Course Overview

This study guide summarizes the foundational concepts and learning objectives for MATH-M 119: Brief Survey of Calculus I, a college-level business calculus course. The course covers essential topics including functions, limits, derivatives, integrals, and their applications in various real-world contexts.

Main Topics

• Functions, Graphs, and Models

• Differentiation

• Exponential and Logarithmic Functions

• Applications of Differentiation

• Integration

• Applications of Integration

Learning Objectives

Modeling with Functions

Students will learn to model applied problems using linear, exponential, and logarithmic functions. This includes identifying appropriate models, defining variables, establishing relationships, and interpreting solutions.

• Key Point 1: Recognize which real-world problems can be solved using linear, exponential, or logarithmic models.

• Key Point 2: Formulate mathematical models by creating variables and deducing relationships.

• Example: Modeling population growth with an exponential function:

Rates of Change and Differentiation

Students will become proficient in calculating and interpreting average and instantaneous rates of change using differential calculus. This includes working with data presented in tables, graphs, text, or formulas.

• Key Point 1: Understand the concept of the derivative as the instantaneous rate of change.

• Key Point 2: Apply differentiation to solve problems involving rates of change.

• Example: The derivative of a revenue function with respect to gives the marginal revenue:

Optimization Applications

Students will learn to model and solve optimization problems using calculus. This involves defining variables, translating constraints, and using derivatives to find optimal values.

• Key Point 1: Set up optimization problems by identifying independent and dependent variables.

• Key Point 2: Use calculus techniques to find maximum or minimum values.

• Example: Maximizing profit by finding the critical points of a profit function where

Integration and Accumulated Change

Students will become proficient in calculating and interpreting accumulated change using integral calculus. This includes working with rates of change presented in various formats.

• Key Point 1: Understand the definite integral as a measure of accumulated change.

• Key Point 2: Apply integration to solve problems involving total change over an interval.

• Example: Calculating total revenue over time by integrating a rate function:

Analytical Reasoning Competency

This course satisfies the Statewide Analytical Reasoning Competency. Students will:

• Interpret information presented in mathematical form

• Represent information mathematically

• Demonstrate skill in mathematical procedures

• Analyze mathematical results for reasonableness

• Clearly explain mathematical representations, solutions, and interpretations

Course Structure and Assessment

• Assignments: Online homework, quizzes, active engagement activities, three tests, and a comprehensive final exam.

• Grading: Final exam (25%), tests (45%), homework (10%), quizzes (10%), active engagement (10%).

• Textbook: Calculus and Its Applications, Bittinger, Pearson Custom Publishing.