MATHEMATICAL TOOLS
Function is a rule of relationship between two variables in which one is assumed to be dependent and the
other independent variable, for example :
MEASUREMENT OF ANGLE AND RELATIONSHIP BETWEEN DEGREES AND
RADIAN
An angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies
along the positive x-axis (Fig.). Angles measured counterclockwise from the positive x-axis are assigned
positive measures ; angles measured clockwise are assigned negative measures.
The trigonometric function of a general angle q are defined in terms of x, y, and r.
RULES FOR FINDING TRIGONOMETRIC RATIO OF ANGLES GREATER THAN 90°
Step 1 ® Identify the quadrant in which angle lies.
sin 2q = 2 sin q cos q ; cos 2q = cos2 q – sin2q = 2cos2 q – 1 = 1 – 2sin2 q
cos2 q =
2
1+ cos2q ; sin2 q =
2
1– cos2q
4. sine rule for triangles 5. cosine rule for triangles
The finite difference between two values of a physical quantity is represented by D notation.
For example
Another name for differentiation is derivative. Suppose y is a function of x or y = f(x)
Differentiation of y with respect to x is denoted by symbol f ’(x)
It is the tan of angle made by a line with the positive direction of x-axis, measured in anticlockwise
direction.
Given an arbitrary function y = f(x) we calculate the average rate of change of y with respect to x over the
interval (x , x + Dx) by dividing the change in value of y, i.e. Dy = f(x + Dx) – f(x), by length of interval Dx over
which the change occurred.
We know that, average rate of change of y w.r.t. x
The geometrical meaning of differentiation is very much useful in the analysis of graphs in physics. To understand
the geometrical meaning of derivatives we should have knowledge of secant and tangent to a curve
RULE NO. 1 : DERIVATIVE OF A CONSTANT
While the derivative of the sum of two functions is the sum of their derivatives, the derivative of the product
of two functions is not the product of their derivatives. For instance,
Just as the derivative of the product of two differentiable functions is not the product of their derivatives, the
derivative of the quotient of two functions is not the quotient of their derivatives.
Because sin x and cos x are differentiable functions of x , the related functions
We now know how to differentiate sin x and x2 – 4, but how do we differentiate a composite like sin (x2 – 4)?
The answer is, with the Chain Rule, which says that the derivative of the composite of two differentiable
functions is the product of their derivatives evaluated at appropriate points
POWER CHAIN RULE
RADIAN VS. DEGREES
We can interpret f ’’ (x) as the slope of the curve y = f ’(x) at the point (x, f ’(x)). In other words, it is the rate
of change of the slope of the original curve y = f (x)
APPLICATION OF DERIVATIVES
DIFFERENTIATION AS A RATE OF CHANGE
Suppose a quantity y depends on another quantity x in a manner
shown in the figure. It becomes maximum at x1 and minimum at
x2. At these points the tangent to the curve is parallel to the x-
axis and hence its slope is tan q = 0. Thus, at a maximum or a
minimum,
In mathematics, for each mathematical operation, there has been defined an inverse operation.
For example- Inverse operation of addition is subtruction, inverse operation of multiplication is division and
inverse operation of square is square root. Similarly there is a inverse operation for differentiation which is
known as integration
INTEGRAL FORMULAS
Indefinite Integral Reversed derivative formula
RULE NO. 1 : CONSTANT MULTIPLE RULE
RULE NO. 3 : RULE OF SUBSTITUTION
DEFINITE INTEGRATION OR INTEGRATION WITH LIMITS
From graph shown in figure if we divide whole area in infinitely small
strips of dx width.
In physics we deal with two type of physical quantity one is scalar and other is vector. Each scalar quantities
has magnitude
If a physical quantity in addition to magnitude -
has a specified direction.
Unit vector is a vector which has a unit magnitude and points in a particular direction. Any vector (A
)
can be written as the product of unit vector (A ˆ ) in that direction and magnitude of the given vector.
MULTIPLICATION OF A VECTOR BY A SCALAR
ADDITION OF VECTORS
Addition of vectors is done by parallelogram law or the triangle law
Resolution along rectangular component :
It is convenient to resolve a general vector along axes of a rectangular
coordinate system using vectors of unit magnitude, which we call as unit
It is always a scalar which is positive if angle between the vectors is acute (i.e. < 90º) and negative if angle between
them is obtuse (i.e. 90º < q £ 180º)
Here q is the angle between the vectors and the direction nˆ is given by the right-hand-thumb rule.
Vector product of two vectors is always a vector perpendicular to the plane containing the two vectors i.e. orthogonal
