Code
Φ-111
Level
Undergraduate
Category
A
Teacher
G. Athanasiu
ECTS
7
Hours
2
Semester
Spring
Display
Yes
Offered
Yes
Teacher Webpage
Goal of the course
The course is addressed to first year students. It covers and expands material presented in the last years of high school, including functions, basic calculus, limits, derivatives and integrals. It also focuses on direct applications of the material covered to a number of simple physics problems.
During the spring semester to course is offered in the form of selfstudy with only one two-hour lecture per week for the students that did not pass it during the fall semester. Emphasis is given in problem solving.
During the spring semester to course is offered in the form of selfstudy with only one two-hour lecture per week for the students that did not pass it during the fall semester. Emphasis is given in problem solving.
Program
Monday, 15:00-17:00, Amphitheater ST
Syllabus
Introductory material: Straight lines, functions and plots, exponential fubctions, inverse functions and logarithms, trigonometric functions and their inverses, parametric equations (0.5 weeks)
Limits and Continuity: Rates of change and limits, computing limits and side limits, infinite limits, continuity, tangent lines (1.5 weeks).
Derivatives: The derivative as a function, the derivative as a rate of change, derivative of a fraction and a negative power, derivatives of trigonometric functions, derivation chain rule, derivative of an implicit function (1 week).
Applications of derivatives: Extrema of functions, mean value theorem,rules for plotting functions, construction of optimization models, differentials, Newton's algorithm, introduction of taylor's formula (2 weeks).
Integration: Indefinite integrals, integration rules, integration by substitution, intergral estimates from finite sums, Riemann sums and definite integrals, mean value theorem, fundamental theorem of integration, evaluation of definite integrals by substitution, numerical integration, main types of integration, integration by parts, partial fractions, trigonometric substitutions, computational algebra systems, Monte Carlo integration, generalized integrals (3 weeks).
Applications of integrals:Calculation of volumes by slicing and rotation around an axis, by using cylindrical shells, calculation of lengths of curves in the plane, springs, pumps, elevators, fluid forces, center of mass and torques. (2 weeks).
Non-algebraic functions: Logarithms, exponentials, derivatives of inverse trigonometric functions, integrals of hyperbolic functions.(1 week)
Infinite Series: Limits of sequences, subsequences, bounded sequences, Picard's method, infinite series, non-negative series, alternating series, absolute convergence, conditiuonal convergence, power series, Taylor and McLaurin series, applications of power series. (2 weeks).
Limits and Continuity: Rates of change and limits, computing limits and side limits, infinite limits, continuity, tangent lines (1.5 weeks).
Derivatives: The derivative as a function, the derivative as a rate of change, derivative of a fraction and a negative power, derivatives of trigonometric functions, derivation chain rule, derivative of an implicit function (1 week).
Applications of derivatives: Extrema of functions, mean value theorem,rules for plotting functions, construction of optimization models, differentials, Newton's algorithm, introduction of taylor's formula (2 weeks).
Integration: Indefinite integrals, integration rules, integration by substitution, intergral estimates from finite sums, Riemann sums and definite integrals, mean value theorem, fundamental theorem of integration, evaluation of definite integrals by substitution, numerical integration, main types of integration, integration by parts, partial fractions, trigonometric substitutions, computational algebra systems, Monte Carlo integration, generalized integrals (3 weeks).
Applications of integrals:Calculation of volumes by slicing and rotation around an axis, by using cylindrical shells, calculation of lengths of curves in the plane, springs, pumps, elevators, fluid forces, center of mass and torques. (2 weeks).
Non-algebraic functions: Logarithms, exponentials, derivatives of inverse trigonometric functions, integrals of hyperbolic functions.(1 week)
Infinite Series: Limits of sequences, subsequences, bounded sequences, Picard's method, infinite series, non-negative series, alternating series, absolute convergence, conditiuonal convergence, power series, Taylor and McLaurin series, applications of power series. (2 weeks).
Bibliography
«THOMAS, Calculus» – J. Haas, Ch. Heil, M.D. Weir., (14th Edition 2018)«higher mathematics», M.R. Spiegel.
«Differential and Integral Calculus» - M. Spivak.
«Differential and Integral Calculus» - M. Spivak.
