Lecture 8: Level curves, partial derivatives, and tangent plane approximation
In this lecture, the focus is on studying functions of several variables and their derivatives. The challenge of visualizing functions of two variables is explored through graphing and contour plots. Contour plots are shown to be an elegant solution to the difficulty of drawing space pictures on a sheet of paper or a blackboard. The contour plot is essentially an xy plot with a bunch of curves representing the elevations on the graph. This method is compared to topographical maps and temperature maps, where the same concept is applied.
A lecture on level curves, partial derivatives, and tangent planes. The beginning of a study of functions of several variables.
Denis Auroux. 18.02 Multivariable Calculus, Fall 2007. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (Accessed March 15, 2009). License: Creative Commons Attribution-Noncommercial-Share Alike.
- Functions of Two Variables Applet - A Java computer simulation applet that helps you visualize what 3-dimensional graphs of variables in two variables look like.
- Lecture Notes - Lecture Notes from this lesson.
- Transcript of Lesson - Written transcript from this lesson.
- Problem Set - A problem set with some problems from this lecture.
- What is an example of a function with two variables?
- How do you graph a function with two variables?
- How do you graph z = -y?
- How do you graph f(x,y) = 1 - x^2 - y^2?
- How do you graph three-dimensional functions?
- What does the graph of z = x^2 + y^2 look like?
- What does the graph of z = y^2 - x^2 look like?
- What is a contour plot of a function?
- What is a level curve of a function?
- How do you graph a function by hand using level curves and contour plots?
- What is a partial derivative?
- How can you find a partial derivative of a function?
- What is the tangent approximation formula?
- What does it mean geometrically to find the partial derivative of a function at a point?
- What is the partial derivative of x^3y + y^2 with respect to x and y?
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Staff Review
This video starts to get into some really interesting parts of Multivariable Calculus. Finally getting to the meat of the course, we see some functions of many variables, which are surfaces in three dimensions. This is a really cool lecture and very important to the essence of this course. Many very interesting graphs are shown and drawn. Also, partial derivatives are introduced in this lecture.
