Chapter 1 : Introduction 11
- If the differential coefficient of a function is zero, the function is either maximum or mini-
mum. Conversely, if the maximum or minimum value of a function is required, then differ-
entiate the function and equate it to zero.
1.23. INTEGRAL CALCULUS
- ∫dxis the sign of integration.
2.17
;^6
17n
xdxn xxxdx
n+
==
∫∫+(i.e., to integration any power of x, add one to the power and divide by the new power).- ∫∫77;dx==x C dx Cx
(i.e., to integrate any constant, multiply the constant by x).
4.()^1
()
(1)n
ax bndx ax b
na+ +
+=
∫ +×(i.e., to integrate any bracket with power, add one to the power and divide by the new power
and also divide by the coefficient of x within the bracket).
1.24.SCALAR QUANTITIES
The scalar quantities (or sometimes known as scalars) are those quantities which have magnitude
only such as length, mass, time, distance, volume, density, temperature, speed etc.
1.25.VECTOR QUANTITIES
The vector quantities (or sometimes known as
vectors) are those quantities which have both magnitude
and direction such as force, displacement, velocity,
acceleration, momentum etc. Following are the important
features of vector quantities :
- Representation of a vector. A vector is
represented by a directed line as shown in
Fig. 1.2. It may be noted that the length OA
represents the magnitude of the vector OA—→. The
direction of the vector is —OA→is from O (i.e.,
starting point) to A (i.e., end point). It is also
known as vector P. - Unit vector. A vector, whose magnitude is unity,
is known as unit vector. - Equal vectors. The vectors, which are parallel to each other and have same direction (i.e.,
same sense) and equal magnitude are known as equal vectors. - Like vectors. The vectors, which are parallel to each other and have same sense but unequal
magnitude, are known as like vectors. - Addition of vectors. Consider two vectors PQ and RS, which are required to be added as
shown in Fig. 1.3. (a).
Fig. 1.2. Vector (^) OA
→
The velocity of this cyclist is an example
of a vector quantity.