Section 3.6
1.Let f (t) = t 2 . Find the relative rate of change of this function.
a. The relative rate of change RRC = f’ (t)/f (t).
RRC = 2t/t 2 = 2/t.
b. Evaluate the relative rate of change when t = 1.
RRC (1) = 2/1 = 2
ln x x 2
c. Evaluate the relative rate of change when t = 10.
RRC (1) = 2/10 = 0.2
2. Let
f (t)  e
t2
Find the relative rate of change of this function.
a. The relative rate of change RRC = f’ (t)/f (t).
RRC 
2te
e
t2
t2
 2t
b. Evaluate the relative rate of change when t = 10.
RRC (10) = 2 10 = 20
3. Let f (t) = 25 (t – 1) .
a. Find the relative rate of change of this function.
This function is f (t) = 25 (t – 1) 1/2 and the relative rate of change RRC = f’ (t)/f (t).
1
1
25  (t  1) 2 (1)
1
2
RRC 

1
2(t  1)
25(t  1) 2
b. Evaluate the relative rate of change when t = 6.
RRC (6) = 1/10 = 0.1
4. ECONOMICS: National Debt. If the national debt of a country (in trillions of
dollars) t years from now is given by the following function, find the relative rate of
change of the debt 10 years from now.
N (t) = 0.5 + 1.1 e 0.01t
The relative rate of change RRC = f’ (t)/f (t).
(1.1)(0.01)e0.01t
RRC 
0.5  1.1e0.01t
(1.1)(0.01)e0.1 0.01215688
RRC(10) 

 0.00709 or 0.71%
0.1
0.5  1.1e
1.71568801
OR use your calculator. Graph the function and find
RRC = f ’ (10)/f (10)
RRC(10) = 0.01215688/1.715688 = 0.0071 = 0.71%
5. GENERAL: Population. The population (in millions) of a city t years from now is given
by
P (t) = 4 + 1.3 e 0.04t .
a. Find the relative rate of change of the population 8 years from now.
The relative rate of change RRC = f’ (t)/f (t). You may use your calculator for this.
See problem 4.
(1.3)(0.04)e0.04t
RRC 
4  1.3e0.04t
(1.3)(0.04)e(0.04)(8) 0.071610643
RRC (8) 

 0.0124 OR 1.24%
(0.04)(8)
4  1.3e
5.790266094
b. Will the relative rate of change ever reach 1.5%?
(1.3)(0.04)e0.04t
Will
ever equal 0.015?
0.04t
4  1.3e
Graph it and look.
In about 15.3 years.
6. For the demand function, D (p) = 200 – 5p;
a. Find the elasticity of demand E (p)
E(p) 
E(p) 
 p  D'(p)
D(p)
 p 5
5p

200  5p 200  5p
b. Determine whether the demand is elastic, inelastic, or unit elastic at
a price of p = 10.
E(10) 
5  10
50

 0.33
200  5  10 150
Demand is inelastic.
7. For the demand function, D (p) = 300 – p 2;
a. Find the elasticity of demand E (p)
E(p) 
 p  D'(p)
D(p)
 p   2p
2p 2
E(p) 

2
300  p
300  p 2
b. Determine whether the demand is elastic, inelastic, or unit elastic at
a price of p = 10.
2  102
200
E(10) 

1
300  102 200
Demand is unitary.
8. For the demand function, D (p) = 300/p;
a. Find the elasticity of demand E (p)
E(p) 
 p  D'(p)
D(p)
NOTE: D (p) = 300 p – 1
 p   300p  2
E(p) 
1
300p  1
b. Determine whether the demand is elastic, inelastic, or unit elastic at
a price of p = 4.
E(4)  1
Demand is unitary.
9. For the demand function, D (p) = 100/p 2 ;
a. Find the elasticity of demand E (p)
E(p) 
 p  D'(p)
D(p)
NOTE: D (p) = 100 p – 2
 p   200p  3
E(p) 
2
2
100p
b. Determine whether the demand is elastic, inelastic, or unit elastic at
a price of p = 10.
E(40)  2
Demand is elastic.
10. AUTOMOBILE SALES - An AUTOMOBILE DEALER IS SELLING CARS AT A PRICE OF
$12,000. The demand function is D(P) = 2(15 – 0.001P)2, where p is the price of a car.
Should the dealer raise or lower the price to increase the revenue?
E(p) 
 p  D'(p)
D(p)
E(p) 
 p  4  (15  0.001p)(  0.001)
0.002p

2(15  0.001p)2
15  0.001p
E(12000) 
0.002(12000)
24

8
15  0.001(12000)
3
Demand is elastic, lower the price.
11. CITY BUS REVENUES – The manager of a city bus line estimates the demand function
to be D (p) = 150,000 (1.75 – p) ½, where p is the fare in dollars. The bus line currently
charges a fare of $1.25, and it plans to raise the fare to increase its revenues. Will the
strategy succeed?
E(p) 
 p  D'(p)
D(p)
 p  150000 1 2  (1.75  p)  1 2 ( 1)
p
E (p) 

2  (1.75  p)
150000(1.75  p)1 2
E (1.25) 
1.25
 1.25
2  (1.75  1.25)
Demand is elastic, the strategy will not work.
12. OIL PRICES – A European oil-producing country estimates that the demand for
its oil (in millions of barrels per day) is D (p) = 3.5 e – 0.06p, where p is the price of a
barrel of oil. To raise its revenues, should it raise or lower its price from its current
level of $120 per barrel?
E(p) 
 p  D'(p)
D(p)
 p  3.5e 0.06p  ( 0.06)
E (p) 
 0.06 p
3.5e 0.06p
E (120)  0.06 (120)  7.2
Demand is elastic, lower the price.