MATH 495
Problem Set 3
1.
2.
If x  denotes the greatest integer < x, then

7
0
2
x  dx =?
If k i i  0,1,2,3,4 are constants such that x 4  k 0  k1 x  1  k 2 x  1  k3 x  1  k 4 x  1 is an
identity in x, what's the value of k 3 ?
2
3
4

3.
If  1  x  1 , find a rational expression for  nx 2 n .
n 1
4.
5.
6.
Graph the level curves of the surface whose equation in cylindrical coordinates is z  r 2 cos2 .
If the variables P, V and T are related by the equation PV = nRT, where n and R are constants, simplify
 V   T   P 
the expression 


.
 T   P   V 
The temperature at each point x, y, z  in a room is given by the equation T x, y, z   9 x 2  3 y 2  6 xyz .
A fly is currently hovering at the point 2,2,2 . Find a direction vector the fly should follow to cool off
as rapidly as possible.
2a
If a is a positive number, what's the value of the double integral 
8.
If a box of 2 dozen ball point pens contains 6 that are defective, what is the probability that 3 pens
selected at random (without replacement) will be defective?
9.
If
10.
11.
4
7
0

0
7.
 2 ay  y 2
x 2  y 2 dxdy ?
is expressed in binary form, what is the digit in the 19 places to the right of the binary point?
ax  ay  z  1
The solution of the system x  ay  az  1 is x, y, z   a, b, a . If a is not an integer find the value of
ax  y  az  1
a  b.
If one arch of the curve 𝑦 = sin 𝑥 is revolved about the x-axis, what is the volume of the generated
solid?
12.
Let A   , a, where  is the empty set. If B is the set of all subsets of A, what's the cardinality of the
set A  B ?
13.
Let f x, y   xyx  y  , where xt, u   t t  u  . What is the value of
14.
If a is a positive constant, what is the maximum value of the function f  x  
f
at the point t , u   1,1 ?
t
ln x
?
xa
15.
Let i = 1,0,0 , j = 0,1,0 , and k = 0,0,1 . If b and c are vectors such that ||b|| =3, b∙c = 5, ||c|| =7, and
bc=8j + 4k find a vector a that satisfies the equation a∙[(a + c)  b] =12.
16.
If y  x 2 , x y  y x  x w for x  0 , then express w as a polynomial function of x.
17.
Consider the family F of circles in the xy-plane,  x  c   y 2  c 2 , that are tangent to the y-axis at the
origin. Find a 1st order differential equation for that all members of the family F satisfy.
18.
Let a be the smallest positive value of x at which the function f x   cos x 2 sin x 2 has a critical point.
What is the exact value of f a  ?
2



19.
In SL(2,Z), the multiplicative group of all 2 by 2 matrices with integer entries whose determinant is 1,
0 1
what is the order of the cyclic subgroup generated by the matrix A  
.
 1  1
20.
A six-sided die (whose faces are numbered 1 through 6, as usual) is known to be counterfeit: The
probability of rolling any even number is twice the probability of rolling any odd number. What is the
probability that if this die is thrown twice, the first roll will be a 5 and the second roll will be a 6?
21.
Given that  e  x dx 
22.
In a survey of 100 undergraduate math majors at a certain university, the following information is
obtained about the courses they are taking during the spring semester:
41 are enrolled in real analysis
44 are enrolled in differential equations
48 are enrolled in linear algebra
11 are enrolled in both real analysis and linear algebra
14 are enrolled in both real analysis and differential equations
19 are enrolled in both differential equations and linear algebra
10 are not enrolled in any of these three courses
How many are enrolled in all three of these courses?
23.
24.

0
2

2
, what is the exact value of the integral 

0
ex
x
dx ?
x
The tangent line to the curve y  x e  at the point 1,1 passes through the point 2, y 0  , where y 0 has
what value?
t 1  t 
If the matrix A  
 has 1 as an eigenvalue, find another eigenvalue for A.
1 2t 