Math 1425.P70
Examples for 4.2 & 4.3
Solve the problem.
1) A special-events promoter sells x tickets and
Name _____________________________________________
3) Hatts and Company determines that its
marginal cost, in dollars per hat, is given by
3
C'(x) = x + 40, for x ≤ 350.
40
has a marginal-revenue function given by
R'(x) = 2x - 1240, where R'(x) is in dollars
per ticket.
This means that the rate of change of total
revenue with respect to the number of tickets
sold, x, is R'(x). Find the total revenue from
the sale of the first 340 tickets.
2) Rejoyne Inc. has a marginal-profit function
Find the total cost of producing the first 260
hats.
4) Accent Woodworkers knows that their
given by
P'(x) = -3x + 140, where P'(x) is in dollars
per unit.
This means that the rate of change of total
profit with respect to the number of units
produced, x, is P'(x). Find the total profit from
the production and sale of the first 30 units.
Examples for 4.2 & 4.3
marginal cost of producing x feet of custom
molding is given by
C'(x) = -0.00003x 2 - 0.04x + 90, for x ≤
1000,
where C'(x) is in cents. Approximate their
total cost, in dollars, of manufacturing 1000
feet of molding, using 5 subintervals over
[0, 1000] and the left endpoint of each
subinterval.
Write summation notation for the expression.
5) 7 + 14 + 21 + 28 + 35
Find the area under the given curve over the indicated
interval.
11) y = 7; [2, 9]
y
8
7
6
5
6) 8 + 16 + 24 + 32 + 40 + 48
4
3
2
1
1
2
3
4
5
6
7
7) f(x1 ) + f(x2 ) + f(x3 ) + f(x4 ) + f(x5 ) + f(x6 )
12) y = 2x + 1; [1, 3]
y
9
8
7
8) g(x1 ) + g(x2 ) + . . . + g(x16)
6
5
4
3
2
1
Express the sum without using summation notation.
6
9) ∑ h(xi)
i=1
1
2
3
4
5
13) y = x2 + 3; [0, 2]
y
9
8
7
6
5
10)
5
∑ 4i
i=1
4
3
2
1
1
Math 1425.P70 Examples for 4.2 & 4.3, page 2
2
3
4
x
x
8
9
10 11 x
14) y =
1
; [0.5, 2]
x
State what the shaded area represents.
17)
y
7
6
5
4
3
2
1
1
2
3
4
x
18)
15) y = ex; [1, 2]
y
9
8
7
6
5
4
3
2
1
1
2
19)
x
3
16) y = (x - 3)2; [2, 4]
y
6
5
4
3
2
1
1
2
3
4
5
6
x
Math 1425.P70 Examples for 4.2 & 4.3, page 3
20)
24) y = 2x + 7; [1, 5]
25) y = x2 - 6x + 9; [2, 4]
21)
26) y = 2x - x2 ; [0, 2]
22)
27) y =
3
; [1, 3]
x3
Find the area under the graph of the function over the
interval given.
23) y = 2x - x2 ; [0, 2]
Math 1425.P70 Examples for 4.2 & 4.3, page 4
28) y = -x2 + 9; [0, 3]
31) A well-drilling company finds that its
marginal profit, in dollars, from drilling a
well that is x feet deep is given by
3
P'(x) = x.
Find the company's profit from drilling a well
that is 230 feet deep.
29) y = x2 + 1; [0, 1]
32) A kitchen remodeling company determines
that the marginal cost, in dollars per foot, of
installing x feet of kitchen countertop is given
by
C'(x) = 7x-1/3.
Solve the problem.
30) A manufacturer determined that its marginal
Find the cost of installing an extra 12 feet of
countertop after 30 feet have already been
ordered.
cost per unit produced is given by the
function
C'(x) = 0.0006x2 - 0.4x + 94.
Find the total cost of producing the 401st unit
through the 500th unit.
33) A company estimates that its sales will grow
continuously at a rate given by the function
S'(t) = 15et,
where S'(t) is the rate at which sales are
increasing, in dollars per day, on day t. Find
the sales from the 2nd day through the 10th
day. (This is the integral from 1 to 10.)
Math 1425.P70 Examples for 4.2 & 4.3, page 5
Answer Key
Testname: 1425_SECTION4_2__4_3
1)
2)
3)
4)
5)
6)
7)
8)
9)
-$306,000
$2850
$7865.00
$668.00
5
∑ 7i
i=1
6
∑ 8i
i=1
6
∑ f(xi)
i=1
16
∑ g(xi)
i=1
h(x 1) + h(x2 ) + h(x3 ) + h(x4) + h(x 5) + h(x 6)
10) 4 + 8 + 12 + 16 + 20
11) 49
12) 10
26
13)
3
14) 1.39
15) e2 - e
16)
2
3
17) Distance traveled in miles
18) Distance traveled in miles
19) Total mass in grams
20) Total number of births
21) Total cost in dollars
22) Total cost
4
23)
3
24) 52
2
25)
3
26)
4
3
27)
4
3
28) 18
4
29)
3
30) $3569.96
31) $1056.89
32) $25.49
33) $330,356.21
Math 1425.P70 Examples for 4.2 & 4.3, page 6