Answer the following questions related to the concept of instantaneous velocity:
a. (a) Define instantaneous velocity and write its mathematical expression in terms of displacement x and time t.
b. (b) Starting from the formula for average velocity, explain mathematically how the instantaneous velocity is obtained by taking an appropriate limit.
c. (c) A displacement–time graph for a particle is a smooth curve. Describe why the slope of the tangent drawn at a point on this graph represents the instantaneous velocity of the particle at that instant.
d. (d) Give one example of a situation in daily life where the distinction between average velocity and instantaneous velocity is important, and justify your choice briefly.