Download - Calculus Challenge.pdf
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Question 1
What is the centroid of the region bounded by the curves and ?
Hint: draw a picture of this region as your first step.
Question 2
If satisfies the differential equation
and , then what is ?
y = x 2 y = 3 x 2
( , ) = (0, 3 )x y 6
( , ) = (0, 2)x y
( , ) = ( , )x y3
2
3
2
( , ) = (0, )x y3
2
( , ) = (0, )x y3
2
( , ) = ( , )x y7
6
3
2
( , ) = (0, )x y8
5
( , ) = (0, 0)x y
x(t)
= 3dx
dtetx
x(0) = 0 x(1)
ln(2e + 1)
ln(2e + 3)
ln(3e + 2)
ln(3e 2)
+ 6e2
+ 1e2
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Question 3
What is the smallest number for which the function
is in as ?
There is no such .
Question 4
Compute the arc-length of the graph of for .
Hint: your expression for the arclength element should admit a "miracle" factorization that
eliminates a certain square root.
ln(2e 1)
6e2
n
cos( x)ex2
2
O( )x n x 0
n = 0
n
n = 3
n = 5
n = 1
n = 6
n = 4
n = 2
y = x 2
4
ln x
21 x e2
dL
e 1
2
3e4
2
11e4
12
+ 11e6
12
+ 3e4
4
+ 74
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Question 5
What is the volume of the solid generated by rotating about the -axis the region defined by the
inequalities:
1. ,
2. , and
3. .
Question 6
You approximate using the series:
+ 7e4
4
+ 1e2
2
7e2
4
y
y 2x 3
y 2xx 0
8
15
3
16
21
6
32
105
4
15
224
15
2
5
ln4
3
ln(1 + x) = (1
n+1
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If you use the first three terms as your approximation that is, using terms up to and including the
term, then what is the bound on your error that comes from observing that the series
above is alternating?
Question 7
Compute .
ln(1 + x) = (1n=1
)n+1x n
n
x 3 E
E