEP examinations. However, because policies on granting credits vary among colleges, you should contact the particular institution from which you wish to receive CLEP credit.
How is my CLEP score determined?
Your CLEP score is based on two calculations. First, your CLEP raw score is figured; this is just the total number of test items you answer correctly. After the test is administered, your raw score is converted to a scaled score through a process called equating. Equating adjusts for minor variations in difficulty across test forms and among test items, and ensures that your score accurately represents your performance on the exam regardless of when or where you take it, or on how well others perform on the same test form.
Your scaled score is the number your college will use to determine if you've performed well enough to earn college credit. Scaled scores for the CLEP exams are delivered on a 20–80 scale. Institutions can set their own scores for granting college credit, but a good passing estimate (based on recommendations from the American Council on Education) is generally a scaled score of 50, which usually requires getting roughly 66% of the questions correct.
For more information on scoring, contact the institution where you wish to be awarded the credit.
Who administers the exam?
CLEP exams are developed by the College Board, administered by Educational Testing Service (ETS), and involve the assistance of educators from throughout the United States. The test development process is designed andimplemented to ensure that the content and difficulty level of the test are appropriate.
When and where is the exam given?
CLEP exams are administered year-round at more than 1,200 test centers in the United States and can be arranged for candidates abroad on request. To find the test center nearest you and to register for the exam, contact the CLEP Program:
CLEP Services
P.O. Box 6600
Princeton, NJ 08541-6600
Phone: (800) 257-9558 (8 a.m. to 6 p.m. ET)
Fax: (610) 628-3726
Website: www.collegeboard.org/clep
The CLEP iBT Platform
To improve the testing experience for both institutions and test-takers, the College Board's CLEP Program has transitioned its 33 exams from the eCBT platform to an Internet-based testing (iBT) platform. All CLEP test-takers may now register for exams and manage their personal account information through the "My Account" feature on the CLEP website. This new feature simplifies the registration process and automatically downloads all pertinent information about the test session, making for a more streamlined check-in.
OPTIONS FOR MILITARY PERSONNEL AND VETERANS
CLEP exams are available free of charge to eligible military personnel and eligible civilian employees. All the CLEP exams are available at test centers on college campuses and military bases. Contact your Educational Services Officer or Navy College Education Specialist for more information. Visit the DANTES or College Board websites for details about CLEP opportunities for military personnel.
Eligible U.S. veterans can claim reimbursement for CLEP exams and administration fees pursuant to provisions of the Veterans Benefits Improvement Act of 2004. For details on eligibility and submitting a claim for reimbursement, visit the U.S. Department of Veterans Affairs website at www.gibill.va.gov.
CLEP can be used in conjunction with the Post-9/11 GI Bill, which applies to veterans returning from the Iraq and Afghanistan theaters of operation. Because the GI Bill provides tuition for up to 36 months, earning college credits with CLEP exams expedites academic progress and degree completion within the funded timeframe.
SSD ACCOMMODATIONS FOR CANDIDATES WITH DISABILITIES
Many test candidates qualify for extra time to take the CLEP exams, but you must make these arrangements in advance. For information, contact:
College Board Services for Students with Disabilities
P.O. Box 8060
Mt. Vernon, IL 62864-0060
Phone: (609) 771-7137 (Monday through Friday, 8 a.m. to 6 p.m. ET)
TTY: (609) 882-4118
Fax: (866)360-0114
E-mail: ssd@info.collegeboard.org
6-WEEK STUDY PLAN
Although our study plan is designed to be used in the six weeks before your exam, it can be condensed to three weeks by combining each two-week period into one.
Be sure to set aside enough time — at least two hours each day — to study. The more time you spend studying, the more prepared and relaxed you will feel on the day of the exam.
TEST-TAKING TIPS
Know the format of the test. Familiarize yourself with the CLEP computer screen beforehand by logging on to the College Board website. Waiting until test day to see what it looks like in the pretest tutorial risks injecting needless anxiety into your testing experience. Also, familiarizing yourself with the directions and format of the exam will save you valuable time on the day of the actual test.
Read all the questions — completely. Make sure you understand each question before looking for the right answer. Reread the question if it doesn't make sense.
Read all of the answers to a question. Just because you think you found the correct response right away, do not assume that it's the best answer. The last answer choice might be the correct answer.
Work quickly and steadily. You will have 90 minutes to answer 44 questions, so work quickly and steadily. Taking the timed practice tests online will help you learn how to budget your time.
Use the process of elimination. Stumped by a question? Don't make a random guess. Eliminate as many of the answer choices as possible. By eliminating just two answer choices, you give yourself a better chance of getting the item correct, since there will only be three choices left from which to make your guess. Remember, your score is based only on the number of questions you answer correctly.
Don't waste time! Don't spend too much time on any one question. Remember, your time is limited and pacing yourself is very important. Work on the easier questions first. Skip the difficult questions and go back to them if you have the time.
Look for clues to answers in other questions. If you skip a question you don't know the answer to, you might find a clue to the answer elsewhere on the test.
Be sure that your answer registers before you go to the next item. Look at the screen to see that your mouse-click causes the pointer to darken the proper oval. If your answer doesn't register, you won't get credit for that question.
THE DAY OF THE EXAM
On test day, you should wake up early (after a good night's rest, of course) and have breakfast. Dress comfortably, so you are not distracted by being too hot or too cold while taking the test. (Note that "hoodies" are not allowed.) Arrive at the test center early. This will allow you to collect your thoughts and relax before the test, and it will also spare you the anxiety that comes with being late.
Before you leave for the test center, make sure you have your admission form and another form of identification, which must contain a recent photograph, your name, and signature (i.e., driver's license, student identification card, or current alien registration card). You may wear a watch. However, you may not wear one that makes noise, because it may disturb the other test-takers. No cell phones, dictionaries, textbooks, notebooks, briefcases, or packages will be permitted, and drinking, smoking, and eating are prohibited.
CHAPTER 2
LIMITS AND CONTINUITY
2.1 INTRODUCTION
Limits are the building blocks of calculus. They are used to establish all major concepts, including continuity, derivatives, and integrals. There are many ways to examine and evaluate limits. Some of the methods that we will review are studying numerical patterns, direct substitution, deducing information from a graph, simplifying prior to substitution, and taking an intuitive approach to the behavior of a particular function. Limits are also closely linked with continuity. Prior to establishing a formal definition of continuity, you should recall from previous courses that any kind of a break in the domain of a function is called a discontinuity. Whether a function is continuous or discontinuous at a point in its domain determines the ease with which a limit at that point may be determined.
Imagine a square with an area of 4 square feet (Figure 2.1). If the consecutive midpoints of each side of the square are connected with segments, a new square is formed with an area of 2 square feet. If the consecutive midpoints of the new square are connected, the resulting area is 1 square foot. Repeating this pattern over and over, the sequence of areas is 4, 2, 1, 1/2, 1/4,.... Even though each new area could continue to be multiplied by 1/2 forever, it can be said that the limit of the areas is 0. Interestingly, you may recognize this sequence from a previous course as an infinite geometric sequence with a common ratio of 1/2 The sum of the infinite number of consecutive areas has a limit given by the formula S∞ = a/1-r, where a is the first term, and r is the common ratio. In this case, S∞ = 4/1-1/2 = 8 square feet. If n is the number of squares whose areas are being summed, the previous conclusion could also be written as [MATHEMATICAL EXPRESSION OMITTED].
2.2 LIMITS AS X APPROACHES A CONSTANT
A NUMERICAL APPROACH
Table 2.1 lists ordered pairs for a certain function, f(x). Examining the table might lead one to believe that the function has a certain limit as x approaches 3. Without knowing the actual function, we might speculate that the limit is 5, but lacking more information, there is no guarantee that for x values even closer to x= 3, the function won't behave erratically and "zoom" off to some extremely different value. Actually, the values given were generated by a known function, f(x) = x2 – 4. This, of course, is a "well-behaved" function — a parabola with no jumps or breaks in its domain. As a result, its limit as x approaches 3 can be determined by simply substituting the value of 3 for x and reporting the resulting function value, in this case, 5. The proper notation is [MATHEMATICAL EXPRESSION OMITTED]. In fact, for every polynomial function and all other functions without undefined points, or discontinuities, in their domain, the limit as x approaches a constant c, can be evaluated by direct substitution. The limit is equal to f(c) and is written as [MATHEMATICAL EXPRESSION OMITTED].
A GRAPHICAL APPROACH
Since many functions are not defined over all real numbers, it is important to know how to deal with limits as x approaches points where there are missing values of the domain. To do so, one must understand one-sided limits.
A general limit exists at a point x = c when the limit as x approaches c from the left equals the limit as x approaches c from the right. The actual function value at x = c does not need to exist. In symbols, the limit as x approaches c from the left is denoted [MATHEMATICAL EXPRESSION OMITTED]. The negative sign above and to the right of the c can be thought of as telling us to examine function values associated with x values less than c. Likewise, the limit as x approaches c from the right is denoted [MATHEMATICAL EXPRESSION OMITTED].
(Continues...)
Excerpted from CLEP Calculus by Gregory Hill. Copyright © 2013 Research & Education Association, Inc.. Excerpted by permission of Research & Education Association.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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