Desription

Knowledge of algebra, elementary functions and trigonometry is assumed. Topics in calculus of functions of one variable, including techniques of differentiation, applications to graphing, optimization (min/max) problems, and an introduction to integration. Main concepts of this course are derivatives (rates of change of a function) and integrals (which, in particular, provide a way to recover a function from the knowledge of its derivative). Knowledge and the ability to work with these concepts is essential for further studies of mathematical subjects, as well as for applications of mathematical techniques in other sciences. This course will focus on understanding calculus concepts, analytical reasoning and developing crucial skills in order to calculate, analyze, interpret and communicate the results clearly.

Intended Learning Outcomes.

At the completion of this course the student will be able to:

• learn the general concept of function and its applications to real-world situations.
• determine the domain and range of functions
• work with exponential, logarithmic and trigonometric functions and their applications in applied problems.
• learn the concepts of the derivative and its underlying concepts such as limits and continuity.
• calculate derivatives using the basic definition, sum and product formulas, quotient rule, chain rule, and power rule.
• analyze functions and sketch curves using first and second derivatives. 
• learn about various applications of the derivative in applied problems.
• demonstrate an understanding of Rolle’s Theorem, Mean-Value Theorem, and the Fundamental Theorem of Calculu
• learn about anti-derivative. 
• calculate definite or indefinite integrals using the power rule or a substitution method.
• use concept of integration to evaluate geometric area and solve other applied problems

Course Contents

1. Functions and their graphs
2. Rates of change and tangents to curves
3. Limit of a function and limit laws, the precise definition of a limit
4. One-sided limits
5. Continuity
6. Limits involving infinity; asymptotes of graphs
7. Differentiation: tangents and derivative at a point, the derivative as a function
8. Differentiation rules, the derivative as a rate of change
9. Derivatives of trigonometric functions, chain rule, implicit differentiation
10. Related rates, linearization and differentials
11. Higher derivatives
12. Applications of derivatives: extreme values of functions
13. Rolls’ theorem, the mean value theorem
14. Monotonic functions and the first derivative test
15. Convexity, point of inflection and second derivative test
16. Concavity and curve sketching
17. Indeterminate forms and L'Hôpital's rule
18. Antiderivatives, Integration: area and estimating with finite sums
19. Sigma notation and limits of finite sums
20. The definite integral, the fundamental theorem of calculus
21. Indefinite integrals and the substitution method
22. Substitution and area between curves
23. Applications of definite integrals: volumes using cross-sections
24. Volumes using cylindrical shells, arc length
25. Areas of surfaces of revolution
26. Work and fluid forces, moments and center of mass

Recommended Books

• G. B. Thomas Jr., M. D. Weir, J. R. Hass, "Thomas Calculus - Early Transcedentals", Pearson, 13th edition, (2013).
• G. B. Thomas, Jr.,M. D.Weir, J. R. Hass, "Thomas Calculus", Pearson, 12th Edition, (2002).
•  J. Stewart, "Calculus Early: Transcendentals", 6th edition, (2008).