final exam

Question 1

Slicing method

Answer 1

Integral of area

Question 2

disk method

Answer 2

pi*integal of radius squared

Question 3

washer method

Answer 3

pi* integral of outer radius square minus inner radius squared

Question 4

shell method

Answer 4

integral of 2pi(r)*height

Question 5

arc length formula

Answer 5

L=integral of sqrrt(1+(f’(x))^2)

Question 6

surface area

Answer 6

integral of 2pi(r)*ArcL

Question 7

integration by parts

Answer 7

uv-Integral(vdu)

Question 8

Int. of cosx

Answer 8

sinx

Question 9

Int. of sinx

Answer 9

-cosx

Question 10

Int. of sec^2(x)

Answer 10

tanx

Question 11

Int. of secx

Answer 11

ln|secx+tanx|

Question 12

Int. of csc^2(x)

Answer 12

-cotx

Question 13

Int. of secx tanx

Answer 13

secx

Question 14

Int. of cscx cotx

Answer 14

-cscx

Question 15

Int. of cscx

Answer 15

-ln|cscx+cotx|

Question 16

Int. of tanx

Answer 16

ln|secx|

Question 17

Int. of cotx

Answer 17

ln|sinx|

Question 18

sin^2(x)+cos^2(x)

Answer 18

1

Question 19

1+cot^2(x)

Answer 19

csc^2(x)

Question 20

tan^2(x) +1

Answer 20

sec^2(x)

Question 21

sin^2(x)

Answer 21

(1-cos(2x))/2

Question 22

cos^2(x)

Answer 22

(1+cos(2x))/2

Question 23

sin(2x)

Answer 23

2sinxcosx

Question 24

Trig substitution 1st step

Answer 24

draw a triangle

Question 25

fraction decomposition

Answer 25

linear factor: A + B
Repeated factor: A/x + B/x-a + C/(x-a)^2
Quadratic Factor: A + (Bx + C)

Question 26

improper integrals

Answer 26

use a limit

Question 27

Geometric Series

Answer 27

a[(1-r^k)/(1-r)]
i. |r|>= 1 diverges
ii. |r| < converges

Question 28

Divergence Test

Answer 28

lim(ak) not 0 then diverges

Question 29

Integral test

Answer 29

f(x) is postive, continuous, decreasing
ak behaves the same as integral

Question 30

p series (1/n^p)

Answer 30

p>1 converges
p<=1 diverges

Question 31

comparison test

Answer 31

bk > ak behave the same

Question 32

direct comparison test

Answer 32

if ak < bk and bk converges, ak converges
if ak>bk and bk diverges, ak diverges

Question 33

sequence convergence test

Answer 33

USE LIMIT

Question 34

limit comparison test

Answer 34

if lim (an/bn) not = 0 then behave the same
if lim (an/bn) = 0 and bn converge, then an converges
if lim (an/bn) = infinity and bn diverges, then an diverges

Question 35

Alternating series test

Answer 35

i. divergence test, lim = 0
ii. a(k+1) < ak
converges

Question 36

Absolute convergence test

Answer 36

if sum |ak| converges, ak converges absolutely

Question 37

Ratio test

Answer 37

r = lim |a(k+1)/ak|
i. r<1 converges
ii. r>1 diverges
iii. r=1 inconclusive

Question 38

root test

Answer 38

p = lim krt(|ak|)
i. p<1 converges
ii. p>1 diverges
iii. p=1 inconclusive

Question 39

convergence of power series

Answer 39

ratio test, |r| <1, solve for x

Question 40

check endpoints

Answer 40

plug in endpoints for x then do a test (alternating series or p test)

Question 41

finding Taylor series

Answer 41

p(x) = c0+c1(x-a)+c2(x-a)^2+c3(x-a)^3+…
c0 = f(a)
c1 = f’(a)/1!
c2 = f’‘(a)/2!
c3 = f’’‘(a)/3!

Question 42

Maclaurin series

Answer 42

taylor series where a=0

Question 43

binomial expansion

Answer 43

sum of (r n) x^n = (r 0)x^0 + (r 1)x + (r 2)x^2 + (r 3)x^3 +… = 1 + rx/1! + r(r-1)x^2 /2! + r(r-1)(r-2)x^3 /3! +…

Question 44

parametric equations

Answer 44

make table with x, y, t and the draw the graph and mark direction

Question 45

eliminate parameter

Answer 45

solve for x or y and substitute

Question 46

finding tangent line

Answer 46

find dy/dx and then plug in point for tangent line y =mx +b

Question 47

critical points

Answer 47

dy/dx = 0 or UND

Question 48

polar coordinate formulas

Answer 48

x^2 + y^2 = r^2
x=rcosO
y=rsinO
tanO = y/x

Question 49

Area of cardiod

Answer 49

1/2 Int. r^2 dO
Use symmetry

Question 50

completing the square

Answer 50

ax^2+bx+c+(b/2)^2-(b/2)^2