A synopsis of what one needs to know to learn (pre-requisites) calculus as well as examples of what calculus is like.
Many Calculus students who enroll in the course in high school sometimes want to know a little bit about what the course is like. Some of these students are not prepared adequately to demonstrate proficiency in the subject. For example, some of these students were never told that they would need to remember some very important pre-requisites. The foundation blocks needed to succeed in calculus include a complete mastery of the rules used in algebra. Added to this foundation block is geometry which is like the cement holding the entire structure together: knowledge of all the basic functions. Functions such as linear, quadratic, cubic, absolute value, logarithmic, exponential, trigonometric, and piece-wise functions which could be a combination of any of the above.
From the thorough understanding of exponents, logarithms, trigonometry, and the formulas for area and volumes of geometric solids and two dimensional areas from algebra two, students have a better chance of understanding how these topics are integrated into calculus. With this comes the possibility that the student can synthesize their understanding of these topics along with the topics which concern one in calculus such as limits, derivatives, tangent lines from secant lines, the classic area problem of irregular shapes, finding the velocity of an object by knowing its position function, or finding the acceleration of an object by knowing its position function. Other topics include how to find the dimensions of a geometric object to maximize or minimize their areas or volumes. One can also minimize or maximize any function that one may have explicitly expressed with the variables defined and with certain given information. This is why algebra is the tool that allows one to work with the equations so one may arrive at the desired solutions.
For the student who understands their algebra and geometry as well as the domain and range of the six main trigonometric functions, the student has a better chance of assimilating calculus concepts some of which discuss the continuity of each of these functions. They also discuss whether two sided limits exist as one approaches an x value on the x – axis from both the right and left of that x value, and whether one is going to the same y-value even if the y value is not explicitly owned by the function at that particular x value. There are many details involving visual pictures or graphs of functions that determine the characteristics of that function. This is the reason that one must understand how functions are inter-related graphically, analytically, and numerically.
Calculus also discusses from an artistic point of view whether the functions mentioned earlier are smooth and continuous when graphed. If they are, then those graphs have tangent lines that can be drawn at every point along their graph. And, this explains in a way, a very important concept in calculus about whether a function is differentiable at every point on its graph. The beauty is that all smooth continuous graphs are differentiable, meaning that one can draw a tangent line at every point along its graph as well as the fact that the function has limits that exist at every point along that graph. These are special and very nice functions from a calculus point of view. By understanding these functions, then one can introduce students to more elaborate functions with subtleties that require much attention to detail.
So, in short, everyone can understand calculus to the degree that they understand the pre-requisites. And, as in anything else, the more time that one devotes to understanding the graphs of the functions discussed above, the better chance one has of truly synthesizing and understanding the subject. Consequently, doing homework problems as well as reading class notes, as well as spending time trying to understand examples from the text or a reference book enables a student to maximize their academic performance as well as understanding of this subject.
