Course Outline
Loosely speaking, Calculus is the Mathematics of change. Its objects of study are (continuous) functions. These are mathematical gadgets that allow to formalize quantities f(x) that depend on another quantity x which is allowed to vary within a certain range of interest. In this setting, it makes sense to ask how the quantity f(x) changes when x takes different values. For example, the position at the time x of the proverbial apple which fell from a tree-branch and hit Newton's head can be described by the formula:
where g is a fixed negative number, called the gravitational acceleration (Sir Isaac Newton discovered the existence of this constant and is also one of the inventor of calculus).
Our first goal in this course will be that of familiriazing ourselves with various characteristics and example of functions.
Sometimes, given a function f(x) (as above, this function could describe the position over time of an object moving on a path), it might be interesting to know what happens to the values f(x) as the quantity x approaches a certain specific value or as it becomes larger and larger. To deal with these kinds of questions we use limits. Our next major goal will be that of studying limits, their properties and how to compute them.
Calculus studies change by considering "instantaneous" change, that is how the quantity f(x) changes when x varies over very tiny ranges of values. If you drove for 2 hours and 190 km, you might conclude that you drove at an average velocity of 95 km/h. But the monitor on your car never displays the average velocity, only the instantaneous one! Derivatives of functions are the mathematical tools to compute instantaneous change: if the function f(x) describes the position of an object at time x, its derivative will describe precisely the velocity at time x of that object. We will then focus on derivatives and on techniques to compute them.
Finally, in many practical circumstances it might be easier to find information about the instantaneous change of an unknown quantity rather than about the quantity itself. This happens all the time in physics where one might find an equation that describes the velocity of an object and would like to understand its trajectory from this information. What one seeks in these situations is an inverse operation to that of taking derivatives, that is, an operation that takes as input a function describing the instantaneous change of a quantity and gives back as output that quantity itself. This is what integrals do, thanks to a very important result called the Fundamental Theorem of Calculus. Our last topic will be the study of integrals, how to compute them and how to use them to solve geometrical problems such as finding the area or volume or certain regions in the plane and the space.
List of topics covered: Review of limits and derivatives of exponential, logarithmic and rational functions. Trigonometric functions and their inverses. The derivatives of the trigonometric functions and their inverses. L'Hospital's rules. The definite integral. Fundamental theorem of Calculus. Simple substitution. Applications including areas of regions and volumes of solids of revolution. This covers Chapter 1-6 of Stewart's book.