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Final Exam for Calculus I - Fall 2004 | MATH 120, Exams of Calculus

Material Type: Exam; Class: CALCULUS I; Subject: Mathematics; University: Clark University; Term: Fall 2004;

Typology: Exams

Pre 2010

Uploaded on 08/07/2009

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Calculus I Math 120 Fall 2004
Final Exam Name: (print neatly)
Instructor: (sign)
1. (20 pts) Find the following limits. If a limit fails to exist, then briefly explain why.
a) lim
x0sin(4x)x
b. lim
x→∞
4x2
5
3x2+ 4x+ 2
c. lim
x0
(2x+ 3) sin(x)
x
d. lim
x2
x+ 2
x2
1 of 7
pf3
pf4
pf5

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Calculus I – Math 120 Fall 2004

Final Exam Name: (print neatly) Instructor: (sign)

  1. (20 pts) Find the following limits. If a limit fails to exist, then briefly explain why.

a) lim x→ 0 sin(4x) − x

b. (^) xlim→∞

4 x^2 − 5 3 x^2 + 4x + 2

c. lim x→ 0

(2x + 3) sin(x) x

d. lim x→ 2

x + 2 x − 2

  1. (25 pts) Find the following derivatives.

a.

d dx

x tan(x)

b.

d dx

x x − 1

)

c.

d dx

sin^3 (2x − π)

d.

d dx

1 − 3 x^2

e.

d^2 dx^2

( 1 − x^3 + x^6

)

  1. (15 pts) A rectangular region is to be enclosed by fencing which costs $8 per foot on the east side (facing a road) and $2 per foot on the other three sides. The total budget for materials is $1000. Answer questions a-c below to determine the dimensions of the largest rectangle that can be enclosed. (^) Enclosure

$2/ft^2

$2/ft^2

$2/ft^2 $8/ft^2

æ -

6

?

y

y

a) Express the total cost in terms of x and y.

b) Express the quantity to be maximized.

c) Use appropriate techniques to find the dimensions of the largest rectangle that can be enclosed.

  1. (25 pts) The function f (x) and its derivatives are given:

f (x) =

x^2 x^2 − 2 x + 2

f ′(x) =

− 2 x(x − 2) (x^2 − 2 x + 2)^2

f ′′(x) =

4(x − 1)(x − 1 −

3)(x − 1 +

(x^2 − 2 x + 2)^3

[Hint: The denominator is always positive.]

a) Find all intervals on which f (x) is increasing, and those on which f (x) is decreasing.

b) Find all critical values and determine whether each is a local max or a local min.

c) Find all inflection points.

Prob Pts

Total