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Basic Calculus

Basic Calculus refers to the simple applications of both differentiation and integration. Before delving into these processes, one must fully understand functions and their corresponding graphs. Y= f(x) would indicate that y is a function of x – this means that the quantity y is dependent on the quantity x, with the condition that one value of x corresponds to only one value of y. In this light, y=f(x) is the dependent variable, while x is the independent variable. Since every x has only one y, they can be considered as an ordered pair. The graph of the function is just an illustration of all the ordered pairs on a xy-plane.

The process of differentiation and integration deals with these functions to analyze the changing relationship between input and output. Consider the following function:

Y = 3x^2

The basic process of differentiation involves finding how y changes with respect to x. This rate of change is known as the derivative (dy/dx). To find the derivative, we differentiate both sides (on the right hand side multiply the coefficient by the exponent, and then subtract one from the exponent):

Y=3x^2

(d/dx)Y = (d/dx)3*x^2

dy/dx = 3*(d/dx)x^2

dy/dx = 3*(2x)

dy/dx = 6x

Thus, it can be seen that the derivative is 6x. So if the value of x is 1, then the rate of change is 6(1) = 6 units. On the other hand, the basic process of integration involves finding F from f(x). To find the integral, we integrate both sides (on the right hand side add one to the exponent, and then divide the function by the new exponent):

Y = 3x^2

F = int(3x^2)

F = (3x^3)/3

F = x^3

Since, integration deals with the accumulation of quantities, using the above function we can figure out the amount that has accumulated between two x values. For example between x=1 and x=3, the amount that has accumulated is 26 units for this given function:

F(3)-F(1) = 27-1 = 26

Thus, understanding basic calculus may prove to be a very practical tool to possess when dealing with changing scenarios.

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