Calculus Refresher Course Fall 2007Review of Calculus I and Calculus II Material(Prepared for the Fall 2006 Algebra/Calculus Review Workshop for engineering majorswho have had a gap of one year or more in their study of mathematics)I. Preview/Overview of CalculusThe calculus sequence at Utah State University consists of three courses: 1) Math 1210 (CalculusI), 2) Math 1220 (Calculus II), and 3) Math 2210 (Multivariable Calculus). All three courses are taught from the same textbook: Calculus: Concepts and Contexts, 3rd Ed. by James Stewart. Honors sections may require different and/or additional texts. Calculus I and the first part of Calculus II focus on the two main branches of calculus (differential calculus and integral calculus) as they pertain to real-valued functions of a single variable and, to a limited degree, parametric equations. In your own time you may wish to perusethe section in the textbook titled A Preview of Calculus that precedes Chapter 1 for a simple, short overview of calculus and its branches.The study of sequences and series is regarded by some as a third branch of calculus. These topics, found in Chapter 8 of the textbook, are studied in Calculus II. The Calculus II course also includes the extension of differential and integral calculus: 1) to functions other than real-valuedfunctions of a single variable (such as vector-valued functions), and 2) to coordinate systems other than the rectangular/Cartesian coordinate system (such as the polar coordinate).The Multivariable Calculus course is a continuation of the extension of differential and integral calculus of functions of a single variable: 1) to real-valued functions of several variables, 2) vector fields, and 3) to the cylindrical, and spherical coordinate systems in three-space.The foundation of all branches of calculus (or the common thread that ties all branches together into one discipline called calculus) is the notion of limits.Differential calculus is founded upon the following definition) for the derivative function f� of afunction f:0( ) ( )( ) limhf x h f xf xh�+ -�= .Question: In terms of the graph of f, what does these represent?Integral calculus is founded upon the following definition for the definite integral ( )baf x dx� of afunction f:*1( ) lim ( )nbianif x dx f x x��== D�� .Note: The key structure of the Riemann sum is a product of f times a small change in x. The sum of these products has widespread applications.1This is called a difference quotientThis is called a Riemann sumCalculus Refresher Course Fall 2007Besides the fact that derivatives and definite integrals are both defined in terms of limits, we will review another remarkable connection between these two branches of calculus established by The Fundamental Theorem of Calculus.The definition of a convergent sequence and the definition of a convergent series are also based on limits. We will spend little, if any, time reviewing this Calculus II material since it is not pertinent to the Multi-variable Calculus material studied in Math 2210.II. The Definition of the Derivative �f of a function f:Recall: 0( ) ( )( ) limhf x h f xf xh�+ -�= . One can see the derivative of a function f is defined in terms of the limit of the quotient ( ) ( )f x h f xh+ -. This quotient is called a difference quotient. This quotient (without the limit) is the slope of a secant line containing the points ))(,( xfx and))(,( hxfhx  on the graph of f. It represents an average rate of change in the function f between these two points. The interpretation/meaning of a difference quotient (and subsequently of its limit, the derivativef�) depends on the context/meaning of the function f. Consider the following three examples of various possible contexts/meanings of the function f. Example 1: Suppose ( )f x gives the distance (ft) of an object, moving in a straight line, from some reference point, as a function of time (sec).In this context the numerator ( ) ( )f x h f x+ - of the difference quotient represents the changein distance of the object in the time period beginning at x seconds and ending at x h+ seconds. The denominator h of the difference quotient represents the time elapsed (or change in time) for the same time period. Therefore, in this context, the difference quotient( ) ( )f x h f xh+ - represents the average velocity (ft per sec) of the object over the time periodbeginning at x seconds and ending at x h+ seconds.Example 2: ( )f x gives the distance (mi) a car travels from some reference point as a function of the amount (gal) of gasoline consumed.In this context the numerator ( ) ( )f x h f x+ - of the difference quotient represents the distance of the car travels from the time x gallons of gas have been consumed to the timex h+ gallons of gas have been consumed. The denominator h of the difference quotient represents the amount of gasoline used during the same time period. Therefore, in this context, the difference quotient ( ) ( )f x h f xh+ - represents the average fuel economy (mi pergal) of the car from the time of x gallons of gas consumption to the time of x h+ gallons of gas consumption.2Calculus Refresher Course Fall 2007Example 3: ( )f x is merely regarded as the y-coordinate (paired with x) of a point on the graph of f.In this context the numerator ( ) ( )f x h f x+ - of the difference quotient represents the vertical change (or “rise”) from the point ( , ( ))x f x to the point ( , ( ))x h f x h+ + on the graphof f. The denominator h of the difference quotient represents the horizontal change (or “run”) from the point ( , ( ))x f x to the point ( , ( ))x h f x h+ + . Therefore, in this context, the difference quotient ( ) ( )f x h f xh+ - represents the slope of a line, called a secant line, from the point ( , ( ))x f x to the point ( , ( ))x h f x h+ + on the graph of f. The difference quotient in each context represents an average rate of change—velocity (mi per gal), fuel economy (mi per gal), slope (“rise” per “run”). There are many other contexts that we could consider. No matter what f represents, the difference quotient ( ) ( )f x h f xh+ - represents some sort of average rate of change. In every case the average rate of change from x to x h+ approximates the “instantaneous rate of change at x”. Difference quotients better approximate the“instantaneous rate of change at x” as the change in the x-coordinate from x to x h+ (i.e., h) gets closer to zero. Therefore,