Calculus II

1. Demonstrate fundamental skills contained in the course, such as integration in one variable, and derivation of several variables.
2. Outline basic definitions and theorems, describe methods, such as necessary and sufficient conditions for extremes, existence of solutions of Cauchy problem, Laplace transform.
3. Explain, connect and interpret basic concepts and theories from the course.
4. Make conclusions by using logical reasoning (analogy, contradiction, implication).
5. Apply mathematical reasoning adequately.
6. Demonstrate modeling and problem solvers skills.
7. Illustrate problem by mathematical model, select and apply appropriate mathematical methods, interpret a solution to ensure it makes sense in the context of the problem.
8. Analyse problems by combining different ares of mathematics.
9. Think critically in solving complex problems.
10. Demonstrate an ability to communicate mathematics, by team work, discussion and written material

The course is divided in two cycles, 4 hours of lessons per week. Two colloquia: one mid-semester and one final.

The course is divided in two cycles, 3 hours of exercises per week.

Regular homeworks.

1. Lectures: Sequences, limits and accumulation points. Exercises: Limits of sequences.
2. Lectures: Series. Convergent and divergent series. Criteria for convergence. Exercises: Determining convergence of series.
3. Lectures: Power series. Taylor series and polynomial approximation. Exercises: Exercises power and Taylor series.
4. Lectures: Vector functions. Velocity and acceleration. Exercises: Derivative of vector functions and applications.
5. Lectures: Calculus of functions of several variables. Limit, continuity, partial and directional derivatives, gradient. Exercises: Exercises partial and directional derivatives, gradient.
6. Lectures: Chain rule, implicit differentiation, higher partial derivatives. Exercises: Exercises chain rule and higher partial derivatives. Second differential.
7. Midterm exam.
8. Lectures: Extrema of functions of two variables. Linear programming. Exercises: linear programming.
9. Lectures: Stationary points and local extrema of two-variable functions. The least squares method. Exercises: Finding local extrema of functions.
10. Lectures: Conditional extrema. Lagrange multiplicator. Exercises: Exercises conditional extrema.
11. Lectures: Differential equations, separate variables. Cauchy problem. Exercises: Differential equations, separate variables.
12. Lectures: Linear differential equations. LDE of higher order and applications. Exercises: Solving linear differential equations.
13. Lectures: Laplace transform. Properties of Laplace transform. Exercises: Laplace transform of functions using definition.
14. Lectures: Application to differential equations and circuits. Exercises: Application to differential equations and circuits.
15. Final exam.