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Material Type: Exam; Class: CALCULUS I; Subject: Mathematics; University: Clark University; Term: Fall 2004;
Typology: Exams
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Final Exam Name: (print neatly) Instructor: (sign)
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
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
)
$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.
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.
