Dans' Mathematica

"Functions are fundamental to the study of calculus, and here we review what they are, their graphs, how they are combined and transformed, and various ways they are classified. A function can be represented by an equation, by a numerical table, by a graph, or by verbal descriptions. The graph of a function is a particularly useful visualization of its features and overall behavior, and we review several ways for obtaining a graph, including the use of graphing calculators and computer graphing s/w. We look at the main types of functions that occur in calculus, with special emphasis (in this chapter) on the exponential functions and their inverses, the logarithmic functions. Trig functions are summarized in App B, along with several other basic topics, including the real number system, Cartesian coordinates in the plane, straight lines, parabolas, and circles." (George Thomas)

"The concept of a limit is a central idea that distinguishes calculus from algebra & trig. It is fundamental to finding the tangent to a curve or the velocity of an object. We develop the limit, first intuitively, and then formally. We use limits to describe the way a function f varies. Some functions vary continuously; small changes in "x" produce only small changes in f(x). Other functions can have values that jump or vary erratically. The notion of a limit gives a precise definition to distinguish between these behaviors. The geometric application of using limits to define the tangent to a curve leads at once to the important concept of the derivative of a function. The derivative quantifies the way a function's values change." (George Thomas)

"This chapter studies some of the important applications of derivatives. We learn how derivatives are used to find extreme values of functions, to determine and analyze the shapes of graphs, to calculate limits of fractions whose numerators and denominators both approach zero or infinity, and to find numerically where a function equals zero. We also consider the process of recovering a function from its derivative. The key to many of these accomplishments is the Mean Value Theorem , a theorem whose corollaries provide the gateway to integral calculus." (George Thomas) In calculus, the mean value theorem states, roughly, that given an arc of a smooth differentiable (continuous) curve, there is at least one point on that arc at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the secant chord connecting the endpoints of the arc. The theorem i...