A basic explanation of the two main ideas in learning calculus 1. Calculus can be learned as it is based on rates of change, and adding things up respectively.
Most of the time the word calculus comes up fear and anxiety sweep a room. In essence, the two main concepts in the study of this subject are differentiation and integration. Everything else concerns ideas, rules, and examples that deal with these two principal concepts. Consequently, by illustrating the two principal concepts to people interested in learning the subject, much relief from anxiety and fear of the subject can be accomplished.
Calculus is concerned with how things change over time. How fast your heart beats per minute at seven am compared to how fast your heart beats at 11pm is an example of something that is changing over time. Things that change over time involve rates of change. Calculus deals with anything that involves rates of change. Therefore, calculus is used in economics, astronomy, air traffic control, radar, engineering, census studies involving population growth, medicine with rates of growth of bacteria, etc. In this way calculus is related to topics that are not directly related. But calculus is the mathematical structure that all these different topics have in common.
Differentiation, or the process of finding derivatives, involves rules for finding explicit formulas which involve rates of change. In calculus one learns the process of finding derivatives for different functions. Functions are algebraic mathematical equations that can be used to simulate real-world problems in different application areas such as medicine, population growth, etc. So, at the most basic level, differentiation involves rules for finding rates of change for equations that represent some real world application.
Since equations involve unknown quantities, and algebra is the study of equations that deal with variables, or unknowns, then algebra is certainly very important in learning to take derivatives or what is called the process of differentiation. Equations are mathematical sentences using mathematical symbols. Word problems written in any language that involve unknown quantities and rates of change can be translated into mathematical sentences using algebra. The words in any language are actually changed into mathematical symbols describing the written word “problem” where each unknown quantity is given a letter which stands for the unknown quantity. This is why algebra is the foundation from which calculus is applied, and is a sentence in English or whatever linguistic language one uses to state the “word problem.”
The second principal concept is very interesting. There are two types of integration: indefinite and definite integration:
(A) Definite integration in its most basic form adds things up. For ease sake, take for
example, the change in position of a car traveling on a highway over time. The rate
of change of the car over time in this case, its velocity, is increasing(the car’s speed
is increasing over the time interval specified). If one wants to add the rate of
change of the car’s velocity over the specified time interval, one takes the integral
of the rate of change of the car’s position over time. This is the process of
integration for this simple case. The answer is going to yield the total distance
traveled by the car over the specified time interval.
A more interesting example provides for the rate of change of the car with respect
to its position on the road to vary. That is, for the first two seconds, let’s allow the
car’s velocity to be increasing. Then, over the next four seconds allow the car’s
velocity to be decreasing. Then, allow the velocity of the car to increase again for
the next two seconds. Consequently, the rate of change of the car with respect to its
position on the road has varied over the 8 seconds. Integrating or using integration
on the velocity of the car over the 8 seconds will yield the car’s net change of
position compared to its initial position at the time one started to measure the
different rates of change of the car’s position with respect to the road.
(B) Indefinite integration finds, if possible, a general formula that gives one a family of
curves whose slope at any time is provided by the answer of the integral also called
the “antiderivative”. For example, if you find the integral of 1, one is trying
to find the equation of a function whose slope at every point is 1. The answer is the
line y = x + K where K is a constant because the slope of y = x + K is always 1.So,
the family of curves y = x + K all have a slope of one anywhere on the line.
This is a simple example of indefinite integration.
In summary, the big picture of calculus is built on two main ideas: differentiation and integration. The examples shown here are a start at understanding the big picture of what calculus is used to accomplish. Any topics that involve rates of change especially those that vary involve differentiation. Integration adds rates of change over specified intervals as well as is applied to things that need to be added. And, indefinite integration is concerned with finding families of curves whose slopes anywhere along the equation being integrated is given by an equation (the answer, or antiderivative). The entire first course on calculus is concerned with the details of these two main topics and their variations.
