Welcome to Math 8.

Math 8 is appropriate for students who have taken Math 3, or have equivalent background, such as a high school AB calculus course. This entails having studied functions and limits, differentiation, and a little bit of integration, including the fundamental theorem of calculus. You do not need to have seen many techniques or applications of integration.

Math 8 includes some single-variable calculus and some multivariable calculus. The course is divided into 3 units.

Unit 1: Single-variable calculus. We cover two major topics, Taylor polynomials, and techniques and applications of integration. You may recall the tangent line approximation from your first calculus course. This is a way of approximating a function by using a linear function (a degree 1 polynomial) that has the same value and derivative as the original function at some chosen point. With Taylor polynomials we extend this idea, using higher degree polynomials that also have the same second and higher derivatives as the original function. Ideally, in the limit as we take polynomials of higher and higher degree, we approach the actual value of the original function. We will learn ways to see when this is true, and to estimate the error in using a Taylor polynomial to approximate a function.

You may have learned one or two applications of integration, such as integrating speed to find distance, and one or two techniques of integration, such as direct substitution (u-substitution). We will learn at least one more useful technique of integration. We will also learn a couple of additional applications of integration, but more importantly, we will learn how to discover new applications of integration to use in new situations. There are many more ways to use integration than can be listed in a textbook.

Unit 2: Vectors and vector-valued functions. An example of the kind of function we will learn to study in this unit is a function representing motion in space, that takes as input a number t, and gives as output the coordinates (x,y,z) of the position of some moving object. We will often use exactly this idea as a good way to visualize these functions, but physics is far from the only application of the things we will be studying. For example, if you are studying an economic system, (x,y,z) could represent the values of three chosen commodities at time t; if you are studying an ecological system, (x,y,z) could represent the populations of three interdependent species. Of course, you are likely to be interested in more than three commodities or species, but the calculus we will learn is not limited to three dimensions.

Unit 3: Functions of several variables. Here an example of the kind of function we will be studying takes as input the coordinates (x,y,z) of a point in space and gives as output the temperature at that point. Again, we will often use examples like this to visualize these functions, but such functions have applications in many fields. An example of the kind of thing we will be able to do is use derivatives to find the point on a given sphere where the temperature is a maximum.