MTH2129: Business Calculus
Tutorial Questions 3
Dr. John O. Mubenwafor
Integral Calculus
Indefinite Integral
1.
Find y if
(a)
dy
=:
dx
4x3
(b)
[Answer: (a) x4 + C (b)
2.
7x3
2x 2
(c)
(d)
1
1
x2
2
1
7 x4
 C (c) x 3  C (d) x   C ]
3
x
4
Integrate the following with respect to x:
3x 4
x  x2
(a)
(b)
(c)
(d)
2
(e)
3  2x
(f)
(g)
4 x3  4 x
(h)
3x 2  5 x  7
(i)
(j)
4 x6  2 x3
(k)
5 x 2  3x 4
(l)
3x
2
x2
7  5 x  3x 2
 3x  4 x3 


x


1
5 x
(n)
x2
3
1
1
3
[Answer: (a) x 5  C (b) x 2  x3  C (c) x 2  C (d) 2x + C
5
2
3
2

2
5
 C (g) x 4  2 x 2  C (h) x 3  x 2  7 x  C
(e) 3x  x 2  C (f)
x
2
5 2
4
1
(i) 7 x  x  x3  C (j) x 7  x 4  C (k) 5x 1  x 3  C
2
7
2
3
10 2
4 3
1
4
3
2
3
x

x

C
x

x

3
x

2
x


C
x
C ]
(l)
(m)
(n)
x
3
3
(m)
3.
4 x3  3x 2  6 x  2 
Evaluate the following
(a)
 5x
(d)
 4x
 (4x
(g)
(j)
0.5
dx
(b)
 8x
dx
(e)
( x
 (9x
1/2
0.2
3
 2x
0.5
 2 x 0.4 ) dx
dx
(h)
(k)
1/3
3
0.5
2
  5 x
1
(c)
5x
 6)dx
(f)
 4 x 0.5 )dx
(i)
(
6
dx
0.5
3
1/4
dx
x 2  4)dx
xdx
 5
1
2 

 x 1.5 dx (l)   3 
dx
5
x

2 x
1

(m)
  3
(p)
 x( x
4 
dx
x
x
2
 1)dx

1 
dx
4 x
(n)
  8
(q)
 ( x  1)( x  2)dx
x
 x 4
dx
2
2
x


(o)
 
(r)
 6x dx
4
 x4  x 
1 x 
(s)
(t)
  x dx
  x3 dx
10 1.5
4 5/4
8
x  C (b) 6x 4/3  C (c)
x  C (d) x3/2  C
[Answer: (a)
3
25
3
2 5/2
3 5/3
10 1.2 10 1.4
x
x  C (h) 6 x1.5  8 x 0.5  C
(e) x  6 x  C (f) x  4 x  C (g)
5
5
3
7
4
2
15 2/3
x  4 x1/2  C
(i) 4x 3/2  C (j) 3x 0.5  C (k) x 0.5  x 0.5  C (l)
5
5
4
16 3/2 1 1/2
x  x  C (o)  x 1/2  2 x 1  C
(m) 2 x 3/2  8 x1/2  C (n)
3
2
3
2
4
2
6
x
3x
x
x
x2 1
(p)

 2 x  C (r) x 5  C (s)
  C (q)
 C
5
4
2
2 x
3
2
2
(t) 2 x1/2  x 3/2  C ]
3
4.
Evaluate the following
(a)
 (3x  5)
(d)
(g)
 (6  x)
 (1  x)
(j)
 (2 x  7)
(m)
 (3
7
dx
(b)
10
dx
(e)
1/2
dx
(h)
3
dx
2  5 x ) dx
(k)
(n)
 (3x  2)
11
dx
3
2
 (4 x  3) dx
 (1  3x) dx
 (4 x  3) dx
1
(c)

(f)
 ( x  3) dx
 (5 x  2) dx
3
(3  7 x)
dx
2
5
(i)
2
(l)
 ( 2 x  8) dx
(o)
 (2 x  3) dx
1
 (4 x  5)
3
dx
4
1
3
3
1
1
3
(3 x  5)8  C (b)
(3 x  2)12  C (c)
(3  7 x) 2/3  C
24
36
14
1
1
2
(4 x  3)5/2  C (f) ( x  3) 1  C (g) (1  x)3/2  C
(d)  (6  x)11  C (e)
11
10
3
1
1
1
1
(3 x  1) 6  C (i)
(5 x  2)5  C (j) (2 x  7) 4  C (k)
(4 x  3)3  C
(h)
18
25
8
12
1
1 1
2
1
 C (o) (2 x  3) 4  C
(l) ( x  8) 4  C (m)  (2  5 x)3/2  C (n)
2
2 2
5
8
8(4 x  5)
[Answer: (a)
2
2.1.2 Integration of Exponential Functions
5.
Integrate each of the following functions with respect to x:
e3 x
e4 x
(a)
(b)
2 e  x
2 e 5 x
(d)
(e)
e(2 x /2)
e45x
(g)
(h)
1
2e (15 x )
e2 x  2 x
(j)
(k)
e
(13 x )
( x 3)
4e
5e
(m)
(n)
(p)
2e x  3e  x
(q)
e x /4  2e x /2
(c)
(f)
(i)
4e  x
e6 x
e(3 x  2)
(l)
6 e 2 x
(o)
e 2 x  e 2 x
4
e3 x  3 x
e
(r)
2e 2 x 1  5e12 x
1
1
2 5 x
1
1 45 x
e (f) e 6 x (g)
e
[Answer: (a) e3 x (b) e 4 x (c) 4e  x (d) 2e  x (e)
4
5
6
5
3
1
2 (15 x )
1
1
e
(h) 2e(2 x /2) (i) e(3 x 2) (j)
(k) e 2 x  3 x (l) 3e 2 x (m) 5e ( x 3)
3
5
2
2e
4 (13 x )
1 2x
5
e
(n)
(o) (e  e 2 x ) (p) 2e x  3e  x (q) 4(e x /4  e x /2 ) (r) e3 x
3
2
3
5
(s) e 2 x 1  e12 x ]
2
(s)
2.1.3 Integration of Logarithmic Functions
6.
Integrate the following with respect to x:
1
1
3x 2
(a)
(b)
(c)
x
2x  5
x3  1
3
1
x 1
(d)
(e)
(f)
2
4  2x
4  3x
x  2x  5
4
1
1
(g)
(h)
(i)
1  2x
2x
3x  1
6
5
4
(j)
(k)
(l)
2  3x
6  7x
1 x
1
3
ln(4  2 x)
[Answer: (a) ln ǀxǀ + C (b) ln(2 x  5) (c) ln( x3  1) (d)
2
2
1
1
1
1
ln(4  3 x) (f) ln( x 2  2 x  5) (g) 2ln(1  2 x) (h) ln(3 x  1) (i) ln( x)
(e)
3
2
3
2
5
ln(6  7 x) (l) 4ln(1  x) ]
(j) 2ln(2  3 x) (k)
7
2.2 Definite Integral
1.
Find the area under the curve y  x  3x 2 between x = 1 and x = 2.
[Answer: 8.5 square units]
3
2.
3.
Find the area under the curve yx 2  1 between x = 1 and x = 2.
[Answer:]
x3
Find the area under the curve y 
between the origin and x = 2.
3
[Answer:]
4.
Find the area under the curve y  x3 between the origin and x = 4.
[Answer:]
5.
Evaluate the following:

3

2
(g)

6
(j)

3
(m)

2
(p)

2
(a)
(d)
1
1
3
1
1

2

4
(h)

2
( x 2  3x)dx
(k)

3
8
)dx
x2
(n)

2
(q)

0
( y  2)( y  3)dy
(b)
 x3  1 
 2  dx
 x 
(e)
( x  3)dx
(3 x 
1
( x 4  2 x 2 )dx
1
2
1
0
1
 3x  1 

 dx
 2x 

3
(f)

3
1
( x  ) 2 dx
x
(i)
 x2  1 
2  x2  dx
3x2 dx
(l)

( x 2  1)dx
(o)

(3x 2  3)dx
(r)

( x  3x2 )dx
(c)
(3x2  2)dx
2
1
2
0
x5 dx
4
3
1
1
(2 x  3)2 dx
(6 x  2)dx
1
2
1
3
(4 x  4)dx
[Answer: (a) ⅔ (b) 8.5 (j) 202/3 ]
6.
Find the value of the following integrals:
(a)

3
(d)

3

1.5
1
1
(5  2 x)dx
(
4
 2 x) dx
5x2
(b)

3
(e)

1
(c)

3
(6  3x  x2 )dx
(f)

12
0.5
1
4
)dx
(3 x 
x
2
0
6
(3x  x 4 )dx
3
dx
x2
1
0.5 (2 x  7)dx
3 x
[Ans: (a) 2 (b) 1.49 (c) –35.1 (d) 8.53 (e) 11.3 (f) 0.25 (g) 4.01 (h) 21.47 (i) –2.75]
(g)
1
(4 x3  x )dx
(h)

4
2
(6 x 
)dx
(i)
2.3 Some Applications of the Integral Calculus
1.
The Wheeler-Dealer Utility Company has determined that its marginal cost function for
the production of 20 units of electric power is:
C1(x) = e2x + x,
where x is measured in thousands of dollars.
Find the cost function and hence, the cost of producing 3 units of power.
1
C ( x)  (e 2 x  x 2 )  19.5 , $225.72]
[Answer:
2
4
2.
Based on economic analysis, a firm has determined that its marginal revenue is given by
the relationship:
r1(x) = –8x + 10. Find the total revenue function, and the demand
function.
[Answer: r(x) = –4x2 + 10x, d(x) = –4x + 10]
3.
The fixed cost of production of a firm is $800, and the marginal cost is:
C1(x) = 0.03x2 + 0.12x + 5.
Find the cost function.
[Answer:]
4.
The rate of change of a certain population, P(t), with respect to time, t, is given by:
3t 2
P1 (t )  25, 000 
5
At time t = 0, the population is P(0) = 50,000.
(i)
Find an expression for P(t);
(ii)
What will be the population when t = 20?
t3
[Answer: (i) P(t )  25, 000t   50, 000 , 551,600]
5
5.
The Xeles Corporation has a fixed overhead cost of $10,000. If its marginal cost function
is given by c1 ( x)  2 x 2  4 x , find the cost function.
[Answer:]
6.
For a group of hospitalized individuals, the discharge rate is given by:
f ( x)  0.008e0.008t dt ,
where f(t) is the proportion discharged per day at the end of t days of
hospitalization.
What proportion of the group is discharged at the end of 100 days?
[Answer: 0.5507]
2.3.2: Consumers’ and Producers’ Surplus
1.
Find the consumers’ surplus at a price level of: (i) $8 and (ii) $4 for the price-demand
equation: d(x) = 20 – 0.05x
[Answer: (i) $1,440 (ii) $2,560]
2.
Find the consumers’ surplus at a price level of $150 for the price-demand equation: d(x)
= 400 – 0.05x
[Answer: $625,000]
3.
If the demand and supply functions for a particular commodity are given by:
d(x) = -x2 + 25
and
s(x) = 2x + 10 respectively, determine the consumers’
surplus at market equilibrium.
[Answer: 18 units]
5
4.
The demand function for a particular commodity is given by d ( x)  (64  x) . Find the
consumers’ surplus when the demand is 40.
[Answer:]
5.
The demand function for a particular product is d ( x)  210 x and the supply equation is
s( x)  2 x 2 , where x is the units demanded or supplied. Determine, to the nearest
thousand dollars, the consumers’ surplus under market equilibrium.
[Answer: $113,000]
6.
Find the consumers’ surplus (to the nearest dollar) at a price level of $2.089 for the pricedemand equation: d(x) = 9 – ln(x + 4)
[Answer: $977]
7.
Find the producers’ surplus at a price level of: (i) $20 and (ii) $4 for the price-supply
equation: s(x) = 2 + 0.0002x2
[Answer: (i) $3,600 (ii) $133]
8.
Find the producers’ surplus at a price level of $67 for the price-supply equation: s(x) = 10
+ 0.1x + 0.0003x2
[Answer: $625,000]
9.
Find the producers’ surplus (to the nearest dollar) at a price level of $26 for the pricesupply equation: s(x) = 5 ln(x + 1)
[Answer:]
10.
Find the equilibrium price and then find the consumers’ surplus and producers’ surplus at
the equilibrium price level, if:
(i)
d(x) = 20 – 0.05x
and
s(x) = 2 + 0.0002x2
2
(ii)
d(x) = 25 – 0.001x
and
s(x) = 5 + 0.1x
[Answer: (i) 10, CS = $1,000, PS = $1,067 (ii) 15, CS = $667, PC = $500]
11.
The demand equation, d(x), and the supply equation, s(x), of a product are given in each
case below. Determine, in each case, the consumers’ surplus and the producers’ surplus
under market equilibrium:
(a)
d(x) = 22 – 0.8x
s(x) = 6 + 1.2x
(c)
d ( x) 
50
x
 4.5
, s( x) 
x5
10
(b)
d(x) = 100(10 – 2x)
s(x) = 50(2x – 1)
(d)
d(x) = 80e–0.001x
s(x) = 30e0.001x
[Answer: (a) CS = 25.6, PS = 38.4 (b) CS = 225, PS = 450 (c) CS = 50ln2 – 25, PS =
1.25 (d) CS = $6,980, PS = $5,041]
6
12.
If the supply function is found to be s( x)  x 2  3x  2 and the demand function is
d ( x)  2 x  16 . Find both the consumers’ and producers’ surpluses at equilibrium.
[Answer:]
13.
The demand and supply functions for a particular commodity are given by:
d(x) = -x2 + 12
and
s(x) = 2x2 + x + 8
respectively. Determine the
consumers’ and producers’ surpluses when market equilibrium prevails.
[Answer: ⅔ units, 15/6 unit]
14.
A manufacturer of electronic sphygmomanometers determines that the demand function
for his product is given by: d ( x)  ( x  8)2 , and the supply function by: s( x)  9 x 2 . The
price p is in dollars and the number of units produced is x. If market equilibrium prevails,
determine:
(i)
the equilibrium price,
(i)
the consumers’ surplus, and
(ii)
the producers’ surplus.
[Answer: (i) $36 (ii) 27 units (iii) 48 units]
15.
Determine the producers’ surplus under market equilibrium, if the demand function is
d ( x)  (40  x)2 and the supply function is s( x)  3x 2  8x  8 . Also, find the
consumers’ surplus when there is demand for 25 units of commodity.
[Answer:]
16.
The supply and demand functions for a product are given to be s( x)  x 4 and
d ( x)  20  x 2 respectively.
(i)
Sketch the graphs of these functions;
(ii)
What is the equilibrium point?
(i)
What are the producers’ and consumers’ surpluses?
[Answer:]
17.
The demand function for a product is: d(q) = 100 – 0.05q, and the supply function is: s(q)
= 10 + 0.1q, where q is the units produced at the price p, per unit (in dollars). Determine
consumers’ surplus and producers’ surplus under market equilibrium. Also find the
equilibrium point.
[Answer: 9000, 18,000; (70, 600)]
18.
The demand function for a product is: d(q) = 0.01q2 – 1.1q + 30, and the supply function
is: s(q) = 0.01q2 + 8, where q is the units produced at the price p, per unit (in dollars).
Determine consumers’ surplus and producers’ surplus when market equilibrium has been
established.
[Answer: CS = 166⅔, PS = 53⅓]
7