Download Formula Sheet Math 122 Miscellaneous Formulae 1. Acceleration

Formula Sheet
Math 122
Miscellaneous Formulae
1. Acceleration due to gravity: 9.8 m/s2 or 32 ft/s2
2. Area of a trapezoid:
h
(b
2 1
+ b2 )
3. Volume of a cylinder: πr2 h
Summation Formulae
4.
n
X
c = cn
n
X
n(n + 1)
i=
2
6.
i=1
i2 =
n(n + 1)(2n + 1)
6
n
X
i3 =
n2 (n + 1)2
4
i=1
i=1
5.
n
X
7.
i=1
Arc Length and Surface Area
Z bp
1 + [f ′ (x)]2 dx
8. The arc length of f (x) from x = a to a = b is
a
9. The surface area of the surface obtained by revolving f (x) from x = a to x = b about the
Z b
p
x-axis is
2πf (x) 1 + [f ′ (x)]2 dx
a
Inverse Trigonometric Functions
10.
11.
12.
13.
d
1
(arcsin x) = √
dx
1 − x2
−1
d
(arccos x) = √
dx
1 − x2
15.
d
−1
(arccot x) =
dx
1 + x2
16.
Z
√
17.
Z
³x´
1
dx
+C
= arctan
a 2 + x2
a
a
18.
Z
dx
1
√
= arcsec
a
x x2 − a 2
1
d
(arctan x) =
dx
1 + x2
1
d
(arcsec x) = √
dx
|x| x2 − 1
−1
d
(arccsc x) = √
14.
dx
|x| x2 − 1
³x´
dx
+C
= arcsin
a
a 2 − x2
Hyperbolic Functions
19. sinh x =
ex − e−x
2
21. tanh x =
sinh x
cosh x
20. cosh x =
ex + e−x
2
22. csch x =
1
sinh x
µ
|x|
a
¶
+C
1
cosh x
cosh x
24. coth x =
sinh x
23. sech x =
25. cosh2 x − sinh2 x = 1
d
26.
(sinh x) = cosh x
dx
d
27.
(cosh x) = sinh x
dx
d
28.
(tanh x) = (sech x)2
dx
Inverse Hyperbolic Functions
√
34. sinh−1 x = ln(x + x2 + 1)
√
35. cosh−1 x = ln(x + x2 − 1), x ≥ 1
µ
¶
1
1+x
−1
36. tanh x = ln
, −1 < x < 1
2
1−x
29.
d
(coth x) = −(csch x)2
dx
30.
d
(sech x) = − sech x tanh x
dx
d
(csch x) = − csch x coth x
dx
Z
32.
sinh x dx = cosh x + C
31.
33.
Z
39.
1
d
(tanh−1 x) =
, −1 < x < 1
dx
1 − x2
40.
1
d
(csch−1 x) = − √
, x 6= 0
dx
|x| 1 + x2
cosh x dx = sinh x + C
37.
1
d
(sinh−1 x) = √
2
dx
x +1
41.
1
d
(sech−1 x) = − √
,0<x<1
dx
x 1 − x2
38.
1
d
(cosh−1 x) = √
,x>1
dx
x2 − 1
42.
d
1
, −1 < x, x > 1
(coth−1 x) =
dx
1 − x2
Trigonometric Identities
43. csc x =
1
sin x
49. sin2 x =
1 − cos(2x)
2
44. sec x =
1
cos x
50. cos2 x =
1 + cos(2x)
2
45. cot x =
1
tan x
51. tan2 x =
1 − cos(2x)
1 + cos(2x)
46. sin2 x + cos2 x = 1
52. sin x sin y = 12 (cos(x − y) − cos(x + y))
47. 1 + tan2 x = sec2 x
53. cos x cos y = 21 (cos(x − y) + cos(x + y))
48. 1 + cot2 x = csc2 x
54. sin x cos y = 12 (sin(x + y) + sin(x − y))