MA 1021 A ’06
Final Exam
Name _________________________
Instructions
This test is closed book. Calculators are not allowed.
Part I - Basic Skills
Problem
1
2
3
4
5
6
7
Total
Value
5
5
5
5
5
5
5
35
Problem
8
9
10
11
12
13
14
Total
Value
12
12
12
15
18
16
15
100
Earned
Part II
Earned
Please Circle your Section
A01 Heinricher, A (PLC)
A02 de Oliveira, G (3:00)
A03 Servatius, B (8:00)
A04 Tashjian, G (8:00)
A05 Onofrei, D (9:00)
A06 Fehribach, J (9:00)
A07 Masamune, J (1:00)
A08 de Oliveira, G (3:00)
A09 Malone, J. J. (10:00)
A10 Abraham, J (10:00)
A11 Lui, R (2:00)
A12 Masamune, J (3:00)
A13 Tashjian, G (12:00)
A14 Abraham, J (11:00)
Part I - Basic Skills
Work the following problems and write your answers in the space provided.
Use the scratch paper provided for your work. You need not simplify your
answers.
1.
Find
dy
dx
if
3
2
y = x + 3x −
4
x
3
+5
Ans. _________________________
2.
Find
dy
dx
if
3
y = x ln x
Ans. _________________________
3.
Find
dy
dx
x2 +1)
(
if y = e
Ans. _________________________
4.
Find an equation for the tangent line to the
curve y
4
3
= x − x at the point (1,0)
Ans. _________________________
5.
Find
dy
dx
if y =
3+ x
1+ x
2
Ans. _________________________
6.
Find
dy
dx
if
(
y = x − 3x + 5
3
2
)
11
Ans. _________________________
7.
Find
dy
dx
if y = sin
3
( x)
Ans. _________________________
Part II
Work all of the following problems. Show your work in the space provided. You
need not simplify your answers, but remember that on this part of the exam your
work and your explanations are graded, not just the final answers
8.
Evaluate each limit or show it does not exist.
a.
x2 − 9
lim 2
x →3 x + x − 12
b.
lim
5 − x + 25
x →0
x
9.
For f ( x )
derivative.
= x2 − 5x ,
find
f ′( x) by using the limit definition of
10.
Find an equation for the tangent line to the curve
at the point
(2, 3)
x 2 + xy + y 2 = 19
11.
A ladder 13 feet long that was leaning against a vertical wall begins to
slip. Its top slides down the wall while its bottom moves along the
ground at a constant speed of 3 ft/s. How fast is the top of the ladder
moving when it is 5 feet above the ground?
12.
In this problem, you will analyze the curve given by
f ( x) = − x3 + 3 x 2
a.
Find all intervals where
f ( x) is increasing
b.
Find all intervals where
f ( x) is decreasing
c.
Find all intervals where
f ( x) is concave up
f ( x) is concave down
d.
Find all intervals where
e.
Find all local maxima and minima of
f.
Find any inflection points of
f ( x)
f ( x)
g.
Sketch a graph of
f ( x)
13.
An open top rectangular box with a square base is to built with wooden sides,
and a metal bottom. It will have a volume of 125 cubic inches. Metal costs six
cents per square inch, while wood costs two cents per square inch. Find the
dimensions of the box which will minimize the cost of materials for the box.
14.
Find the derivative of the following functions
c.
a.
f ( x) = e x + x e + (e)( x)
b.
f ( x) = x sin x
f ( x ) = sin ( e
5x
)
****** End of exam ******