Download Reference Sheet

Calculus Reference Sheet
𝑑
𝑢′
[log 𝑎 𝑢] =
(ln 𝑎)𝑢
𝑑𝑥
𝑑 𝑢
𝑢′ 𝑣 − 𝑢𝑣 ′
[ ]=
𝑑𝑥 𝑣
𝑣2
𝑑
[𝑢𝑣] = 𝑢′ 𝑣 + 𝑢𝑣′
𝑑𝑥
𝑑
𝑢
[|𝑢|] =
(𝑢′ )
|𝑢|
𝑑𝑥
𝑑 𝑢
[𝑎 ] = (ln 𝑎)𝑎𝑢 𝑢′
𝑑𝑥
𝑑
𝑢′
[arcsin 𝑢] =
𝑑𝑥
√1 − 𝑢2
𝑑
−𝑢′
[arccos 𝑢] =
𝑑𝑥
√1 − 𝑢2
′
𝑑
𝑢
[ln 𝑢] =
𝑑𝑥
𝑢
𝑑 𝑢
[𝑒 ] = 𝑒 𝑢 𝑢′
𝑑𝑥
Area of a Triangle
1
𝐴𝑟𝑒𝑎 = 𝑏ℎ
2
1
𝐴𝑟𝑒𝑎 = 𝑏𝑐 ∙ sin 𝐴
2
𝐴𝑟𝑒𝑎 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
1
𝑤ℎ𝑒𝑟𝑒 𝑠 = (𝑎 + 𝑏 + 𝑐)
2
Reciprocal Identities:
sin 𝑥 =
1
csc 𝑥
csc 𝑥 =
1
sin 𝑥
cos 𝑥 =
1
sec 𝑥
sec 𝑥 =
1
cos 𝑥
1
tan 𝑥 = cot 𝑥
1
cot 𝑥 = tan 𝑥
Odd-Even Identities:
csc(−𝑥) = − csc 𝑥
cos(−𝑥) = cos 𝑥
sec(−𝑥) = sec 𝑥
tan(−𝑥) = − tan 𝑥
cot(−𝑥) = − cot 𝑥
Sum and Difference Identities:
sin(𝑢 ± 𝑣) = sin 𝑢 cos 𝑣
± cos 𝑢 sin 𝑣
cos(𝑢 ± 𝑣) = cos 𝑢 cos 𝑣
∓ sin 𝑢 sin 𝑣
tan(𝑢 ± 𝑣) =
𝑑
𝑢′
[arctan 𝑢] =
𝑑𝑥
1 + 𝑢2
Arc Length:
𝑠 = 𝑟𝜃
(θ must be in radians)
𝑑
[csc 𝑢] = −(csc 𝑢 cot 𝑢)𝑢′
𝑑𝑥
𝑑
[sec 𝑢] = (sec 𝑢 tan 𝑢)𝑢′
𝑑𝑥
𝑑
[cot 𝑢] = −(csc 2 𝑢)𝑢′
𝑑𝑥
Circle:
𝐴𝑟𝑒𝑎 = 𝜋𝑟 2
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝜋𝑑
Sphere:
Cone:
4
𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3
3
tan 𝑢 ± tan 𝑣
1 ∓ tan 𝑢 tan 𝑣
1
𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ
3
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 4𝜋𝑟 2
Quotient Identities:
tan 𝑥 =
Pythagorean Identities:
sin 𝑥
cos 𝑥
cos 𝑥
cot 𝑥 =
sin 𝑥
Law of Sines:
sin(−𝑥) = − sin 𝑥
𝑑
[sin 𝑢] = (cos 𝑢)𝑢′
𝑑𝑥
𝑑
[cos 𝑢] = −(sin 𝑢)𝑢′
𝑑𝑥
𝑑
[tan 𝑢] = (sec 2 𝑢)𝑢′
𝑑𝑥
sin 𝐴 sin 𝐵 sin 𝐶
=
=
𝑎
𝑏
𝑐
Area of a Trapezoid:
𝑏1 + 𝑏2
𝐴𝑟𝑒𝑎 =
(ℎ)
2
Area of a Prism:
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 ∗ ℎ𝑒𝑖𝑔ℎ𝑡
sin2 𝑥 + cos2 𝑥 = 1
tan2 𝑥 + 1 = sec 2 𝑥
1 + cot 2 𝑥 = csc 2 𝑥
Law of Cosines:
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 ∙ cos 𝐴
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 ∙ cos 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 ∙ cos 𝐶
Double Angle Identities:
sin 2𝑢 = 2 sin 𝑢 cos 𝑢
cos 2𝑢 = cos2 𝑢 − sin2 𝑢
cos 2𝑢 = 2 cos2 𝑢 − 1
cos 2𝑢 = 1 − 2sin2 𝑢
tan 2𝑢 =
2 tan 𝑢
1 − tan2 𝑢
Conics: Parabolas
2
Std. Eqn
(𝑥 − ℎ) = 4𝑝(𝑦 − 𝑘)
(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ)
Opens
Focus
Up/down
(h, k+p)
Left/right
(h+p, k)
Directrix
axis
y=k-p
x=h
x=h-p
y=k
Conics: Ellipses
(𝑥 − ℎ)
(𝑦 − 𝑘)2 (𝑥 − ℎ)2
(𝑦 − 𝑘)
+
=
1
+
=1
𝑎2
𝑏2
𝑎2
𝑏2
y = k (horizontal)
x = h (vertical)
(ℎ ± 𝑐, 𝑘)
(ℎ, 𝑘 ± 𝑐)
(ℎ ± 𝑎, 𝑘)
(ℎ, 𝑘 ± 𝑎)
𝑎2 = 𝑏 2 + 𝑐 2
𝑎2 = 𝑏 2 + 𝑐 2
Conics: Hyperbolas
2
2
(𝑥 − ℎ)
(𝑦 − 𝑘)2 (𝑥 − ℎ)2
(𝑦 − 𝑘)
−
=
1
−
=1
𝑎2
𝑏2
𝑎2
𝑏2
y = k (horizontal)
x = h (vertical)
2
Std. Eqn
a>b
Focal axis
Foci
vertices
pythagorean
Std. Eqn
a>b
Focal axis
2
Foci
(ℎ ± 𝑐, 𝑘)
(ℎ, 𝑘 ± 𝑐)
vertices
pythagorean
Asymptotes
(ℎ ± 𝑎, 𝑘)
𝑐 2 = 𝑎2 + 𝑏 2
𝑏
𝑦 = ± (𝑥 − ℎ) + 𝑘
𝑎
(ℎ, 𝑘 ± 𝑎)
𝑐 2 = 𝑎2 + 𝑏 2
𝑎
𝑦 = ± (𝑥 − ℎ) + 𝑘
𝑏
Center is at (h, k)
2a = length of major axis
2b = length of minor axis
𝑒 = 𝑐⁄𝑎
2
𝑐 = 𝑎2 − 𝑏 2
Center is at (h, k)
a = dist. From center
to vertex
c = dist. From center
to focus
𝑒 = 𝑐⁄𝑎
∫ 𝑘𝑓(𝑢)𝑑𝑢 = 𝑘 ∫ 𝑓(𝑢)𝑑𝑢
∫ sin 𝑢 𝑑𝑢 = − cos 𝑢 + 𝑐
∫ cot 𝑢 𝑑𝑢 = ln | sin 𝑢| + 𝑐
∫[𝑓(𝑢) ± 𝑔(𝑢)] 𝑑𝑢
∫ cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝑐
∫ csc 2 𝑢 𝑑𝑢 = − cot 𝑢 + 𝑐
∫ tan 𝑢 𝑑𝑢 = − ln|cos 𝑢| + 𝑐
∫ sec 2 𝑢 𝑑𝑢 = tan 𝑢 + 𝑐
∫ csc 𝑢 𝑑𝑢 = ln|sin 𝑢| + 𝑐
𝑢
= arcsin ( ) + 𝑐
𝑎
√𝑎2 − 𝑢2
𝑑𝑢
1
|𝑢|
∫
= arcsec ( ) + 𝑐
𝑎
𝑢√𝑢2 − 𝑎2 a
= ∫ 𝑓(𝑢)𝑑𝑢 ± ∫ 𝑔(𝑢)𝑑𝑢
∫ 𝑑𝑢 = 𝑢 + 𝑐
1
∫ 𝑎𝑢 𝑑𝑢 = ( ) 𝑎𝑢 + 𝑐
𝑙𝑛𝑎
𝑢
𝑢
∫ 𝑒 𝑑𝑢 = 𝑒 + 𝑐
∫ sec 𝑢 𝑑𝑢 = ln|sec 𝑢 + tan 𝑢| + 𝑐
∫
∫
𝑑𝑢
𝑑𝑢
1
𝑢
= ( ) arctan ( ) + 𝑐
𝑎2 + 𝑢 2
𝑎
𝑎