Calculus 1

The course offers a general view to some important ideas and techniques of differentiation and integration, and reveals the relationship between them. The fundamental objects that we deal with in calculus are functions. We discus the basic ideas concerning functions, their graphs, and ways of transforming and combining them.

We will see how to interpret derivatives as slopes and rates of change, and also develop rules for finding derivatives. These differentiation rules enable us to calculate with relative ease the derivatives of polynomials, rational functions, algebraic functions, exponential and logarithmic functions, and trigonometric and inverse trigonometric functions.

The course shows how to use the integral to solve problems concerning volumes, lengths of curves, population predictions, work, consumer surplus and many others. There is a connection between integral calculus and differential calculus. The Fundamental Theorem of Calculus relates the integral to the derivative, and we will see in this course that it greatly simplifies the solution of many problems.

• Local max, min

• Absolute max, min

• Concavity

• Newton's Iterations

• L'hospital's rule

• Linearization

• 9.1 | Finding the areas between the curves

• 9.2 | Substitution rule

• 11.1 | Trigonometric Integration

• 11.2 | Integration by Parts

• 10.1 | Volumes as a Riemann's sum

• 10.2 | Volumes using Cylindrical Disks (Slices)

• 10.3 | Volumes using Cylindrical Shells

• 11.3 | Integration of Rational functions

• 11.4 | Integration of Radicals

• Average value of a function

• Mean value theorem

• Finding the Length of curves

• Surface areas of cylinders, cones

• Approximation of the surface area with bands

• Example: surface area of a sphere

• curves defined by parametric equations

• the area under a parametric curves

• derivatives of a parametric equation

• guides of constructing a tangent lines

• arc length of a parametric curve

• surface area of revolution of a parametric curve

• Points in Polar Coordinates

• Change of coordinates

• Curves in Polar Coordinates

• Derivatives of Polar Equations

• Tangent lines to Polar Curves