1
136.151: Test #1
20 minutes
1.
Name:_
Student Number: _
1. Consider the following three lines.
l1 : 2 y = 2 x + 1
l2 : y - x = 10
l3 : y = - x + 4
Which of these lines are mutually parallel, which are mutually perpendicular? Why?
Solution.
1
so that the
1
136.151: Test #1
20 minutes
1.
Name:_
Student Number: _
1. Consider the following three lines.
l1 : y = 2 x + 1
l2 : y - 2 x = 10
l3 : 2 y = - x + 4
Which of these lines are mutually parallel, which are mutually perpendicular? Why?
Solution.
For l1 we a
1
136.151: Test #4
20 minutes
2,3,4.
Name:_
1. Is the graph of the function y =
Justify your answer.
Student Number: _
1
, x 0 concave up or concave down?
x +1
1
1
-1
and y =
. Since
is positive for
2
3
( x + 1)
( x + 1)
( x + 1) 3
x 0, it follows that th
1
136.151: Test #1
20 minutes
1.
Name:_
Student Number: _
1. Consider the following three lines.
l1 : 2 y = 2 x + 1
l2 : y - x = 10
l3 : y = - x + 4
Which of these lines are mutually parallel, which are mutually perpendicular? Why?
Solution.
1
so that the
1
136.151: Test #2 Solutions
2,3,4.
1. Evaluate the limit or show it does not exist. In the latter case check if the limit is + , -
or neither. Identify any horizontal or vertical asymptotes from the two limits below (Do not
compute other limits; just ext
1
136.151: Test #3
Solutions
2,3,4.
Name:_
Student Number: _
1. Differentiate (with respect to x):
(a) ln( tan x )
(b) e cos 2 x
Solutions.
(ln(
(a)
tan x ) =
)
1
1
1
tan x 2 tan x cos2 x
cos 2 x
) = 2(- sin 2 x )e cos 2x
(b) (e
2. Use derivatives to sho
1
136.151: Test #1 Solutions
20 minutes
2,3,4
Name:_
Student Number: _
1. Consider the following three lines.
l1 : 2y = 2x + 1
l2 : y + x = 10
l3 : y = 2 x
Find the slopes of each of the lines. Which of these lines are mutually parallel,
which are mutuall
1
136.151: Test #2 Solutions
2,3,4.
1. Evaluate the limit or show it does not exist. In the latter case check if the limit is + , -
or neither. Identify any horizontal or vertical asymptotes from the two limits below (Do not
compute other limits; just ext
1. 20 minutes 136.151 Test 3 Oct. 24 2000
Student name: _ Student number: _
Value
[6] 1. Give an example of a function
derivative at the point when
function.
One example is
. Provide a formula as well as the graph of your
. The graph of that function is g
1. 136.151: Test #1
20 minutes
Name: _ Student Number: _
1. Find an equation of the line passing through a point (2,3) and perpendicular to the
line
. Show your work.
Solution:
The line through (2,3) and with slope m has an equation
. So,
we need to find
1. 20 minutes 136.151 Test 2 Oct. 10 2000
Student name: _ Student number: _
Value
1. Evaluate each of the following limits if they exist. If it does not exist,
indicate why not.
[4] (a)
[5] (b)
2. The graph of the function
undefined at
and at
.
[4] (a) Fi
Midterm Solutions
1. Evaluate the limit (or explain why it does not exist)
(a)
(b)
2. Is the function
differentiable at
answer using only the definition of derivative.
? Justify your
The right hand side derivative at x=0 is
. We
find
(we have used
that he
1. 136.151 Test 5
20 minutes Nov. 28 2000
Student name: _ Student number: _
Value
[6] 1. Find the absolute extrema of the function
[0,2].
in the interval
and the critical points come from solving
two solutions of the last equation only
(so we ignore
;
. O
1. 136.151 Test 4
20 minutes Nov. 14 2000
[8] 1.
(a) Find
if
your answer.
. Do not simplify
(b) Find
.
(a)
(b)
[6] 2. Find
answer.
Set
to get
So
if
. Then
. Do not simplify your
.Differentiate implicitly
.
[6] 3. Show that the function
for
.
First we find
ANTECEDENTES
La industria de hidrocarburos es una de las actividades que ms consecuencias
negativas trae al medio ambiente en Colombia, se dice que el 70% de la
contaminacin de los suelos en el pas es debido al derrame y explotacin de
hidrocarburos, esta