7/17/97

Final Exam: \Cheat Sheet"

Z

Z

1

2

6. csc xdx = cot x

dx =lnjxj 1.

x

Z

Z

x

a

x

7. sec x tan xdx = sec x

2. a dx =

ln a

Z

Z

8. csc x cot xdx = csc x

3. sin xdx = cos x

Z

Z

1 x dx

1

= tan 9.

4. cos xdx = sin x

2 2

x + a a a

Z

Z

dx x

1

2

p

= sin 10.

5. sec xdx = tan x

2 2

a

a x

Z Z

d b

[f y  g y ] dy [f x g x] dx. In terms of y : A = 11. Area in terms of x: A =

c a

Z Z

d b

2 2

 [g y ] dy  [f x] dx. Around the y -axis: V = 12. Volume around the x-axis: V =

c a

Z

b

13. Cylindrical Shells: V = 2xf x dx where 0  a< b

a

Z Z

b b

b

0 0

14. f xg x dx = f xg x] f xg x dx

a

a a

R

m 2 2

n 2 2

15. HowtoEvaluate sin x cos xdx.:Ifm or n are o dd use sin x =1 cos x and cos x =1 sin x,

2

1 1

2

resp ectively and then apply substitution. Otherwise, use sin x = 1 cos 2x or cos x = 1 + cos 2x

2 2

1

or sin x cos x = sin 2x.

2

R

m n 2 2 2

16. HowtoEvaluate tan x sec xdx.:Ifm is o dd or n is even, use tan x = sec x 1 and sec x =

2

1 + tan x, resp ectively and then apply substitution.

17. Table of Trigonometric Substitutions

Expression Substitution Identity

p

2

 

2

2 2

  a x x = a sin , 1 sin = cos

2 2

p

 

2 2

2 2

< < a + x x = a tan , 1 + tan = sec

2 2

p

 3

2 2

2 2

x a x = a sec 0  < or   < sec 1 = tan

2 2

 "

n n1

X X

x

f x  and T = f x +f x  f x +2 18. M =
x

i n i n 0 n

2

i=1 i=1

x

19. S = [f x +4f x +2f x +4f x +


+2f x + 4f x + f x ]

n 0 1 2 3 n2 n1 n

3

3 3 5

K b a K b a K b a

2 2 4

20. jE j and jE j and jE j

T M S

2 2 4

12n 24n 180n

Z Z Z Z

1 t b b

21. f x dx = lim f x dx and f x dx = lim f x dx

t!1 t!1

a a 1 t

P

1

22. Test for Divergence. If lim a do es not exist or if lim a 6= 0, then the series a is

n!1 n n!1 n n

n=1

divergent.

23. Remainder Estimate for the Integral Test. If a converges by the Integral Test and R = s s ,

n n n

Z Z

1 1

then f x dx  R  f x dx

n

n+1 n 10

P P

24. Limit Comparison Test. Supp ose that a and b are series with p ositive terms. a If

n n

a a

n n

= c>0, then either b oth series converge or b oth diverge. b If lim = 0 and lim

n!1 n!1

b b

P n P n P P

a

n

b converges, then a also converges. c If lim = 1 and b diverges, then a also

n n n!1 n n

b

n

diverges.

P

n1

25. Alternating Series Estimation Theorem. If s = 1 b is the sum of an alternating series

n

that satis es a and b ab ove, then jR j = js s jb

n n n+1

P

1 a

n+1

a is absolutely .IfL exists and L<1, then the series 26. The Ratio Test Let L = lim

n n!1

n=1

a

n

P

1

a is divergent. convergent. If L exists and L> 1 or is in nity, then the series

n

n=1

p

P

1

n

27. The Ro ot Test Let L = lim j = L.IfL exists and L<1, then the series a is ja

n!1 n n

n=1

P

1

absolutely convergent. If L exists and L>1 or is in nity, then the series a is divergent.

n

n=1

P

1

n

28. Theorem. If f x= c x a has radius of convergence R> 0 then f x is di erentiable on

n

n=0

Z

1 1

n+1

X X

x a

0 n1

, each a R; a + R and a f x= nc x a and b f x dx = C + c

n n

n +1

n=1 n=0

with radius of convergence R.

n

29. Theorem. If f has a p ower series representation at a, that is, if f x=c x a for jx aj

n

n

f a

then its co ecients are of the form

n!

30. Taylor's Formula. If f has n + 1 derivatives in an interval I that contains the number a, then for x

in I there is a number z strictly b etween x and a such that the remainder term in the Taylor series

n+1

f z 

n+1

can b e expressed as R x= x a

n

n + 1!

1 1 1

2n+1 n

X X X

x 1 x

n x 1 n

31. = x ,1; 1 ,1; 1 ,[1; 1] e = tan x = 1

1 x n! 2n +1

n=0 n=0 n=0

1 1

2n+1 2n

X X

x x

n n

,1; 1 ,1; 1 sin x = cos x = 1 1

2n + 1! 2n!

n=0 n=0

1

X

k

k n

32. The Binomial Series If k is any real numb er and jxj < 1, then 1 + x = x where

n

n=0

k k 1


k n+1

k k

= =1. for n  1 and

n 0 n!

dy dy =dt dx

33. = if 6=0.

dx dx=dt dt

0

34. The Arc Length Formula. If f is continuous on [a; b], then the length of the curve y = f x,

Z

b

p

0 2

a  x  b,is L = 1+ [f x] dx.

a

35. Theorem. If a curve C is describ ed by the parametric equations x = f t, y = g t,  t  ,

0 0

where f and g are continuous on [ ; ] and C is traversed exactly once as t increases from to ,

s

Z

2 2

dx dy

then the length of C is L = + dt.

dt dt

s

Z

2 2

dx dy

2y + dt 36. Surface Area. S =

dt dt

y

2 2 2

. 37. Polar co ordinate conversion: x = r cos , y = r sin , r = x + y , and tan =

x

s

Z Z

2

b b

1 dr

2

2

38. For p olar co ordinates: A = r d and L = r + d

2 d

a a 11