MTH 1: Calculus I
- Page ID
- 93332
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This course is the first in the three-course calculus sequence intended for majors in math, engineering, and physical sciences. The course covers elements of analytic geometry, derivatives, limits and continuity, differentiation of algebraic and trigonometric functions, and the definite integral.
- 1: Functions and Graphs
- In this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functions, and we show the properties of their graphs. We provide examples of equations with terms involving these functions and illustrate the algebraic techniques necessary to solve them. In short, this chapter provides the foundation for the material to come.
- 2: Limits
- The idea of a limit is central to all of calculus. We begin this chapter by examining why limits are so important. Then, we go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value.
- 3: Derivatives
- Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, and it is critical to analysis in business and health as well. In this chapter, we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter.
- 3.1: Introducing the Derivative
- 3.2: Differentiation Rules
- 3.3: The Product and Quotient Rules
- 3.4: Derivatives of Trigonometric Functions
- 3.5: Derivatives as Rates of Change
- 3.6: The Chain Rule
- 3.7: Implicit Differentiation
- 3.8: Related Rates
- 3.9: Derivatives of Inverse Functions
- 3.10: Derivatives of Exponential and Logarithmic Functions
- 3.11: Chapter 3 Review Exercises
- 4: Applications of Derivatives
- In this chapter we look at how derivatives are used to find maximum and minimum values of functions. As a result, we will be able to solve applied optimization problems, such as maximizing revenue and minimizing surface area. In addition, we examine how derivatives are used to evaluate complicated limits, to approximate roots of functions, and to provide accurate graphs of functions.
- 6: Applications of Integration
- In this chapter, we use definite integrals to calculate the force exerted on the dam when the reservoir is full and we examine how changing water levels affect that force. Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter. From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world.
- 6.1: Numerical Integration
- 6.2: Areas Between Curves
- 6.3: Determining Volumes by Slicing
- 6.4: Volumes of Revolution - Cylindrical Shells
- 6.5: Arc Length of a Curve and Surface Area
- 6.6: Physical Applications of Integration
- 6.7: Moments and Centers of Mass
- 6.8: Integrals, Exponential Functions, and Logarithms
- 6.9: Exponential Growth and Decay
- 6.10: Calculus of the Hyperbolic Functions
- 6.11: Chapter 6 Review Exercises