NYU Polytechnic School Of Engineering MA 1154 WORKSHEET # 6

NYU Polytechnic School Of Engineering
MA 1154
WORKSHEET # 6
Date:
Print Name:
Signature:
Section:
Instructor: Dr. Manocha
ID #:
Directions: Complete all questions clearly and neatly. You must show all work to have credit.
Unclear work will not be graded. THIS IS A CRUCIAL HOMEWORK UNDERSTAND IT
WELL FOR YOUR NEXT EXAM.
Problem Possible
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Total
100
Points
Your signature:
(1) Find the integral
(a)
(b)
(c)
R
2x ln(x2 + 1)
dx
x2 + 1
R
1
x3
R
x
√
dx
2x + 1
3/4
1
dx
−1
x2
Your signature:
(2) Solve the following initial value problems.
(a)
1
dy
=
dx
x ln(x)
where y = 2 when x = e
(b)
dy
2x
=
dx
1 + 3x2
where y = 5
when x = 0
Your signature:
(3) The price p (dollars per unit) of a particular commodity is increasing at the rate
20x
p0 (x) =
(7 − x)2
where x hundred units of the commodity are supplied to the market. The manufacturer supplies 200 units (x = 2) when the price is $2 per unit.
(a) Find the supply function p(x).
(b) What price corresponds to a supply of 500 units?
Your signature:
(4) The owner of a fast food chain determines that if x thousand units of a new meal item
are supplied, then the marginal price at that level of supply is given by
x
p0 (x) =
dollars per meal
(x + 3)2
where p(x) is the price (in dollars) per unit at which all x thousand meal units will be
sold. Currently, 5, 000 units are being supplied at a price of $2.20 per unit.
(a) Find the supply (price) function p(x).
(b) If 10, 000 meal units are supplied to restaurants in the chain, what unit price
should be charged so that all the units will be sold?
Your signature:
(5) The table gives the coordinates (x, f (x)) of points on the graph of a function f over the
interval
a ≤ x ≤ b. In each case, estimate the value of the indicated definite integral
Rb
f (x) dx by forming a Riemann sum using right endpoints.
a
(a)
R
2
f (x) dx
1
x
1 1.2 1.4 1.6
1.8
2.0
f (x) 1.1 1.4 0.8 −0.3 −1.4 −1.1
(b)
R
2
f (x) dx
0
x
0 0.4 0.8 1.2 1.6 2.0
f (x) 1.1 1.7 2.3 2.5 2.4 2.1
Your signature:
(6) Evaluate each of the definite integral using the fundamental the theorem of calculus.
(a)
R
2
1
(b)
R
4
1
(c)
R
x2
dx
(x3 + 1)2
√
( x − 1)3/2
√
dx
x
1/2
1/3
e1/x
dx
x2
Your signature:
(7) Find the area of each of the region R that lies under the curve y = f (x) over the
interval a ≤ x ≤ b.
(a) Under y = √
3
, over −8 ≤ x ≤ 0.
9 − 2x
2
(b) Under y = xe−x , over 0 ≤ x ≤ 3.
Your signature:
(8) The marginal cost of producing a certain commodity is C 0 (q) = 6q + 1 dollars per unit
when q units are being produced.
(a) What is the total cost of producing the first 10 units.
(b) What is the cost of producing the next 10 units?
Your signature:
(9) The output of a factory is changing at the rate
Q0 (t) = 2t3 − 3t2 + 10t + 3 units/hour
where t is the number of hours after the morning shift begins at 8 A.M. How many
units are produced between 10 A.M and noon?
Your signature:
(10) A study indicates that t months from now the population of a certain town will be
growing at the rate of P 0 (t) = 5 + 3t2/3 people per month. By how much will the
population of the town increase over the next 8 months?