*Graph of logarithmic function
The function y = logb x is called a logarithmic function, it is not defined for x ≤ 0.
It is a decreasing function for 0 < b & increasing for b > 1. The y – axis is an asymptote of the curve
y = logb x. The x intercept is the point (1, 0).
*Parabola
Parabola is an open figure & can be defined as the locus of a point P which moves in a plane so that its distance from a fixed point (called Focus, F) bears a constant ratio (called eccentricity, e = 1) to its distance from a fixed straight line (called directrix).
*Shape of y versus x graph
Let a physical quantity y depends on other physical quantity x, this dependence is called a mathematical function & usually expressed as
y = f (x) (read as y is a function of x).
The dependence of y on x can be visualised by drawing a graph of function y = f (x). While plotting graphs the independent varaible is taken on horizontal line & dependent on vertical line. Shape of y versus x graph depends on the type of function.
Following types of y versus x graphs are frequently encountered in Physics.
*Characteristic & Mantisa
The integral part of the logarithm is called Characteristic and the decimal part called Mantisa. Characteristic may be positive, zero or negative. Negative characteristic is, it is represented it with a bar. The mantisa should be positive, if not, we try to make it by rearrangement as illustrated below
log10 N = – 4.5678,
= (- 4 -1) + (1- 0.5678)
= – 5 + (0.4322).
= .4322
*Arithmetic operations
*Bounded functions
If a function attains finite values only, then it is said to be bounded otherwise unbounded.
Only sinθ & cosθ are bounded functions, secθ, cosθ, tanθ & cotθ are unbounded.
*Transformation formulae
sinA + sinB = 2sinC cosD
sinA – sinB = 2cosC sinD
cosA + cosB = 2cosC cosD
cosA – cosB = – 2sinC sinD
^A long horizontal fixed wire carries a current of 100 A. Directly above and parallel to it is another wire carrying a current of 20 A and weighing 0.04 N/m. The separation between the two wires so that the upper wire is just supported by the magnetic repulsion is
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