Cambridge Additional Mathematics

368 Applications of differential calculus (Chapter 14)

Opening problem

Michael rides up a hill and down the other side to his friend’s house. The dots on the graph show

Michael’s position at various timest.

The distance Michael has travelled at various times is given by the function

s(t)=1: 2 t^3 ¡ 30 t^2 + 285t metres for 06 t 619 minutes.

Things to think about:

a Can you find a function for Michael’sspeedat any timet?

b Michael’saccelerationis the rate at which his speed is

changing with respect to time. How can we interpret

s^00 (t)?

c Can you find Michael’s speed and acceleration at the time

t=15minutes?

d At what point do you think the hill was steepest? How

far had Michael travelled to this point?

In the previous chapter we saw how to differentiate many types of functions.

In this chapter we will use derivatives to find:

² tangents and normals to curves

² turning points which are local minima and maxima.

We will then look at applying these techniques to real world problems including:

² kinematics (motion problems of displacement, velocity, and acceleration)

² rates of change

² optimisation (maxima and minima).

DEMO

s( )m

2500

2000

1500

1000

500

O 5 10 15 t(min)

st( ) = 1 2 - 30 + 285. t 32 t t

t=0 t=5

t=10

t=15 t=17 t=19

Michael’s place friend’s house

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\368CamAdd_14.cdr Wednesday, 8 January 2014 10:10:35 AM BRIAN