Rates of Change
Suppose that the dependent variable y is a function of the variable t.
Absolute
Relative ∆y
∆y
∆y ∆t =
y
Average
∆t
y
∆t
dy
y0
dy
Instantaneous
where y 0 =
dt
y
dt
If y = f (t) and if, as t changes from t = t1 to t = t2 , we have that y changes from y1 = f (t1 ) to
y2 = f (t2 ), then ∆t = t2 − t1 and ∆y = y2 − y1 , so these rates of change are calculated as follows:
Absolute
Relative
y2 − y1
y2 − y1 y2 − y1
y1
Average
(t2 − t1 ) =
t
−
t
y
t
−
t
2
1
1
2
1
dy y0
f 0 (t1 )
0
Instantaneous
= f (t1 )
=
dt t=t1
y
f (t1 )
Absolute rates of change are expressed in terms of y-units per t-unit; for example, if the amount
of money in an account increases from 1000 dollars to 1400 dollars in ten years, then it increases
at an average rate of 40 dollars per year during that particular ten-year interval. This will be the
average rate when the actual amount of money in the account during the ten-year interval varies
in any way whatsoever as long as the starting value and ending value are as specified. The average
rate of change corresponds to the constant rate of change (slope) of a linear function with the given
starting and ending values.
Relative rates of change are often expressed in terms of the percentage change of y per unit
change in t; for example, if the amount of money in an account goes from 1000 dollars to 1400
dollars during a ten-year period, then the amount increases at an average rate of 400 dollars per
1000 dollars invested per 10 years, or 40 percent per decade, which could be expressed as an average
relative growth rate of 4 percent per year during that particular ten-year interval.
Caution: But this does not correspond to growth at a rate of 4 percent per year as usually
.
understood. At a rate of 4%/yr, $1000 would grow to 1000(1 + 0.04)10 = 1480.24 dollars, by
the compound amount formula.. So what is the equivalent constant instantaneous relative rate of
growth that yields $1400 after 10 years? Let r be this rate. Then
dy/dt
= r,
y
so
y = y0 ert = 1000ert .
.
Thus 1400 = 1000er·10 , whence r = (1/10) ln(1.400) = 0.0336, or 3.36% per year. In actuarial
science, this instantaneous relative rate of change is known as the force of interest. In financial
situations, this would typically be expressed instead in terms of the APR, the annual percentage rate. Let a = the APR expressed as a decimal. Then by the compound amount formula,
.
1400 = 1000(1 + a)10 , so a = 1.4001/10 − 1 = 0.0342, or 3.42% per year.
1
Percentage v. percentage points: Suppose 30% of likely voters favored our candidate in
the previous poll. If support for our candidate increases by 25 percent, then 37.5% of likely voters
favor our candidate, while if support for our candidate increases by 25 percentage points, then 55%
of likely voters favor our candidate.
The instantaneous rates are the limits, as ∆t → 0, of the corresponding average rates of change.
For example, if water is pouring into a tank so that there are y = f (t) = 10(1 − e−2t ) liters in
the tank at time t seconds, then at time t = 0.5 second the rate at which the volume of water is
increasing is f 0 (0.5) = 10(−e−2·(0.5) · (−2)) = 20e−1 ≈ 7.358 liters per second. The relative rate
of increase at time t = 0.5 is f 0 (0.5)/f (0.5) = 20e−1 /[10(1 − e−2·(0.5) )] ≈ 7.358/6.321 ≈ 1.164, or
about 116.4 percent per second. Note that an increase of 100 percent corresponds to a doubling of
the amount.
Key facts:
If the instantaneous rate of change of a quantity is constant over an interval, then the quantity is
a linear function of the independent variable:
y 0 = c ⇒ y = ct + b.
If the instantaneous relative rate of change of a quantity is constant over an interval, then the
quantity is an exponential function of the independent variable:
y 0 /y = c ⇒ y = y0 ect .
Exercises. Given
Year t Amount A
2012
1,000.00
2013
1,030.00
2014
1,060.90
2015
1,092.73
2016
1,125.51
2017
1,159.27
(1) Calculate these five-year average rates of change:
(a) Absolute
(b) Relative
(c) APR
(2) A table does not provide enough information to calculate exact instantaneous rates of change
with certainty, but we may use table values to estimate instantaneous rates. Estimate the
following instantaneous rates of change at time t = 2014:
(a) Absolute
(b) Relative
(3) We can calculate instantaneous rates from a formula. Here the table amounts come from
the formula A = 1000(1.03)t−2012 . Using this formula, calculate to six decimal places
(a) the absolute instantaneous rate of change of A with respect to t at t = 2014.
(b) the relative instantaneous rate of change of A at t = 2014 as a percentage/year.