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    Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)
    Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)
    Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)
    Ebook1,040 pages3 hours

    Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)

    By Patrick Jones

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    About this ebook

    Practice your way to a higher grade in Calculus!

    Calculus is a hands-on skill. You’ve gotta use it or lose it. And the best way to get the practice you need to develop your mathematical talents is Calculus: 1001 Practice Problems For Dummies.

    The perfect companion to Calculus For Dummies—and your class— this book offers readers challenging practice problems with step-by-step and detailed answer explanations and narrative walkthroughs. You’ll get free access to all 1,001 practice problems online so you can create your own study sets for extra-focused learning.

    Readers will also find:

    • A useful course supplement and resource for students in high school and college taking Calculus I
    • Free, one-year access to all practice problems online, for on-the-go study and practice
    • An excellent preparatory resource for faster-paced college classes

    Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice) is an essential resource for high school and college students looking for more practice and extra help with this challenging math subject.

    Calculus: 1001 Practice Problems For Dummies (9781119883654) was previously published as 1,001 Calculus Practice Problems For Dummies (9781118496718). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.

    LanguageEnglish
    PublisherWiley
    Release dateMay 5, 2022
    ISBN9781119883678
    Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)
    Author

    Patrick Jones

    Patrick Jones lives in Minneapolis and is the author of many novels including the Support and Defend series. A former librarian, Jones received lifetime achievement awards from the American Library Association and the Catholic Library Association.

    Read more from Patrick Jones

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      Book preview

      Calculus - Patrick Jones

      Introduction

      This book is intended for a variety of calculus students. Perhaps you want a supplement to your current calculus class or you’re looking to brush up on a course you took long ago. Or maybe you’re teaching yourself and need a comprehensive book of extra practice problems.

      The 1,001 questions in this book cover calculus concepts that a high school student would encounter in a calculus course in preparation for the AP exam. It also covers most of the concepts that a calculus student could expect to see in the first two semesters of a three-semester calculus course. The types of questions are questions that I regularly assigned when teaching both as homework questions or are questions that a student could’ve expected to see on a quiz or test.

      Jump around the book as you like. You can find a robust algebra and trigonometry review at the beginning of the book to make sure that you’re prepared for calculus. The number-one reason students have difficulty in calculus is not calculus itself but having a weak background in algebra and trigonometry. If you’re rusty on the fundamentals, spend time on those first two chapters before jumping into the rest of the text!

      As with many things worth doing in life, there’s no shortcut to becoming proficient in mathematics. However, by practicing the problems in this book, you’ll be on your way to becoming a much stronger calculus student.

      What You’ll Find

      The 1,001 calculus practice problems in the book are divided into 15 chapters, with each chapter providing practice of the mechanical side of calculus or of applications of calculus. Some of the questions have a diagram or graph that you need in order to answer the question.

      The end of the book provides thorough and detailed solutions to all the problems. If you get an answer wrong, try again before reading the solution! Knowing what not to do is often a great starting point in discovering the correct approach, so don’t worry if you don’t immediately solve each question; some problems can be quite challenging.

      Beyond the Book

      In addition to what you’re reading right now, this book comes with a free, access-anywhere Cheat Sheet that includes tips and other goodies you may want to have at your fingertips. To get this Cheat Sheet, simply go to www.dummies.com and type Calculus 1001 Dummies Cheat Sheet into the Search box.

      The online practice that comes free with this book offers you the same 1,001 questions and answers that are available here, presented in a multiple-choice format. The beauty of the online problems is that you can customize your online practice to focus on the topic areas that give you trouble. If you’re short on time and want to maximize your study, you can specify the quantity of problems you want to practice, pick your topics, and go. You can practice a few hundred problems in one sitting or just a couple dozen, and whether you can focus on a few types of problems or a mix of several types. Regardless of the combination you create, the online program keeps track of the questions you get right and wrong so you can monitor your progress and spend time studying exactly what you need.

      To gain access to the online practice, you simply have to register. Just follow these steps:

      Register your book or ebook at Dummies.com to get your PIN. Go towww.dummies.com/go/getaccess.

      Select your product from the dropdown list on that page.

      Follow the prompts to validate your product, and then check your email for a confirmation message that includes your PIN and instructions for logging in.

      If you don’t receive this email within two hours, please check your spam folder before contacting us through our Technical Support website at http://support.wiley.com or by phone at 877-762-2974.

      Now you’re ready to go! You can come back to the practice material as often as you want — simply log in with the username and password you created during your initial login. No need to enter the access code a second time.

      Your registration is good for one year from the day you activate your PIN.

      Where to Go for Additional Help

      Calculus is hard, so don’t become overwhelmed if a particular topic isn’t immediately easy to you. This book has many practice problems of varying difficulty, so you can focus on those problems that are most appropriate for you.

      In addition to getting help from your friends, teachers, or coworkers, you can find a variety of great materials online. If you have internet access, a simple search often turns up a treasure trove of information. You can also head to www.dummies.com to see the many articles and books that can help you in your studies.

      Calculus: 1001 Practice Problems For Dummies gives you just that — 1,001 practice questions and answers in order for you to practice your calculus skills. If you need more in-depth study and direction for your calculus courses, you may want to try out the following For Dummies products (or their companion workbooks):

      Calculus For Dummies: This book provides instruction parallel to the 1,001 calculus practice problems found here.

      Calculus II For Dummies: This book provides content similar to what you may encounter in a second-semester college calculus course.

      Pre-Calculus For Dummies: Use this book to brush up on the foundational skills and concepts you need for calculus — solving polynomials, graphing functions, using trig identities, and the like.

      Trigonometry For Dummies: Try this book if you need a refresher on trigonometry.

      Part 1

      The Questions

      IN THIS PART …

      The only way to become proficient in math is through a lot of practice. Fortunately, you have now 1,001 practice opportunities right in front of you. These questions cover a variety of calculus-related concepts and range in difficulty from easy to hard. Master these problems, and you’ll be well on your way to a very solid calculus foundation.

      Here are the types of problems that you can expect to see:

      Algebra review (Chapter 1)

      Trigonometry review (Chapter 2)

      Limits and continuity (Chapter 3)

      Derivative fundamentals (Chapters 4 through 7)

      Applications of derivatives (Chapter 8)

      Antiderivative basics (Chapters 9 and 10)

      Applications of antiderivatives (Chapter 11)

      Antiderivatives of other common functions and L’Hôpital’s rule (Chapter 12)

      More integration techniques (Chapters 13 and 14)

      Improper integrals, the trapezoid rule, and Simpson’s rule (Chapter 15)

      Chapter 1

      Algebra Review

      Performing well in calculus is impossible without a solid algebra foundation. Many calculus problems that you encounter involve a calculus concept but then require many, many steps of algebraic simplification. Having a strong algebra background will allow you to focus on the calculus concepts and not get lost in the mechanical manipulation that's required to solve the problem.

      The Problems You’ll Work On

      In this chapter, you see a variety of algebra problems:

      Simplifying exponents and radicals

      Finding the inverse of a function

      Understanding and transforming graphs of common functions

      Finding the domain and range of a function using a graph

      Combining and simplifying polynomial expressions

      What to Watch Out For

      Don't let common mistakes trip you up. Some of the following suggestions may be helpful:

      Be careful when using properties of exponents. For example, when multiplying like bases, you add the exponents, and when dividing like bases, you subtract the exponents.

      Factor thoroughly in order to simplify expressions.

      Check your solutions for equations and inequalities if you're unsure of your answer. Some solutions may be extraneous!

      It's easy to forget some algebra techniques, so don't worry if you don't remember everything! Review, review, review.

      Simplifying Fractions

      1–13 Simplify the given fractions by adding, subtracting, multiplying, and/or dividing.

      1. math

      2. math

      3. math

      4. math

      5. math

      6. math

      7. math

      8. math

      9. math

      10. math

      11. math

      12. math

      13. math

      Simplifying Radicals

      14–18 Simplify the given radicals. Assume all variables are positive.

      14. math

      15. math

      16. math

      17. math

      18. math

      Writing Exponents Using Radical Notation

      19–20 Convert between exponential and radical notation.

      19. Convert math to radical notation. ( Note: The final answer can have more than one radical sign.)

      20. Convert math to exponential notation.

      The Horizontal Line Test

      21–23 Use the horizontal line test to identify one-to-one functions.

      21. Use the horizontal line test to determine which of the following functions is a one-to-one function and therefore has an inverse.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      22. Use the horizontal line test to determine which of the following functions is a one-to-one function and therefore has an inverse.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      23. Use the horizontal line test to determine which of the following functions is a one-to-one function and therefore has an inverse.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      Find Inverses Algebraically

      24–29 Find the inverse of the one-to-one function algebraically.

      24. math

      25. math

      26. math

      27. math

      28. math

      29. math

      The Domain and Range of a Function and Its Inverse

      30–32 Solve the given question related to a function and its inverse.

      30. The set of points math is on the graph of f ( x ), which is a one-to-one function. Which points belong to the graph of math ?

      31. f ( x ) is a one-to-one function with domain math and range math . What are the domain and range of math ?

      32. Suppose that f ( x ) is a one-to-one function. What is an expression for the inverse of math ?

      Linear Equations

      33–37 Solve the given linear equation.

      33. math

      34. math

      35. math

      36. math

      37. math

      Quadratic Equations

      38–43 Solve the quadratic equation.

      38. Solve math .

      39. Solve math by completing the square.

      40. Solve math by completing the square.

      41. Solve math .

      42. Solve math .

      43. Solve math .

      Solving Polynomial Equations by Factoring

      44–47 Solve the polynomial equation by factoring.

      44. math

      45. math

      46. math

      47. math

      Absolute Value Equations

      48–51 Solve the given absolute value equation.

      48. math

      49. math

      50. math

      51. math

      Solving Rational Equations

      52–55 Solve the given rational equation.

      52. math

      53. math

      54. math

      55. math

      Polynomial and Rational Inequalities

      56–59 Solve the given polynomial or rational inequality.

      56. x ² - 4 x - 32 < 0

      57. math

      58. math

      59. math

      Absolute Value Inequalities

      60–62 Solve the absolute value inequality.

      60. math

      61. math

      62. math

      Graphing Common Functions

      63–77 Solve the given question related to graphing common functions.

      63. What is the slope of the line that goes through the points math and math ?

      64. What is the equation of the line that has a slope of 4 and goes through the point math ?

      65. What is the equation of the line that goes through the points math and math ?

      66. Find the equation of the line that goes through the point math and is parallel to the line math .

      67. Find the equation of the line that goes through the point math and is perpendicular to the line that goes through the points math and math .

      68. What is the equation of the graph of math after you stretch it vertically by a factor of 2, shift the graph 3 units to the right, and then shift it 4 units upward?

      69. Find the vertex form of the parabola that passes through the point math and has a vertex at math .

      70. Find the vertex form of the parabola that passes through the point math and has a vertex at math .

      71. A parabola has the vertex form math . What is the vertex form of this parabola if it’s shifted 6 units to the right and 2 units down?

      72. What is the equation of the graph of math after you compress the graph horizontally by a factor of 2, reflect it across the y -axis, and shift it down 5 units?

      73. What is the equation of the graph of math after you stretch the graph horizontally by a factor of 5, reflect it across the x -axis, and shift it up 3 units?

      74. Find the equation of the third-degree polynomial that goes through the points math , math , math , and math .

      75. Find the equation of the fourth-degree polynomial that goes through the point math and has the roots –1, 2, and 3, where 3 is a repeated root.

      76. A parabola crosses the x -axis at the points math and math . If the point math is on the parabola, what is the equation of the parabola?

      77. A parabola crosses the x -axis at the points math and math , and the point math is on the parabola. What is the equation of the parabola?

      Domain and Range from a Graph

      78–80 Find the domain and range of the function with the given graph.

      78.

      Graph shows y = f(x) to find the the domain and range of the function.

      79.

      Graph shows y = f(x) to find the the domain and range of the function.

      80.

      Graph shows y = f(x) to find the the domain and range of the function.

      End Behavior of Polynomials

      81–82 Find the end behavior of the given polynomial. That is, find math and math .

      81. math

      82. math

      Adding Polynomials

      83–87 Add the given polynomials.

      83. math

      84. math

      85. math

      86. math

      87. math

      Subtracting Polynomials

      88–92 Subtract the given polynomials.

      88. math

      89. math

      90. math

      91. math

      92.

      math

      Multiplying Polynomials

      93–97 Multiply the given polynomials.

      93. math

      94. math

      95. math

      96. math

      97. math

      Long Division of Polynomials

      98–102 Use polynomial long division to divide.

      98. math

      99. math

      100. math

      101. math

      102. math

      Chapter 2

      Trigonometry Review

      In addition to having a strong algebra background, you need a strong trigonometry skill set for calculus. You want to know the graphs of the trigonometric functions and to be able to evaluate trigonometric functions quickly. Many calculus problems require one or more trigonometric identities, so make sure you have more than a few of them memorized or at least can derive them quickly.

      The Problems You’ll Work On

      In this chapter, you solve a variety of fundamental trigonometric problems that cover topics such as the following:

      Understanding the trigonometric functions in relation to right triangles

      Finding degree and radian measure

      Finding angles on the unit circle

      Proving identities

      Finding the amplitude, period, and phase shift of a periodic function

      Working with inverse trigonometric functions

      Solving trigonometric equations with and without using inverses

      What to Watch Out For

      Remember the following when working on the trigonometry review questions:

      Being able to evaluate the trigonometric functions at common angles is very important since they appear often in problems. Having them memorized will be extremely useful!

      Watch out when solving equations using inverse trigonometric functions. Calculators give only a single solution to the equation, but the equation may have many more (sometimes infinitely many solutions), depending on the given interval. Thinking about solutions on the unit circle is often a good way to visualize the other solutions.

      Although you may be most familiar with using degrees to measure angles, radians are used almost exclusively in calculus, so learn to love radian measure.

      Memorizing many trigonometric identities is a good idea because they appear often in calculus problems.

      Basic Trigonometry

      103–104 Evaluate math , math , and math for the given right triangle. Remember to rationalize denominators that contain radicals.

      103.

      Geometric representation of a right triangle.

      104.

      Geometric representation of a right triangle.

      105–108 Evaluate the trig function. Remember to rationalize denominators that contain radicals.

      105. Given math , where math , find math .

      106. Given math , where math , find math .

      107. Given math , where math and math , find math .

      108. Given math , where math , find math .

      Converting Degree Measure to Radian Measure

      109–112 Convert the given degree measure to radian measure.

      109. 135°

      110. math

      111. 36°

      112. math

      Converting Radian Measure to Degree Measure

      113–116 Convert the given radian measure to degree measure.

      113. math rad

      114. math rad

      115. math rad

      116. math rad

      Finding Angles in the Coordinate Plane

      117–119 Choose the angle that most closely resembles the angle in the given diagram.

      117. Using the diagram, find the angle measure that most closely resembles the angle math .

      Geometric representation of the angle theta.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      118. Using the diagram, find the angle measure that most closely resembles the angle math .

      Geometric representation of the angle theta.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      119. Using the diagram, find the angle measure that most closely resembles the angle math .

      Geometric representation of the angle theta.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      Finding Common Trigonometric Values

      120–124 Find math , math , and math for the given angle measure. Remember to rationalize denominators that contain radicals.

      120. math

      121. math

      122. math

      123. math

      124. math

      Simplifying Trigonometric Expressions

      125–132 Determine which expression is equivalent to the given one.

      125. math

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      126. math

      (A) 1

      (B) math

      (C) math

      (D) math

      (E) math

      127. math

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      128. math

      (A) cos x

      (B) sin x

      (C) csc x

      (D) sec x

      (E) tan x

      129. math

      (A) math

      (B) math

      (C) csc x

      (D) math

      (E) math

      130. math

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      131. math

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      132. math

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      Solving Trigonometric Equations

      133–144 Solve the given trigonometric equations. Find all solutions in the interval math .

      133. math

      134. math

      135. math

      136. math

      137. math

      138. math

      139. math

      140. math

      141. math

      142. math

      143. math

      144. math

      Amplitude, Period, Phase Shift, and Midline

      145–148 Determine the amplitude, the period, the phase shift, and the midline of the function.

      145. math

      146. math

      147. math

      148. math

      Equations of Periodic Functions

      149–154 Choose the equation that describes the given periodic function.

      149.

      Geometric representation of Equations of Periodic Functions.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      150.

      Geometric representation of Equations of Periodic Functions.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      151.

      Geometric representation of Equations of Periodic Functions.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      152.

      Geometric representation of Equations of Periodic Functions.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      153.

      Geometric representation of Equations of Periodic Functions.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      154.

      Geometric representation of Equations of Periodic Functions.

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      Inverse Trigonometric Function Basics

      155–160 Evaluate the inverse trigonometric function for the given value.

      155. Find the value of math .

      156. Find the value of arctan math .

      157. Find the value of math .

      158. Find the value of math .

      159. Find the value of math .

      160. Find the value of math .

      Solving Trigonometric Equations Using Inverses

      161–166 Solve the given trigonometric equation using inverses. Find all solutions in the interval math .

      161. math

      162. math

      163. math

      164. math

      165. math

      166. math

      Chapter 3

      Limits and Rates of Change

      Limits are the foundation of calculus. Being able to work with limits and to understand them conceptually is crucial, because key ideas and definitions in calculus make use of limits. This chapter examines a variety of limit problems and makes the intuitive idea of continuity formal by using limits. Many later problems also involve the use of limits, so although limits may go away for a while during your calculus studies, they’ll return!

      The Problems You’ll Work On

      In this chapter, you encounter a variety of problems involving limits:

      Using graphs to find limits

      Finding left-hand and right-hand limits

      Determining infinite limits and limits at infinity

      Practicing many algebraic techniques to evaluate limits of the form 0/0

      Determining where a function is continuous

      What to Watch Out For

      You can use a variety of techniques to evaluate limits, and you want to be familiar with them all! Remember the following tips:

      When substituting in the limiting value, a value of zero in the denominator of a fraction doesn't automatically mean that the limit does not exist! For example, if the function has a removable discontinuity, the limit still exists!

      Be careful with signs, as you may have to include a negative when evaluating limits at infinity involving radicals (especially when the variable approaches negative infinity). It’s easy to make a limit positive when it should have been negative!

      Know and understand the definition of continuity, which says the following: A function f(x) is continuous at a if math .

      Finding Limits from Graphs

      167–172 Use the graph to find the indicated limit.

      167.

      A graph is shown to find the limits.math

      168.

      A graph is shown to find the limits.math

      169.

      A graph is shown to find the limits.math

      170.

      A graph is shown to find the limits.math

      171.

      A graph is shown to find the limits.math

      172.

      A graph is shown to evaluate the limits.math

      Evaluating Limits

      173–192 Evaluate the given limit.

      173. math

      174. math

      175. math

      176. math

      177. math

      178. math

      179. math

      180. math

      181. math

      182. math

      183. math

      184. math

      185. math

      186. math

      187. math

      188. math

      189. math

      190. math

      191. math

      192. math

      Applying the Squeeze Theorem

      193–198 Use the squeeze theorem to evaluate the given limit.

      193. If math for all x in math , find math .

      194. If math for math , find math .

      195. If math for math , evaluate math .

      196. Find the limit: math .

      197. Find the limit: math .

      198. Find the limit: math .

      Evaluating Trigonometric Limits

      199–206 Evaluate the given trigonometric limit. Recall that math and that math .

      199. math

      200. math

      201. math

      202. math

      203. math

      204. math

      205. math

      206. math

      Infinite Limits

      207–211 Find the indicated limit using the given graph.

      207.

      A graph is shown to find the infinite limits.math

      208.

      A graph is shown to find the infinite limits.math

      209.

      A graph is shown to find the infinite limits.math

      210.

      A graph is shown to find the infinite limits.math

      211.

      A graph is shown to find the infinite limits.math

      212−231 Find the indicated limit.

      212. math

      213. math

      214. math

      215. math

      216. math

      217. math

      218. math

      219. math

      220. math

      221. math

      222. math

      223. math

      224. math

      225. math

      226. math

      227. math

      228. math

      229. math

      230. math

      231. math

      Limits from Graphs

      232–235 Find the indicated limit using the given graph.

      232.

      A graph is shown to find the limits.math

      233.

      A graph is shown to find the limits.math

      234.

      A graph is shown to find the limits.math

      235.

      A graph is shown to find the limits.math

      Limits at Infinity

      236–247 Find the indicated limit.

      236. math

      237. math

      238. math

      239. math

      240. math

      241. math

      242. math

      243. math

      244. math

      245. math

      246. math

      247. math

      Horizontal Asymptotes

      248–251 Find any horizontal asymptotes of the given function.

      248. math

      249. math

      250. math

      251. math

      Classifying Discontinuities

      252–255 Use the graph to find all discontinuities and classify each one as a jump discontinuity, a removable discontinuity, or an infinite discontinuity.

      252.

      A graph is shown to Classifying Discontinuities.

      253.

      A graph is shown to Classifying Discontinuities.

      254.

      A graph is shown to Classifying Discontinuities.

      255.

      A graph is shown to Classifying Discontinuities.

      Continuity and Discontinuities

      256–261 Determine whether the function is continuous at the given value of a. If it’s continuous, state the value at f(a). If it isn’t continuous, classify the discontinuity as a jump, removable, or infinite discontinuity.

      256. math

      where math

      257. math

      where math

      258. math

      where math

      259. math

      where math

      260. math

      where math

      261. math

      where math

      262–265 Determine whether the function is continuous at the given values of a. If it isn’t continuous, classify each discontinuity as a jump, removable, or infinite discontinuity.

      262. math

      where math and math

      263. math

      where math and math

      264. math

      where math and math

      265. math

      where math and math

      Making a Function Continuous

      266–267 Determine the value of c that makes the given function continuous everywhere.

      266. math

      267. math

      The Intermediate Value Theorem

      268–271 Determine which of the given intervals is guaranteed to contain a root of the function by the intermediate value theorem.

      268. By checking only the endpoints of each interval, determine which interval contains a root of the function math by the intermediate value theorem:

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      269. By checking only the endpoints of each interval, determine which interval contains a root of the function math by the intermediate value theorem:

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      270. By checking only the endpoints of each interval, determine which interval contains a solution to the equation math according to the intermediate value theorem:

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      271. By checking only the endpoints of each interval, determine which interval contains a solution to the equation math according to the intermediate value theorem:

      (A) math

      (B) math

      (C) math

      (D) math

      (E) math

      Chapter 4

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