Inverse Functions
• Projects & Applications – Richardson’s Arms Race 83
2.4 The Technical Defi nition of a Limit 116
• Projects & Applications – The Apothem and Limits at Infi nity; Implementing the Bisection Algorithm 144
Derivatives of Power
3.3 Derivatives of Trigonometric and Exponential Functions 169 3.4 The Derivative as a Function 180
3.5 The Product Rule & Quotient Rule 192
• Projects & Applications – Parabolic Refl ectors;
The Cumulative Area Function 233 iii
Chapter 4 Applications of the Derivative 236
4.5 Mean Value Theorem & L’Hopital’s Rule 281
Newton’s Method; Optimization and Apothem;
Energy and Force; Quantum Mechanics 318
5.2 Area and the Defi nite Integral 343
5.5 More About the Fundamental Theorem 384
• Projects & Applications – Numerical Integration with
Spreadsheets; An Introduction to Stirling’s Approximation; Pollution and Pulmonary Distress 398
• Projects & Applications – The Gamma Function; The Seasons 482
7.2 Solids of Revolution & the Method of Slicing 501
7.5 Center of Mass and Radius of Gyration 537
• Projects & Applications – Ideal Gas, Energy, and the n-Dimensional Ball; Asteroid Mining; Stopping on a Dime 581
8.3 Integral Test and Estimating Sums 617
8.5 Absolute Convergence, Ratio & Root Tests 639
8.7 Elementary Operations with Power Series 660
• Projects & Applications – Revisiting Stirling’s Approximation; Weighing Exploitation Against Retaliation 702
• Projects & Applications – Planetary Orbits 768 v vi
10.3 Slope Fields and Euler’s Method 787
• Projects & Applications – Crosswind Landing; Pursuit Curve 835
Appendix B Review of Algebraic Techniques 853
Appendix E The World We Live In 884
As you study calculus, you may have questions about its principles, real-world applications, and professional uses To address these inquiries, we have anticipated and answered many common questions to help you understand formulas and techniques Our aim is to enhance your skills and intuition while illustrating how mathematics describes and transforms our world To achieve this, we need your engagement, particularly through reading Therefore, we want to provide you with useful advice on how to effectively read technical content.
The most important thing we can tell you is that learning technical content takes time, and often requires substantial mental effort, so fi rst and foremost, be patient.
A scan-then-dig strategy is effective for reading technical writing, as the initial scan helps familiarize you with the organization of topics and terms, creating a mental roadmap The subsequent pass allows you to delve deeper into the main ideas and develop essential skills related to the content.
Learning technical material, much like the physical act of digging, demands active engagement and considerable effort, even from seasoned professionals Recently, while exploring a book on the applications of physics in medical sciences, one reader experienced a moment of realization, leading to intense note-taking and a series of enlightening discoveries that ultimately clarified the complex relationship being discussed.
To truly master a subject, active engagement is essential Always keep a pencil and paper nearby while reading to check calculations, which will enhance your understanding of how different concepts interact Work through the "Try It Yourself" examples provided to ensure your reading is interactive If your answers differ from ours and you're unsure why, full solutions are available online for clarification.
To excel in quizzes and real-world applications of calculus, it's essential to think creatively about combining learned concepts and techniques This creative mathematical thinking cannot be developed by merely observing others; it requires personal practice While effective reading strategies have been previously discussed, this advice specifically focuses on transforming homework exercises into powerful learning tools.
Many students find a solutions manual beneficial for their studies, and we have collaborated with Dr Justin Young to create an effective one for this text However, it's essential to use the manual as a tool for learning rather than a dependency After completing an exercise with the manual's guidance, challenge yourself by attempting a similar problem independently Relying too heavily on the manual may lead to a false sense of understanding, resulting in a disconnect between homework success and exam performance.
To enhance your understanding of new concepts, it’s crucial to discuss what you’re learning Articulating your ideas to others not only helps solidify your knowledge but also allows for valuable feedback Engaging in conversations with your teacher can further refine your understanding, as they can respond to your explanations and provide guidance on the techniques you describe.
The project was inspired by a former student's critique of calculus texts, which he described as good reference books that fail to teach effectively After reviewing popular texts, we recognized that while they are accurate, their technical writing style is challenging for students unfamiliar with the material Furthermore, these texts often lack practical motivation, making it difficult for modern learners to engage with the content Consequently, both students and teachers tend to skim rather than thoroughly read these books, which, although useful as reference materials later on, serve primarily as cumbersome exercise sets during the initial learning phase.
A calculus textbook should ignite curiosity and captivate students by presenting material in engaging and relevant ways It must address a range of questions, from practical "How do we do this?" to deeper inquiries like "Why do we do this?" and "How can I make sense of this?" Additionally, the book should enhance the classroom experience by fostering discussion, encouraging discovery, and developing intuition, ultimately showcasing the power, beauty, and versatility of calculus This book aims to fulfill that vision.
This article targets students pursuing physical and medical sciences, mathematics, engineering, or computer science, while also offering insights and exercises for those interested in business or social sciences Recognizing that many calculus students are focused on practical applications, we incorporate real-world examples and discussions of relevant applications throughout the learning process.
The concept of limits is illustrated through a thought experiment involving a planet's position relative to the sun, while derivatives are introduced using clinical data on artery radius The motivation for derivatives of trigonometric functions comes from the vibration of a guitar string, and Riemann sums are employed to determine the net displacement of an accelerating vehicle using radar gun data Additionally, the idea of sequences is inspired by medication concentrations in a patient's bloodstream, among other examples.
We understand the importance of presenting applications at an elementary level, allowing newcomers to grasp fundamental concepts without distraction from the mathematical subject Additionally, we have discovered that students engage more deeply when topics are contextualized, as this approach offers them a broader perspective of the rich connections that extend beyond their current learning.
Engaging students in calculus involves linking mathematical concepts to their real-life experiences and observations This approach helps them grasp the material by relating it to familiar aspects of their lives, transforming calculus from a collection of abstract facts and formulas into a meaningful subject Here are some effective examples to illustrate this connection.
Newton’s method: We begin our discussion of Newton’s method by asking the question,
A model rocket descending at 6 feet per second from a height of 30 feet will take 5 seconds to land Students typically arrive at this answer based on their understanding of motion, illustrating their assumption of a constant rate of change This concept aligns with Newton's method, highlighting the importance of recognizing uniform motion in physics.
The Mean Value Theorem illustrates a fundamental concept in calculus through a relatable analogy of a race Imagine racing a friend who maintains a constant speed from start to finish while you vary your pace If you both finish simultaneously, it implies that at some point during the race, your speeds must have been equal This scenario demonstrates that it's impossible for you to have consistently run faster or slower than your friend throughout the entire race, as one of you would have inevitably won This reasoning leads us to the important principles outlined in the Mean Value Theorem.