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An Overview of Calculus: Understanding Functions, Rates of Change, and Areas Under Curves

This resource introduces the foundational concepts of calculus, focusing on how to analyze changing quantities represented as functions. It outlines the key areas of study, including derivatives, integrals, and limits, and provides sample problems that illustrate calculating rates of change and areas beneath curves. Additionally, it addresses the importance of self-assessment in choosing the appropriate calculus course level. Dive into essential calculus concepts and enhance your mathematical understanding.

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An Overview of Calculus: Understanding Functions, Rates of Change, and Areas Under Curves

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  1. Calculus: Where are you? • A. I’ve seen no calculus. • B. I saw just a little in Pre-calculus. • C. I’ve had less than a year. • D. I’ve had a year. • E. I’ve had more than a year.

  2. A QCQ (Quick Calculus Question) • What is the slope of the tangent line to the curve f (x) = x2 at the point (3, 9)? • A. 9 • B. 2x • C. 6 • D. 18 • E. Haven’t the foggiest

  3. Another QCQ • What is the area under the curve f (x) = x2 between x = 0 and x = 2? • A. 4 square units • B. 8/3 square units • C. 11/2 square units • D. 7 square units • E. I’m clueless

  4. Calculus – The Big Picture(1/21/09) • Calculus is the study of quantities which are changing. • These quantities are most often expressed as functions. • We may know a function and seek information about its rate of change, or conversely we may know about its rate of change and seek information about that function.

  5. Calculus: Limits, Derivatives, and Integrals • Given a function, its rate of change is called its derivative (Calc I). • Given a rate of change, its corresponding function is called an antiderivative (Calc I and II). • Given lots of values of a function, we can add them all up by using the integral (Calc I and II). • All of these concepts involve the concept of limit (Calc I).

  6. Some Sample Problems • Given a linear function, say y = 3 x + 1, what is its rate of change when x = 2? When x = 4? • Given a quadratic function, say y = x 2, what is its rate of change when x = 2? When x = 4? • Given an exponential function, say y = 2x, what is its rate of change when x = 2? When x = 4?

  7. More samples • What is the area below y = 3 x + 1 between x = 2 and x = 4? • What is the area below y = x2between x = 2 and x = 4? • What is the area below y = 2xbetween x = 2 and x = 4?

  8. A final clicker question today • Right now, I feel that MA 111 • A. is probably about right for me. • B. may be too hard – I’m worried about my preparation. • C. may be too easy - I may know most of this stuff already. • D. …. Not sure yet how I feel.

  9. Assignment for Friday • If you are unsure if MA 111 is right for you, go to http://www.skidmore.edu/academics/mcsand take thecalculus placement exam. You need your Skidmore username and password to access the exam. • You must decide by next Monday if you will stay in MA 111, or if you want to move up (113). No “down” is available in the spring. • Obtain the textbook and read To the Student (page xxiii) and A Preview of Calculus (pages 2-9).

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