Cambridge Additional Mathematics

GRAPHING

PACKAGE

Applications of differential calculus (Chapter 14) 389

c

dA

dt

=^16 t

¡^12

)

d^2 A

dt^2

=¡ 121 t

¡^32

=¡

1

12 t

p

t

)

d^2 A

dt^2

< 0 for all 56 t 618 :

This means that while the ability to understand spatial concepts increases with age, the rate of

increase slows down with age.

You are encouraged to use technology to graph each function you need to consider.

This is often useful in interpreting results.

EXERCISE 14D

1 The estimated future profits of a small business are given byP(t)=2t^2 ¡ 12 t+118 thousand dollars,

wheretis the time in years from now.

a What is the current annual profit? b Find dP

dt

and state its units.

c Explain the significance of

dP

dt

.

d For what values oftwill the profit:

i decrease ii increase on the previous year?

e What is the minimum profit and when does it occur?

f Find

dP

dt

when t=4, 10 , and 25. What do these figures represent?

2 The quantity of a chemical in human skin which is responsible for its ‘elasticity’ is given by

Q= 100¡ 10

p

t wheretis the age of a person in years.

a FindQat:

i t=0 ii t=25 iii t= 100years.

b At what rate is the quantity of the chemical changing at the age of:

i 25 years ii 50 years?

c Show that the quantity of the chemical is decreasing for all t> 0.

3 The height ofpinus radiata, grown in ideal conditions, is

given by H=20¡

97 : 5

t+5

metres, wheretis the number of

years after the tree was planted from an established seedling.

a How high was the tree at the time of its planting?

b Find the height of the tree after 4 , 8 , and 12 years.

c Find the rate at which the tree is growing after 0 , 5 , and

10 years.

d Show that

dH

dt

> 0 for all t> 0.

What is the significance of this result?

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_14\389CamAdd_14.cdr Monday, 7 April 2014 10:46:31 AM BRIAN