Preface Introduction
These notes are an adjunct to the open source text for the course MATH 120 Introductory Calculus, Calculus, Volume 1 2 from OpenStax 3 They summarize the key ideas, examples and results from each section of that text that we will cover, along with additional examples and recommended homework exercises.
We start by restating some key ideal and objectives from the syllabus; see there for more about course organization, important dates, assessment, and so on.
Course Objectives and Student Learning Outcomes.
The main goal of this course is for students to learn the basic concepts and skills of solving mathematical and scientific problems described by functions that vary "smoothly" (with no jumps, breaks or sharp corners in their graphs), and to solve problems whose solutions can at best only be approximated with algebra, geometry and trigonometry (like the areas of most regions), but can be solved exactly with the methods of calculus.
Applications include the description of motion in terms of velocity and acceleration, models of population growth, chemical reaction rates and growth of the value of an investment, and optimization problems such as minimizing the cost of a task or maximizing what can be achieved with a fixed amount of resources. This material is covered in the first five chapters of the text, with a few sections omitted or left until Calculus 2 (Math 220).
Students are expected to do not only the quizzes, assignments and class exercises, but also to review each section of the text after it has been covered in class and to attempt the exercises set for each section. This is because, more broadly, a majority of the learning in this or any college course comes through students' efforts outside of class meetings.
By the end of the course, students should be able to:
- Calculate a wide variety of limits, including derivatives using the limit definition and limits computed using l'Hospital's rule;
- Demonstrate understanding of the main theorems of one-variable calculus (including the Intermediate and Mean Value Theorems, and the Fundamental Theorem of Calculus) by using them to answer questions;
- Compute derivatives of functions with formulas involving elementary polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions;
- Use information about the derivative(s) or antiderivative of a function (in graphical or symbolic form) to understand a function's behavior and sketch its graph;
- Construct models and use them to solve related rates and optimization problems;
- Recognize functions defined by integrals and find their derivatives;
- Approximate the values of integrals geometrically or by using Riemann sums; and
- Evaluate integrals by finding simple antiderivatives and by applying the method of substitution.
These outcomes will be assessed on the final exam.
General Education Student Learning Outcomes.
This course can be used to satisfy some general education requirements, for which there are some standard goals. Students are expected to display a thorough understanding of the topics covered. In particular, upon completion of the course, students will be able to
- model phenomena in mathematical terms,
- solve problems using these models, and
- demonstrate an understanding of the supporting theory behind the models apart from any particular application.
These outcomes will be assessed on the final exam.
Calculators.
It might be useful to have a graphing calculator, and the standard recommendation is the Texas Instruments TI-84 Plus. However, many choices of free "calculating device" can work too, including websites, phone apps and computer software, and I will demonstrate some of them. One of my favorites is the Desmos Graphing Calculator 4 ; in addition to that website interface, this is available as a free app for iPhones, iPads 5 , and Android 6 devices.
Such tools may be used for some homework and in-class exercises, but not on tests or the final exam.
openstax.org/details/books/calculus-volume-1/openstax.orgwww.desmos.com/calculator/apps.apple.com/us/app/desmos-graphing-calculator/id653517540play.google.com/store/apps/details?id=com.desmos.calculatorwww.desmos.com
