There are countless examples in the real world where quantities vary with time, or with respect to some
other variable.
For example: ² temperature varies continuously
² the height of a tree varies as it grows
² the prices of stocks and shares vary with each day’s trading.
We have already seen that if y=f(x) then f^0 (x) or
dy
dx
is the gradient of the tangent to y=f(x) at
the given point.
dy
dx
gives therate of change inywith respect tox.
We can therefore use the derivative of a function to tell us therateat which something is happening.
For example:
²
dH
dt
or H^0 (t) could be the instantaneous rate of ascent of a person in a Ferris wheel.
It might have units metres per second or m s¡^1.
²
dC
dt
or C^0 (t) could be a person’s instantaneous rate of change in lung capacity.
It might have units litres per second or L s¡^1.
Example 15 Self Tutor
According to a psychologist, the ability of a person to understand spatial concepts is given byA=^13
p
t
wheretis the age in years, 56 t 618.
a Find the rate of improvement in ability to understand spatial concepts when a person is:
i 9 years old ii 16 years old.
b Show that
dA
dt
> 0 for 56 t 618. Comment on the significance of this result.
c Show that
d^2 A
dt^2
< 0 for 56 t 618. Comment on the significance of this result.
a A=^13
p
t=^13 t
1
2
)
dA
dt
=^16 t
¡^12
=
1
6
p
t
i When t=9,
dA
dt
= 181
) the rate of improvement is
1
18 units per year for a^9 year old.
ii When t=16,
dA
dt
= 241
) the rate of improvement is
1
24 units per year for a^16 year old.
b Since
p
tis never negative,
1
6
p
t
is never negative
)
dA
dt
> 0 for all 56 t 618.
This means that the ability to understand spatial concepts increases with age.
D Rates of change
388 Applications of differential calculus (Chapter 14)
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\388CamAdd_14.cdr Monday, 7 April 2014 10:43:09 AM BRIAN