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MATH 495
Problem Set 1
1.
Every point on the parabola y  2 x  1 is equidistant from the y-axis and what other point?
2.
If the polynomial px is divided by x  1 the remainder is 1, and if it divided by x  1 the remainder is
-1. Find the remainder when px is divided by x 2  1 .
3.
If the sinh x  
4.
 x 2  6x  8

If f x    x 3  2 x 2  2 x  4 if x  2 is continuous find the value for k.

k
if x  2
5.
What is the domain of the function arcsin ln x ?
6.
sinh x 
e x  ex
e x  ex
, cosh x  
and tanh x  
find a formula for tanh 1 x  .
cosh x 
2
2


 x 2  x  3 if x  1
For what values of m and b will the function f x   
be differentiable for all values
mx

b
if
x

1

of x?
7.
What is the slope of the tangent line to the curve xyx  y   x  y 4 at the point 1,1 ?
8.
If f x   2 x  1  x  1 what is the value of f ' 0 ?
9.
At what value of x in the interval 0, 32  does the function f  x   
10.
How many zeros are at the end of the number 100!?
11.
If L is the least common multiple of 1001 and 10101, which of the numbers 6,11,17,22, and 33 is the
sum of the digits of L? Can this be determined without actually computing L?
12.
Let x1 and x 2 be the two smallest positive integers so that the expression 85x1  12x2 is a multiple of
19. Find x1  x2 .
13.
Let G be a group generated by the elements x and y that have the following relations: x 2  y 3 , y 6  1 ,
and x 1 yx  y 1 . Express in simplest form the inverse of the element z  x 2 yx 3 y 3 .
14.
Let R be a ring; an element x in R is said to be idempotent if x 2  x . How many idempotent elements
does the ring ℤ20 contain?
15.
2
2x
x
sin t
t
dt attain its maximum value?
1 2 3
If r is the rank and d is the determinant of the matrix 4 5 6 what is r-d?
7 8 9
16.
17.
18.
7
5
For what value of x will the matrix 
8

0
20.
0
x
0
1
1
0
not be invertible?
1

1
a 
Let the vector v =  b  in ℝ3 and the linear operator 𝑇: ℝ3 → ℝ3 is given by T(u)=vx where  denotes
 
 c 
the cross product of the vectors. Then T(u)=Au for a matrix A. Find A.
Let y  f x  be a solution of the equation
f 1 ?
19.
6
4
7
0
dy
x2
such that y  0 when x  0 . What is the value of
 2
dx x  1
A population of bacteria grows at a rate proportional to the number present. After two hours, the
population has tripled. After two more hours elapse, the population has increased by a factor of k. What
is the value of k?
Every curve in a certain family, y  f x, C  , has the following property: The area of the region in the
first quadrant bounded above by the curve from 0,0 to x,0 and bounded below by the x-axis is 13 the
area of the rectangle with vertices at 0,0 and x, y  . Find f x, C .