The Fundamental Theorem of Calculus
In Week 1 we discussed why it is sometimes of interest to be able calulate the area enclosed between the graph y=f(x) of some function f and the X-axis, over some interval with specified left and right endpoints. The Fundamental Theorem of Calculus (FToC) connects this problem to the theory of differential calculus, and gives us a strategy to solve it. If we imagine the left endpoint of our interval to be fixed, and consider the right end point to be a variable x, then the area changes as we move the right endpoint around, it is a function of x. The FToC asserts that the derivative of this function is the f(x) whose graph defines the region. It means that IF we can find a function whose derivative is f(x), we can calculate our definite integral. So our focus will move in the next couple of weeks to methods for finding a function with a given derivative.
In this week’s lectures we’ll discuss the meaning and significance of the Fundamental Theorem of Calculus, and start thinking about how to use it to calculate definite integrals. In the process we’ll introduce the useful concept of the indefinite integral.
Relevant sections of the Lecture notes this week are Sections 1.3 and the first four pages or so of Section 1.4 (which is quite long and will occupy us at least for Week 3).
Our lectures take place on Wednesday and Thursday, January 19 and 20, at 10.00 in the Kirwan Lecture Theatre. Video versions (from last year) are posted here, along with slides.
Here is Lecture 3.
Slides for Lecture 3 without annotation, and annotated.
Here is Lecture 4.
Slides for Lecture 4 without annotation, and annotated.
Supplementary video tutorial on FToC
Here is a short video with some examples (from recent exam papers) of questions about the Fundamental Theorem of Calculus.
Notes from this little tutorial.
Weekly Problem 2
The weekly problems are just for fun. They have nothing much to do with our curriculum. Please send me an email if you have a solution that you would like to share with the class!