University of the People 3 credits MATH
This course offers an interdisciplinary introduction to differential calculus, an essential component of mathematics with wide-ranging applications across various fields. It facilitates a thorough grasp of polynomials, exponential, logarithmic, and trigonometric functions. Each family of functions will include an exploration of related limits, derivatives and integrals. Throughout the course, learners will enhance their independent learning capabilities, problem-solving skills, and precision in mathematical reasoning and writing. Participants will also develop the ability to express solutions using geometric, symbolic, and analytical methods applicable to theoretical and practical mathematical problems. Learning Objectives and Outcomes: By the end of this course students will be able to: Analyze the graph of polynomial, rational, exponential, logarithmic, and trigonometric functions; critically assess their limits and continuity. Employ various differential rules, including the chain rule and implicit differentiation, to compute derivatives of polynomial, rational, exponential, logarithmic, trigonometric, composite, and implicitly defined functions. Utilize derivative techniques to analyze functions for extreme values; employ L'Hopital's Rule to solve equations and address complex limit problems. Apply derivatives to solve real-world problems involving related rates and optimization. Explain the Fundamental Theorem of Calculus and compute indefinite and definite integrals to find areas under curves. Course Schedule and Topics This course will cover the following topics in eight learning sessions, with one Unit per week. The Final Exam will take place during Week/Unit 9 (UoPeople time). Week 1: Unit 1 - Functions and Graphs Week 2: Unit 2 - Limits and Continuity Week 3: Unit 3 - Fundamentals of Differentiation Week 4: Unit 4 - Advanced Differentiation Techniques Week 5: Unit 5 - Extreme values, Mean Value Theorem (MVT) and L'Hôpital's Rule Week 6: Unit 6 - Applications of Derivatives Week 7: Unit 7 - Introduction to Integrals Week 8: Unit 8 - Definite Integrals and Their Applications Learning Guide The following is an outline of how this course will be conducted, with suggested best practices for students. Unit 1 - Functions and Graphs Describe different types of functions. Analyze graphs of functions to identify key features such as intervals of increase or decrease. Unit 2 - Limits and Continuity Describe the limit definition of a derivative. Apply limit laws to calculate the limits of functions. Analyze the continuity of functions at specific points using limits. Unit 3 - Fundamentals of Differentiation Explain the concept of rates of change and how they are represented by derivatives. Apply differentiation rules to find the derivatives of various functions (such as Polynomial, Rational, Trigonometric, Exponential and Logarithmic Functions). Unit 4 - Advanced Differentiation Techniques Implement the chain rule to differentiate composite functions. Solve problems using implicit differentiation to find derivatives of implicitly defined functions and determine the equation of a tangent and normal line. Unit 5 - Extreme values, Mean Value Theorem (MVT) and L'Hôpital's Rule Explore critical points and extreme values of functions using derivative tests. Explain the application of the Mean Value Theorem in understanding the behavior of functions over intervals. Apply L'Hôpital's Rule to evaluate limits of indeterminate forms. Unit 6 - Applications of Derivatives Apply derivatives to solve related rates problems involving real-world scenarios. Analyze optimization problems to find the maximum and minimum values of functions. Unit 7 - Introduction to Integrals Explain the concept of antiderivatives and demonstrate how to compute indefinite integrals. Apply the Fundamental Theorem of Calculus to connect the processes of differentiation and integration. Perform basic integration techniques, including substitution, to solve integrals. Unit 8 - Definite Integrals and Their Applications Evaluate definite integrals using proper limits of integration and appropriate techniques. Apply definite integrals to calculate the area under the curve.
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