Going Graphically

You can go graphically to analyze a set of data and underlying process.

Graphical methods are pictorial and readily surveyable.

Following examples show use of graphic methods in solving different problems.

In many applications in calculus, the behavior of a function(or behavior of certain physical quantity) is studied from its graph.

When the value of a function increases as the value of the independent variable increases, the lines that are tangent to the graph of the function have positive slope, and the function is said to be increasing.

Conversely, when the values of the function decrease with increasing values of the independent variable, the tangent lines have negative slope, and the function is said to be decreasing.

Precisely at the point where the function changes from increasing to decreasing or from decreasing to increasing the tangent line is horizontal(has slope 0) and the derivative is zero.

The terms maxima and minima refer to extreme values of a function ,that is , the maximum and minimum values the function attains.

Functions may have relative(or local) maximum and minimum. Relative means relative to local or nearby values of the function.

Finding the maxima and minima, both absolute and relative of various functions represents an important class of problem and we come across such type of problems in economics, business and engineering where it is desired to find maximum or minimum value of a particular quantity.

Refer to my blog “Best way to do something” 28^th Feb 2011(https://blogs.siliconindia.com/Academics)

Another example of going graphically is finding solution of algebraic equations.

Graphical representation of function makes it easy to find an approximate solution to an equation in one unknown or to a system of two equations in two unknowns.

The solution of a system of two equations, is found by reading coordinates of points common to both the curves(intersection of two curves describing two different phenomena) and yields very useful information.

Similarly graphical solution of an inequality can be seen in pictorial way.

The advantage of a graphical representation compared to tables lies in its pictorialness and surveyability. A proper choice of scale is very important here. Finding intermediate values of a function from its graph is called interpolation and widely used.

Many cases the shape of the graph or shape of portion of the graph tells about important behavior of physical quantity. For example graph of direct proportionality is a straight line.

Graphs are compared with standard shapes viz: concave, convex, exponential , parabola, hyperbola ,elliptical, periodic. These show important  features which are  valuable for analyzing the process.

Expressing location of points choosing  an appropriate   coordinate system viz: cartesian, spherical, cylindrical and transformation from one coordinate system to another often simplifies the problem and problem becomes easy and surveyable.