Mathematics Notes by Success Tutorials: Calculus
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Math 31: Calculus
Introduction
Calculus was developed in the seventeenth century by Gottfried Wilhelm Leibniz (1646 - 1716) and Isaac Newton (1642 - 1727). Liebniz, a German, was a self-taught mathematician and invented calculus independently of Newton. Although they personally admired each others work, an intense conflict arose between England and continental Europe over whether Newton or Liebniz had invented calculus. In truth, they invented calculus independently of each other.
Calculus is a mathematical tool which has many applications. Some of the main areas of application for us are:
(1) To find the slope of the tangent to a curve at any specific point on the curve. We have studied the meaning of slope, as in the slope of a line. What if we were studying a curve and we wanted to find the slope of the curve at a specific point. Remember that the slope of the curve is constantly changing as we move up or down the curve.
This relates to the problem of finding the velocity and acceleration of a body in motion given the formula for its distance travelled vs time. It turns out that if we have the formula for the distance a body travels in any given time, call this s(t), the velocity of the body at any time t is just the slope of the graph of s(t) at time t. We can call this velocity s'(t). If we plot the graph of s'(t) we can then measure its slope and get the acceleration of the body at any time t. We can call this s"(t). So, as is obvious to any newt, the velocity of the body at any time t is just the slope of s(t) at that time t. The acceleration of the body at any time t is no more than the slope of the slope of s(t) at any time t.
The way to find the velocity of the body at any time t is to differentiate s(t) and then substitute in the specific value of t which we want to evaluate s'(t) at. To find the acceleration of the body at any time t we simply differentiate s(t) twice, yielding s"(t), and then substitute in the value of t which we wish to evaluate. What could be easier.
(2) To find the length of a curve, the area of a region or the volume of a solid. These curves, regions and solids must be able to be represented at functions, not just as pictures.
(3) To find the maximum or minimum value of a quantity which we have the function for. We did this to some degree in math 30 when we looked at a curve and tried to find the maximum or minimum value or the curve. Calculus makes this easier, as we would expect.
Differentiation
Calculus can be divided into two main areas: differentiation and integration. Integration is the opposite of differentiation.
Differentiation is the process of finding the slope of a curve at a specific point. There are rules for differentiation but the theory behind it is simple. When we differentiate a function we are finding what is called the derivative of the function. Below, we will find out where the derivative of function f comes from. Again, the derivative of a function is what you get when you differentiate the function. It all comes from the study of limits of functions. Integration comes from the study of limits also. We will look at integration later but it does make you want to scream, "Why is calculus so easy!!! How can I impress my friends by studying calculus when calculus is so easy!! Arrgggghhhh!!!!". My advice to you is, "Sshhh, don't tell them".
