Math 50a --- Differential Calculus
Catalog Description
A study of limits, continuity, and derivatives of algebraic, transcendental, and trigonometric functions. Applications of the derivative include optimization, related rates, examples from the natural and social sciences, and graphing of functions. The course introduces the integral and the connection between the integral and derivative.
Special notes or advisories: A graphing calculator is required.
Prerequisites
Math 30 and Math 25 (or the equivalent) with a grade of "C" or better, or appropriate score on assessment exam
Describe representative skills without which the student would be highly unlikely to succeed: Students must be well-grounded in both college algebra and trigonometry to be successful in this first course in calculus.
Course Learning Outcomes
What should the student be able to do as a result of taking this course? State some of the objectives in terms of specific, measurable student accomplishments.
- Read, write, and speak accurately about mathematical ideas and use correct mathematical notation.
- Use graphing technology to visualize functions, explore mathematical concepts, and verify their work.
- Apply differentiation techniques and theory to functions of one variable.
- Apply the concept of the derivative and integral to solve real-world problems and applications.
- Demonstrate the fundamental relationship between the derivative and the integral.
- Use numerical, graphical, symbolic, and verbal representations to solve problems and communicate with others.
Course Content
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Themes: What themes, if any, are threaded throughout the learning experiences in this course?
- Problem solving
- Writing
- Technology
- Communication
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Concepts: What concepts do students need to understand to demonstrate course outcomes?
- The use of the graphing calculator and/or mathematical software as a fundamental problem-solving tool.
- The presentation of mathematical solutions in a logical coherent structure, including the use of fundamental writing skills, grammar, and punctuation.
- Limits and their role in the major theorems of calculus.
- Continuity and its role in the major theorems of calculus.
- Differentiation and its role in the major theorems of calculus.
- Integration and its role in the major theorems of calculus.
- The connection between integration and differentiation.
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Issues: What primary issues or problems, if any, must students understand to achieve course outcomes.
- The appropriate use of technology in the problem-solving process.
- The connection between mathematics and the real world.
- The role of the student in becoming a successful learner.
- The recognition that the problem-solving skills learned in this class are applicable in future mathematics classes and classes in related fields, such as physics, engineering, etc.
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Skills: What skills must students master to demonstrate course outcomes?
- Limits and Continuity
- Find the tangent line to the graph of a function as the limiting position of a secant line.
- Find the instantaneous rate of change of a function as the limit of the average rate of change.
- Determine the limit of a function through descriptive tables and analysis of graphs.
- Use limit laws and symbolic manipulation to compute the limits of functions.
- Determine the limit of a function using the formal definition of the limit.
- Determine the continuity of a function through analysis of its graph and through the use of the formal definition of continuity.
- Apply continuity theorems to problems in mathematics and applications.
- Use limits that involve infinities to determine horizontal and vertical asymptotes.
- Differentiation
- Use the derivative to determine the instantaneous rate of change of a function.
- Use the formal definition of the derivative to compute the derivative of a function.
- Use the differentiation laws to compute the derivative of a function symbolically.
- Use the chain rule to differentiate compositions of functions.
- Use the method of implicit differentiation to determine the derivative of the inverse of a function.
- Compute the derivatives of trigonometric, logarithmic, and exponential functions.
- Use the method of logarithmic differentiation to compute the derivative of a power function.
- Apply the derivative as a rate of change to problems in the natural and social sciences (motion, growth and decay, population ecology, etc.).
- Solve applications that involve related rates.
- Use the differential to produce a linear approximation of a function.
- Time permitting:
- Differentiate hyperbolic functions.
- Graphing functions
- Apply the Mean Value Theorem to problems in mathematics and applications.
- Use the first derivative test to determine where a function is increasing or decreasing.
- Use the second derivative test to determine the concavity of a function.
- Apply the first and second derivative tests to determine the extrema of a function.
- Apply the Extreme Value Theorem to determine the extrema of a continuous function on a closed and bounded interval.
- Use l'Hopital's rule to find limits and to assist in sketching the graph of a function.
- Time permitting:
- Use Newton's method to approximate the zeros of a function.
- Optimization
- Create a mathematical model from a word problem. Identify constraints on the model.
- Use the various derivative tests to optimize (maximize or minimize) one quantity of the model.
- Integration
- Compute the antiderivative of a function.
- Select the antiderivative from a family of antiderivatives that satisfies a given initial condition.
- Apply the antiderivative to problems involving motion.
- Use a Riemann sum to approximate a definite integral.
- Determine the definite integral of a function as the limit of a Riemann sum.
- Use the Fundamental Theorem of Calculus to evaluate definite integrals.
- Use the substitution technique to evaluate definite integrals.
- Limits and Continuity
Representative Learning Activities
What will the students be doing (i.e., Listening to lectures, participating in discussions and/or group activities, attending a field trip, etc.)? Relate the activities directly to the Course Learning Outcomes.
- Listening to lectures.
- Participating in group activities and/or assignments.
- Participating in class assignments and/or discussions.
- Completing homework assignments.
- Using the graphing calculator and/or mathematical software to complete activities designed to foster a deeper level of understanding of the concepts and skills developed in this class.
Assessment Tasks
How will the student show evidence of achieving the Course Learning Outcomes? Indicate which assessments (if any) are required for all sections.
- Representative assessment tasks:
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- In-class examinations and/or quizzes.
- Homework assignments.
- Take-home examinations and/or quizzes allow the instructor to include questions and/or exercises that require the use of the graphing calculator and/or mathematical software to supplement the analysis. Extra time allows the students to develop their writing and presentation skills.
- Writing assignments designed to develop communication of mathematical concepts.
- Group and/or individual projects and presentations.
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Required assessments for all sections – to include but not limited to:
- Homework assignments.
- At least two proctored, closed book examinations, plus a comprehensive final examination.
Examples of Appropriate Texts or Other Readings
- Author: Stewart; Title: Calculus, Early Transcendentals, Sixth Edition; Date: 2007