This document discusses the concept of derivatives and rates of change. It provides examples of calculating first and second derivatives of functions, using derivatives to find velocity and acceleration from displacement-time graphs, and using derivatives to determine rates of change in various contexts like population growth. The key ideas are that the derivative represents the rate of change of one variable with respect to another, and it can be used to calculate velocities, accelerations, and other instantaneous rates of change from functional relationships.

1. Gradient as rate of change
By the end of the lesson you will be able to:
• Calculate the second derivative of a function.
• Understand and use the derivative as rate of
change.

2. also represents the second derivative of y
with respect to x.
The derivative of f ' (x) is denoted f ''(x) and it is
called the second derivative of f(x) with respect to x.

3. Given y = 3 x4
­ 5 x, find the first and second
derivatives of y.

4. Find f ''(2) if

5. A ball is thrown from the top of a building. The
distance it has fallen after t seconds is recorded and
the following graph of distance against time is
obtained.
Calculate the average speed in the time
interval 2 ≤ t ≤4.
distance travelled
time taken
average speed =

6. Calculate the average speed in the time interval 2 ≤ t ≤4.

7. Calculate the average speed in the time interval 2 ≤ t ≤3.
How can we calculate the speed at t = 2 sec ?

8. The velocity of the ball at any instant in time is given by
the gradient of the displacement­time graph.

9. The gradient represents the rate of change of
displacement with respect to time.
rate of change derivative
rate: comparison between two quantities with different units
gives the rate of change in y with respect to x.

10. If displacement is s(t)
instantaneous velocity
v (t) = s ' (t) =
instantaneous acceleration
a(t) = v ' ( t )=
Velocity is the rate of change of
displacement with respect to time.
Acceleration is the rate of change of
velocity with respect to time.

11. The displacement of a body at time t seconds is
given in metres by s= 2t3
­ t2
+ 2 . Find:
a) the velocity of the body at time t.
b) the initial velocity of the body.
c) the acceleration after 1 second.
d) the time at which the acceleration is zero.

12. The derivate represents the rate of change of a
variable with respect to another variable.
Rate of change area circle.ggb

13. The derivate represents the rate of change of a
variable with respect to another variable.

14. The number of deer , N, in a reserve is modelled by
the function where t is the number
of years after the reserve started. Find the rate of
change at which the population is increasing 2 years
after the program started.

15. Book page 366: Ex 12D
Ex. 1 to 9

16. Attachments
Rate of change area circle.ggb