During the seventeenth century, European mathematicians were at work on four major problems. These four problems gave birth to the subject of Calculus. The problems were the tangent line problem, the velocity and acceleration problem, the minimum and maximum problem, and the area problem. Each of these four problems involves the idea of limits.
The tangent line problem
There is a given function (f), and a point (P} on its graph. The idea of this problem is to find the equation of the tangent line to the graph at that point. This problem is equivalent to finding the slope of the tangent line at that point. This may be approximated by using a line through the point of tangency and a second point on the curve (Q)—this gives us a secant line.
As point Q approaches point P, the secant line will become a better and better approximation of the tangent line. This uses the concept of limits—the limit as Q approaches P will give you the slope of the tangent line. In other words, choosing points closer and closer to the point of tangency would give you more accurate approximations. The derivative of a function gives us the slope of the tangent line to the function.
Although partial solutions to this problem were given by Pierre de Fermat (1601-1665), Rene Descartes (1596-1650), Christian Huygens (1629-1695), and Isaac Barrow (1630-1677), credit for the first general solution is usually given to Sir Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716).
The velocity and acceleration problem
The velocity and acceleration of a particle can be found by using Calculus. This was one of the problems faced by mathematicians in the seventeenth century. The derivative of a function can not only be used to determine slopes, but also to determine the rate of change between two variables. This may be used to describe the motion of an object moving in a straight line. This is the position function, which, if differentiated (or the derivative of it is found) gives us the velocity function. In other words, the velocity function is the derivative of the position function. You may also find the acceleration function by finding the derivative of the velocity function. So the velocity and acceleration problem helped in the development of Calculus.
The minimum and maximum problem
What if we want to examine a function by finding where it is increasing? Where it is decreasing? What is the behavior of its concavity? When does it have a maximum point? Where does it have a minimum point? All of these questions were answered with the development of Calculus. The minimum or maximum of the function must occur at a critical point, or a critical number. If we find the derivative of a function, its zeros are called critical numbers.
Now, we must analyze the behavior of the function. The values over which the derivative is positive equates into the actual function increasing. When the derivative is negative, the function is decreasing. If the function is increasing, and then changes to decreasing, that point is a relative maximum of the function. Similarly, if the function is decreasing, and then changes to increasing, that point is a relative minimum. An easier way to analyze the minimum and maximum problem is to graph the derivative. If the point to the left of the critical number is a negative, and the point to the right of it is a positive, then the critical number is a minimum of the function. Similarly, if the point to the left of the critical point is a positive, and the point to the right is a negative, the point is a maximum of the function.
We may also analyze concavity. If the second derivative of the function is positive over a given interval, then the function is concave up over that given interval. If the second derivative is negative, then the function is concave down.
The area problem
This classic Calculus problem is used to find the area of a plane region that is bounded by the graphs of functions. Like the tangent line problem, the limit concept is applied here. To approximate the area of the plane region underneath the graph, one may break the region up into several rectangles, and sum up the values of the rectangles. This method is a form of the Riemann Sums. This would give an approximation of the area of the graph. Now, if the amount of rectangles is increased, the approximation will become more and more precise. The area will therefore be the sum of the areas of the rectangles as the number of rectangles increases without bound. In other words, the limit as the number of rectangles approaches infinity, will give you the area of the region. This eventually leads into the idea of integration.