CAMBRIDGE INTERNATIONAL EXAMINATIONS
Cambridge International Advanced Subsidiary and Advanced Level
MARK SCHEME for the May/June 2015 series
9708 ECONOMICS
9708/42
Paper 4 (Data Response and Essays (Supplement),
maximum raw mark 70
This mark scheme is pub
Momentum and impulse
If we first start by looking at the definition of momentum, it is defined as being the product of mass
and velocity, that is M = mv. Since momentum depends on velocity, it is a vector quantity.
How is momentum related to force? Well i
Math 81 - Calculus with Business Applications Lehigh University
Exam 2 Ver A, October 31, 2013
Short version with no spacing.
1: [24, 4 each]: You do not need to show work for the following multiple choice problems.
Circle your answer clearly.
(i)
If f (x
Math 81 - Calculus with Business Applications Lehigh University Exam 2 ver A November 11, 2011
Short version with no spacing.
1: [24, 4 each]: You do not need to show work for the following multiple choice problems.
Circle your answer clearly.
Find lim+ l
Math 81 - Calculus with Business Applications Lehigh University
Exam 2 November 8, 2012
Short version with no spacing.
1: [4 each]: You do not need to show work for the following multiple choice problems.
Circle your answer clearly.
(i) Assume that f (c)
Math 81 - Calculus with Business Applications Lehigh University
Exam 1 Ver A, September 26,
2013
Short version with no spacing.
1: [24, 4 each]: You do not need to show work for the following multiple choice problems.
Circle your answer clearly.
If Q(K) =
Math 81 - Calculus with Business Applications Lehigh University
Exam 1 October 4, 2012
Short version with no spacing.
1: [24, 4 each]: You do not need to show work for the following multiple choice problems.
Circle your answer clearly.
If f (x) = x2 + 2 a
Homework #4
Math 23, Fall 2016
Due Thursday, September 29 or Friday, September 30 in class.
1. Suppose that, at time t = 0, a particle with mass 3 has position vector ~r(0) = 4~ ~k and
velocity ~v (0) = 5~ 13~k. The particle is then subjected to a constan
Homework #3
Math 23, Fall 2016
Due Thursday, September 22 or Friday, September 23 in class.
1. At what point do the curves
~r1 (t) = t, 3 t, t2
and ~r2 (s) = 1 s, s + 2, s2 1
intersect? Find the cosine of the angle of intersection.
2. Find a vector functi
Homework #6
Math 23, Fall 2016
Due Thursday, October 13 or Friday, October 14 in class.
Problems 1-3 are worth 2 points each; and Problem 4 is worth 4 points, a total of 10 points.
1. If
w = xy + y 2 z; where x = st, y = s cos t and z = set ,
find
2. For
Homework #5
Math 23, Fall 2016
Not to be handed in, but please do it (and check your answers) before the first exam.
1. Determine each of the following limits (if they exist). You should justify your answer.
(a)
z
cos 3x2 y y 2 z +
3
(x,y,z)(0,0,)
lim
(b)
Homework #2
Math 23, Fall 2016
Due Thursday, September 15 or Friday, September 16 in class.
1. Find the area of the parallelogram with vertices (1, 3, 0), (1, 1, 1), (2, 1, 1), and (4, 1, 2).
2. For each of the following pairs of lines, determine whether
Scratch paper for multiple choice problems
tear o, use and do not turn in
1
LEHIGH U MATH 81 - CALCULUS WITH BUSINESS
APPLICATIONS
FINAL EXAM DEC 11, 2012
December 11, 2012
Name
Section
Grading
1.
11.
2.
12.
3.
13.
4.
14.
5.
15.
6.
16.
7.
17.
8.
18.
9.
19
Math 81 Calculus with Business Applications Lehigh University Final Exam Dec 19, 2011
1: [20, 4 each]: There are 5 parts on this page. Circle your answers clearly.
You do not need to show work for the following multiple choice problems.
(i) If U (t) = at3
Math 81 - Calculus with Business Applications Lehigh University
Final Exam
December 11, 2012
Short version with no spacing.
1: [24, 4 each] Circle your answers clearly.
You do not need to show work for the following multiple choice problems.
b
(i) If U (t
Indefinite Integrals I
k dx
kx
C k u( x ) dx
ku( x ) dx
where k is a real constant .
[u( x ) v ( x )]dx
n
u( x )dx
v ( x )dx
1
[u( x)]
u ( x)dx
[u( x)]n n 1
C, if n
C C
1
sin u( x ) u ( x )dx cos u( x ) u ( x )dx
cos u( x ) sin u( x )
Derivatives III
Dx Arcsin u( x ) 1 Dx Arccos u( x ) 1
Dx Arctan u( x )
1 ux 1 ux
2 2
u ( x)
u ( x)
1 1 ux
1 1 u x
2
u ( x)
D x Arccot u( x )
2
u ( x)
Dx Arcsec u( x ) ux Dx Arccsc u( x ) ux
1 ux 1 ux
2 2
u ( x) 1 u ( x) 1
D x sinh u( x )
Integrals of Rational Functions
Guidelines for Partial Fraction Decomposition of
f (x) g(x)
1. If the degree of f ( x ) is not lower than the degree of g ( x ) , use long division to obtain the proper form. 2. Express g ( x ) as a product of linear
Indefinite Integrals III
1 1 ux
1 1 ux
2
2
u x dx
Arcsin u x
C
u x dx
Arc tan u x
C
1 ux ux
2
u x dx 1
Arcsec u x
C
sinh u x
u x dx
cosh u x
C
cosh u x
u x dx
sinh u x
C
sech2 u x
u x dx
tanh u x
C
csch2 u x
u x dx
coth u x
Trigonometric Substitutions
For Expression in the Integrand
Use Trigonometric Substitution
To Obtain
a 1 sin 2 a cos
a2
x2
x
a sin ,
2
2
a2
x2
x
a tan ,
2 0
2
or
a 1 tan 2
a sec
x2
a2
x
a sec ,
2 3 2
a sec 2
1
a tan
sin
x a
January 26, 2005
Announcements Math 125 A&B Joint Marathon Review Session! Today (Wednesday) 5:30 7:30 PM in GUG 317 . (Thanks to Bill, Kiana and Allen.) Sections covered by the midterm: 4.10, 5.1-5.5, 6.1-6.3. One sheet (8 1 11) of handwritten
February 23, 2005 Announcements:
1. Office hours today: A. Chen - 2:30-4:30 MSC K. Ross - 4:30-6:30 MSC T. Toro - 1:30-3:30 MSC 2. Bring your scientific calculator to the midterm tomorrow, you will need it. Recall graphing calculators are NOT all
Formulae for Calculus 151 p. 1
Differentiation Definition: f (a + h) - f (a) f (a) = lim h0 h General formulas: Product: (uv) = u v + uv Quotient: (u/v) = [u v - uv ]/v 2 Chain rule: [f (u)] = f (u) u Constant multiple: (cu) = cu Special functions:
Sequences
Theorems on Sequences: I. Modeling Theorem Let {an } be a sequence, let f (n) an , and suppose that f ( x) exists for every real number x 1 . (i) lim f ( x) L , then lim f (n) L .
x n
(ii) lim f ( x)
x
(or
) , then lim f (n)
n
(or
).
Positive-Term Series
Theorems on Convergence or Divergence of a Positive-Term Series 1. If
a n is a positive-term series and if there exists a number M such that
Sn a1 a2 an
M for every n, then the series converges and has a sum S
diverges. 2. Inte