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

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

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

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