Calculus :It is the study of rate of change of a function with respect to variation of domain of a function.
Since function can always be drawn in x-y Cartesian plane,the study of function reduces to the study of curves in the x-y Cartesian plane and hence calculus comes into picture to study the rate of change of range with respect to domain.First we will analyses some function and use calculus to study it.
GRAPHS OF FUNCTIONS
If f is a real-valued function of a real variable, then the graph of f in the xy-plane is
defined to be the graph of the equation y = f(x). For example, the graph of the function
f(x) = x is the graph of the equation y = x, shown in Figure 0.1.4. That figure also shows
the graphs of some other basic functions that may already be familiar to you.
If f is a real-valued function of a real variable, then the graph of f in the xy-plane is
defined to be the graph of the equation y = f(x). For example, the graph of the function
f(x) = x is the graph of the equation y = x, shown in Figure 0.1.4. That figure also shows
the graphs of some other basic functions that may already be familiar to you.
Graphs can provide valuable visual information about a function. For example, since
the graph of a function f in the xy-plane is the graph of the equation y = f(x), the points
on the graph of f are of the form (x, f(x)); that is, the y-coordinate of a point on the graph
of f is the value of f at the corresponding x-coordinate
the graph of a function f in the xy-plane is the graph of the equation y = f(x), the points
on the graph of f are of the form (x, f(x)); that is, the y-coordinate of a point on the graph
of f is the value of f at the corresponding x-coordinate
The y-coordinate of a point on the graph of y = f(x) is the value of f at the corresponding x-coordinate.
The values of x for which f(x) = 0 are the x-coordinates of the points where the graph of f intersects the
x-axis (Figure 0.1.6). These values are called the zeros of f , the roots of f(x) = 0, or the x-intercepts of the graph of y = f(x).
x-axis (Figure 0.1.6). These values are called the zeros of f , the roots of f(x) = 0, or the x-intercepts of the graph of y = f(x).
Zeros of a function is the domain where the function intersect the x axis.
So looking at the above figure we can say x=x1, x=0,x=x2 and x=x3 are the zeros of the function means
f(x1)=f(0)=f(x2)=f(x3)=0.
and x1,0,x2,x3 are called solution of the equation y=f(x)=0.
The graph of the equation x2 + y2 = 25 is a circle of radius 5 centered at the origin which is not a function because a function for unique domain should have unique range but here for one x, y1=y2=y.
THE ABSOLUTE VALUE FUNCTION
Recall that the absolute value or magnitude of a real number x is defined by
|x| = x, x ≥ Recall that the absolute value or magnitude of a real number x is defined by
−x, x < 0 , its corresponding graph is as shown in the following figure.
many to one function and symmetrical about x axis which property is called even function ie
f(x)=f(-x) , even function.
Another example of even function is cosine function which is symmetrical about y-axis such that f(x)=f(-x).
Example:Sketch the graph of the function defined piecewise by the formula
The formula for f changes at the points x = −1 and x = 1. (We call these the breakpoints for the formula.) Agood procedure for graphing functions defined piecewise is to graph the function separately over the open intervals determined by the breakpoints, and then graph f at the breakpoints themselves. For the function f in this example the graph is the horizontal ray y = 0 on the interval (−infinity ,−1], it is the semicircle y = √1 − x2 on the interval (−1, 1), and it is the ray y = x on the interval [1,+infinity ). The formula for f specifies that the equation y = 0 applies at the breakpoint −1 [so y = f(−1) = 0], and it specifies that the equation y = x applies at the breakpoint 1 [so y = f(1) = 1]. The graph of f is shown in
the function is discontinuous at x=1.
DOMAIN AND RANGE
Find the domain of this function.
By observing the graph of arccos(x) function where -1 <=x<=1.
Slope of a function.
This is an example of increasing function whose slope =Tan(theta)>0.so we call it an increasing function.
