Everything Maths Grade 12

7.6 CHAPTER 7. DIFFERENTIAL CALCULUS

taneous rate of change isthe derivative. When average rate of change is required, it will be specifically
referred to as average rate of change.

Velocity is one of the most common forms of rate of change. Again, average velocity = average rate
of change and instantaneous velocity = instantaneous rate of change = derivative. Velocity refers to
the increase of distance(s) for a correspondingincrease in time (t). Thenotation commonly used for
this is:

v(t) =

ds

dt

= s�(t)

where s�(t) is the position function.Acceleration is the change in velocity for a corresponding increase
in time. Therefore, acceleration is the derivativeof velocity

a(t) = v�(t)

This implies that acceleration is the second derivative of the distance(s).

Example 15: Rate of Change

QUESTION

The height (in metres) of a golf ball that is hit into the air after t seconds, is given by h(t) =

20 t− 5 t^2. Determine

1. the average velocityof the ball during the first two seconds


2. the velocity of the ball after 1 ,5 s


3. the time at which thevelocity is zero


4. the velocity at whichthe ball hits the ground


5. the acceleration of the ball

SOLUTION

Step 1 : Average velocity

vave =

h(2)−h(0)

2 − 0

=

[20(2)− 5(2)^2 ]− [20(0)− 5(0)^2 ]

2

=

40 − 20

2

= 10 m· s−^1

Step 2 : Instantaneous Velocity

v(t) =

dh

dt

= 20− 10 t