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Math 225 Final Exam

For full credit, show your work. An answer not supported by work will not receive full points,

and might not receive any.

This is a graph of f ( x )=( x5 )

s

+2

1) Find the Center of Gravity equation to find the balancing point of the shaded area.

2) Find the volume of the shape that would be created, if the shaded area was rotated around the

x-axis.

3) Solve each of the following equations:

7x=14 1, XXXXXXXXXXt=1,300

log x=25 ln x3=3

4) If f ( x )=10 e2x find

a) the derivative of f(x)

b) the anti-derivative of f(x)

5) If g ( x )=ln 4 x find

a) the derivative of g(x)

b) the anti-derivative of g(x)

6) If t ( x )= tan−1 x then

a) Find t (1)

b) Find t ' (1 )

c) Find ∫

0

1

t ( x )dx

7) Demonstrate why ∫ 1√4−x2

dx=sin−1 x

2

+C

“Because the chart says so” is not a valid answer.

8) Another Calculus student, after doing Problem 7, writes the following

“This probably means ∫ 1√4+x2

dx=sinh−1 x

2

+C”

without checking the table in the back. Clearly, she didn’t have the correct answer memorized.

How did she know this was the correct formula?

9) Mandatory Integration By Parts Problem #1.

∫√x ln x dx=¿

10) Mandatory Integration By Parts Problem #2.

∫ ( ln x )2dx=¿

11) ∫ 5 x−12x (x−4 ) dx=¿

HALFTIME

TAKE A

BREATH

12) Find lim

x→∞

x ln x

x+ ln x or show that it does not exist

13) Find lim

x→0+¿ xx ¿

¿ or show that it does not exist.

14) Find ∫

1

∞ ln x

x

dx or show that it does not exist.

15) Find the general solution to the differential equation

y '= y+xy

16) If the SEQUENCE {an} is given by

an=

4n+1

n2+5n+6

then does the sequence converge? If so, to what value?

17) Explain how you know the SERIES {sn} that is the sequence of partial sums from {an} in

Problem 16 will diverge.

18) Find the fraction (in simplest terms) that is equal to

1+ 1

3

+ 1

9

+ 1

27

+…

without using a calculator.

19) The Root Test and Ratio Test both find a limit when used, and they both say the SERIES

converges if the limit exists and does what?

20) Imagine an infinite pattern of alternating shaded/non-shaded squares, each of which has sides

half the size of the previous, like this:

If the original square was ten by ten (having an area of 100) what is the area of only the shaded

parts?

21) Well, here it is: you knew this was coming.

If f ( x )=e−x then find the MacLaurin series, to at least the x3 term, that approximates f(x).

22) Using your answer to Problem 21, find

1

√e

=e−1 /2 by hand.

Math 225 Final Exam

Math 225 Final Exam

For full credit, show your work. An answer not supported by work will not receive full points,

and might not receive any.

This is a graph of f ( x )=( x5 )

s

+2

1) Find the Center of Gravity equation to find the balancing point of the shaded area.

2) Find the volume of the shape that would be created, if the shaded area was rotated around the

x-axis.

3) Solve each of the following equations:

7x=14 1, XXXXXXXXXXt=1,300

log x=25 ln x3=3

4) If f ( x )=10 e2x find

a) the derivative of f(x)

b) the anti-derivative of f(x)

5) If g ( x )=ln 4 x find

a) the derivative of g(x)

b) the anti-derivative of g(x)

6) If t ( x )= tan−1 x then

a) Find t (1)

b) Find t ' (1 )

c) Find ∫

0

1

t ( x )dx

7) Demonstrate why ∫ 1√4−x2

dx=sin−1 x

2

+C

“Because the chart says so” is not a valid answer.

8) Another Calculus student, after doing Problem 7, writes the following

“This probably means ∫ 1√4+x2

dx=sinh−1 x

2

+C”

without checking the table in the back. Clearly, she didn’t have the correct answer memorized.

How did she know this was the correct formula?

9) Mandatory Integration By Parts Problem #1.

∫√x ln x dx=¿

10) Mandatory Integration By Parts Problem #2.

∫ ( ln x )2dx=¿

11) ∫ 5 x−12x (x−4 ) dx=¿

HALFTIME

TAKE A

BREATH

12) Find lim

x→∞

x ln x

x+ ln x or show that it does not exist

13) Find lim

x→0+¿ xx ¿

¿ or show that it does not exist.

14) Find ∫

1

∞ ln x

x

dx or show that it does not exist.

15) Find the general solution to the differential equation

y '= y+xy

16) If the SEQUENCE {an} is given by

an=

4n+1

n2+5n+6

then does the sequence converge? If so, to what value?

17) Explain how you know the SERIES {sn} that is the sequence of partial sums from {an} in

Problem 16 will diverge.

18) Find the fraction (in simplest terms) that is equal to

1+ 1

3

+ 1

9

+ 1

27

+…

without using a calculator.

19) The Root Test and Ratio Test both find a limit when used, and they both say the SERIES

converges if the limit exists and does what?

20) Imagine an infinite pattern of alternating shaded/non-shaded squares, each of which has sides

half the size of the previous, like this:

If the original square was ten by ten (having an area of 100) what is the area of only the shaded

parts?

21) Well, here it is: you knew this was coming.

If f ( x )=e−x then find the MacLaurin series, to at least the x3 term, that approximates f(x).

22) Using your answer to Problem 21, find

1

√e

=e−1 /2 by hand.

Math 225 Final Exam

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