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In our last video, we talked about de Casteljau Bezier curves, mostly cubics, for design work. In this lecture we discuss another application of cubic splinesto the interpolation problem: finding a smooth curve passing through a finite number of points in the (x,y) plane. Our approach to this question is somewhat novel, and focusses on the use of what we call Taylor coefficient vectors. A given cubic polynomial in our space P^3 has a 4-vector of Taylor coefficients at any point, and the relations between two such Taylor vectors is given by a linear transformation, essentially a Pascal matrix (see WLA23). So our strategy is to create the cubic spline one segment at a time, transferring the knowledge of the Taylor coefficient vector at one endpoint to the other. Although we are using calculus ideas, we develop them independently, so the viewer is not required to have had prior knowledge of calculus. ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at / njwildberger Your support would be much appreciated.