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)
e45x
(g)
(h)
1
2e (15 x )
e2 x 2 x
(j)
(k)
e
(13 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 5e12 x
1
1
2 5 x
1
1 45 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 (15 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 (13 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 e12 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) 0.008e0.008t 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)
x5
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
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