The word problem is:
Sam has $500 in the savings account at the beginning of summer. He withdraws $25 each week for
his weekly expenditures. He wants to save at least $200 in the account by the end of summer. How
many weeks can Sam withdraw money from his
Let x denote the number of racing bikes manufactured, y denote the number of touring bikes
manufactured and z denote the number of mountain bikes manufactured.
Then according to the question constraints are
17x + 27y + 34z
91800
12x + 21y + 15z
42000
x,
Soln 1. P(x) = R(x) C(x)
Thus, the profit function is positive where R(x) is greater than C(x). We observe from the graph that
the profit function is positive at [12, 25]
The profit function is negative where R(x) is less than C(x). We observe from the gr
4)
f(x) =
3
x + 2 x
Slope of the tangent line is given by
d (f ( x )
dx
3
d (f ( x ) d ( x + 2 x)
=
dx
dx
=
d ( x3 ) d ( 2 x)
+
dx
dx
=
3 x2 +
2
2x
1
2
3
x
+
=
2 x
At the point (2,10) slope of the tangent line
(
=
d (f ( x )
1
) =3(2)2+
dx x=2
2(2)
3( 4)
Define the following functions f, g, and h:
3
3
f : cfw_0, 1 cfw_ 0, 1 . The output of f is obtained by taking the input string and replacing the first
bit by 1, regardless of whether the first bit is a 0 or 1. For example, f(001) = 101 and f(110) =
110
1.
lim ( f ( x )+ g ( x ) )=2
If
and
x c
(A)4
lim ( f ( x )g ( x ) ) =2
x c
(B )2
(C) 0
( D ) not enough information
of the above
(E )
lim ( f ( x )+ g ( x ) )=2
ANSWER:
x c
lim ( f ( x ) ) +lim (g ( x ) )=2
x c
x c
Solving for
lim ( f ( x )
and
x c
and
1. Determine analytically if the following functions are even, odd, or neither.
a. f(x) = 5x 3 2x
b. f(x) = x 2 + 3x 2
ANSWER. a) This is an odd function.
f(-x) = 5 (x )32 (x ) =5 x 3 +2 x =( 5 x 32 x )=f (x)
Thus, it is an odd function.
b) This is
f(-x)
SOLUTION
Let G represents glazed donuts and C represents creme-filled frosted donuts.
Then according to the given data n(G) = number of respondents who said that they would buy glazed
donuts
= 1415
n(C) = number of respondents who said that they would buy
1) Graph of y = sin(x) and tangent line.
Tangent line at (0, 0): y = sin(x)
y = cos(x)
When x = 0, slope is y(0) = cos(0) = 1
The equation of a tangent line: y = mx + b = x + b
Solving for b, if y = sinx then at x = 0, y = 0 => b = 0
Thus, equation of tan
SOLUTIONS
12. As x 0
One-sided limits:
h= 0
f ( 0h ) = lim
h 0
f ( x )= lim
h0
x0
lim
sin ( h)= sin ( 0 )=0
f ( 0+ h )= lim
h 0
f ( x ) = lim
h0
x 0+
lim
Two sided limit:
Since both the one sided limits are equal therefore,
f ( x )= 0
lim
x 0
As
3.
Cardinality of a set is the number of elements in the set.
1. S =
cfw_ , cfw_ , cfw_ ,cfw_ ,cfw_ cfw_
There are 3 elements in this set,
, cfw_ , cfw_ cfw_ , cfw_ cfw_
Thus, cardinality of S is 3.
2. S is a set of 15 elements.
So, the number of
1) Eulers formula states that for any real number x
which is equal to
1
ix
e =cos ( x ) +i sin ( x ) where i is imaginary
, cos(x) and sin(x) are trigonometric functions.
Polar Form: Let z be a non zero complex number, then
z=a+i b . This is represented
VECTOR ADDITION
Add the following vectors and determine the resultant vector. u = 3.0 m/s, 45 deg and v = 5.0 m/s,
135 deg
For these type of problems follow these steps:
Step 1: Find the horizontal and vertical components of both the vectors u and v.
For
1
1 1 1
= + +
R T R 1 R 2 R3
1)
1
1
1
1
=
R 3 RT R 1 R2
R1 R2RT R 2RT R1
R T R 1 R2
R 3=
RT R 1 R2
R1 R 2RT R2 R T R1
x ( x2 x 2 ) + x 3x 3+ x 2 +12x 2 + x3 x 3+ 3 ( 7 x 3 x+5 )11
2) P(x) =
2
3
=
16 x3 + x 22 x +16
( nR T 4 )
3
R T
8
16
4 16
16
n R (T )
Test #2
1) n(U) = 60
2) n( A B =43
3) n(AC) = 37
4)
5) Total choices = 10+5
Possibilities = 157=170859375
6) 72
7) 840
8) 15
9) Total alphabets = 26
Total digits = 10
Possibilities = 36P5
= 45239040
If O and 0 are excluded.
Total choices = 34
Possibilitie
1) a) Mathematical expression for the initial condition is
T ( 0 ) =160
b) The domain of T(t) is
2)
Graph B
100
90
80
70
60
Temperature (Deg F)
50
40
30
20
10
0
0
200 400 600 800 1000 1200 1400 1600 1800 2000
Time
Graph B
Temperature (Deg F)
100
90
80
70
ANSWER
3
3
4
34 2 3
-(1)
For this we will follow the BODMAS rule, which says we solve in order of Bracket Of Division
Multiplication Addition Subtraction.
3
So first we will solve the mixed fraction 3 4 =
(34)+3
4
=
12+3
4
=
15
4
Putting this in (1)
3
4
3
UNIT 8
1) Expected value = 1
2) Expected value = -5
3) Expected value = 3/2
4) Expected value = 4
5) r1 = 1/6
r2 = 5/6
c1 = 5/6
c2 = 1/6
v = 16/3
The game is favourable to the row player.
6) r1 = 5/7
r2 = 2/7
c1 = 3/14
c2 = 11/14
v = 27/7
The game is favo
1) infinity < x < 1/3
1/3 < x < infinity
inflection point x = 1/3
2) Global maximum: (4.5, 5)
Global minimum: (0, 1.5)
Local minimum: (6, 1.5)
3) (a) x4
(b) x2 and x6
(c) x1
(d) x7
4) Global maximum: (4, 16777172)
Global minimum: (0.99, -10.00)
5) Global
CALC 3
1) (0, 9)
[4, INF)
(2, INF)
2) 2i + (-3)j + 0k
3)
5,1,
1
3
4) T
F
F
F
5) D
F
C
E
A
B
6)
t , 2t 2 ,2 t 2+12 t 4
7) x(t) =
23
5
cos ( t )
2
2
y(t) =
23
3
sin ( t ) +
2
2
z(t) =
3
8)
2 t
1
,
, 24 e6 t
2
2
5+t
19t
2 2
,
2
2
9) A)
B)
C)
10)
23
23
s (
1) A)
It is given that my speed = v
My friends speed = w
Here, v and w are constant.
We know that distance = speed x time
Let t be time.
=> my distance = vt and my friends distance = wt
Using the distance formula, we get
Distance between me and my friend,
Let length of the ice cream cake be l and width be b.
Then according to the question
l = w + 5.9
- (1)
The perimeter is given to be 51 ft.
The formula of the perimeter is P = 2w + 2l
Using these two, we get
2w + 2l = 51
2w + 2(w + 5.9) = 51
2w + 2w + 11.8
1) A point given in the graph is ( 4 ,2
a tan
=> 2 =
( 4 )=a
=> a = 2
2) Comparing the graph with the graph of tan(x), we get a = 4 and using the point ( 12 , 4
=> 4 = 4tan(b 12
=> tan(b 12 = 1
=> b 12 = 4
=> b =
( 4 )( 12 )
=3
3) a = 4
4) 1 is correc
DERIVATIVES
1)
3x 7
g ( x)
5x 2
and evaluate at x = 2
Let, p(x) = 3x -7 and
q(x) = 5x +2
We will use the quotient rule which states that the derivative of g(x) = p(x)/q(x) is given by
g (x )=
p' ( x )=
q(x ) p ( x) p( x )q (x )
, where q(x) 0 for any x.
22videolecture/notes
Inboththenotesandvideo,choosesrightendpointrectanglesindemonstrating
integrationbylimn.Butisthereaparticularreasonforusingrightendpointinstead
pfmidpointorleftendpointrectangles?
Solution:
For better approximation we have to mid poin
1. Find the equation of the angle bisector of the angles between the lines 2x + y = 5 and
x 2y = 1.
Solution:
The equations of the angle bisectors of the angles between the lines 2x + y 5 = 0 and x
2y 1 = 0 are
2x y 5 x 2 y 1
22 12
12 22
2x y 5 x 2
Mathematical Methods for Economics: Vectors and Vector Operations
DC-1
Semester-II
Paper-IV: Mathematical methods for Economics-II
Lesson: Vectors And Vector Operations
Lesson Developer: Sanjeev Kumar
College/Department: Dyal Singh College, University of
Mathematical expectation discrete
DC-1
Semester-II
Paper-III: Statistical Methods in Economics-I
Lesson: Mathematical expectation discrete
Lesson Developer: Chandra Goswami
College/Department: Department of Economics, Dyal
Singh College, University of Del