𝑑𝑦
𝑑𝑡
𝑑𝑥
𝑑𝑡
𝑑2 𝑦
𝑑𝑦 ′
)
𝑑𝑥
𝑑𝑥
𝑑𝑡
(

𝑑𝑦

𝑆 = 2𝜋 ∫𝛼 𝑥√( 𝑑𝑡 ) + ( 𝑑𝑡 ) , if rotating about the 𝑦 − 𝑎𝑥𝑖𝑠, put 𝑦 instead of 𝑥 if rotating about the 𝑥 − 𝑎𝑥𝑖𝑠

to find the horizontal tangent to a parametric curve, 𝑑𝑡 = 0, for vertical, 𝑑𝑡 𝐷𝑁𝐸

if there is a value of 𝑡 that makes 𝑑𝑥 both zero and 𝐷𝑁𝐸, then use lim 𝑑𝑥 ,

𝑑𝑦

Symmetry test: 𝑎𝑏𝑜𝑢𝑡 ? −𝑎𝑥𝑖𝑠 )

sin(𝜃 ± 𝛼 ) = 𝑠𝑖𝑛𝜃 cos 𝛼 ± 𝑠𝑖𝑛𝛼 cos 𝜃

𝑟 2 = 𝑥 2 + 𝑦 2 , 𝜃 = tan−1 (𝑥 )

𝐴=

In graph, 𝑟 = 𝑎 is the same as 𝑟 = −𝑎

𝐿 = ∫𝑎 √𝑟 2 + (𝑑𝜃) 𝑑𝜃

⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
𝐴𝐶 = −𝐶𝐴

Slope ≡ 𝑟𝑢𝑛

𝑎 ⋅ 𝑎 = |𝑎 |2

Angle b\w two vectors is cos(𝜃) = |𝑎||𝑏|

To get a vector which is parallel to a vector, get the unit vector.

Two vectors are parallel if and only if the ratio of their components are equal, 𝑎 =< 𝑎, 𝑏, 𝑐 >, 𝑏⃗ =< 𝑑, 𝑒, 𝑓 >, 𝑑 = 𝑒 = 𝑓

Volume of parallelepiped is 𝑉 = |𝑎 ⋅ (𝑏⃗ × 𝑐)|

𝑎 ⋅ (𝑏⃗ + 𝑐) = 𝑎 ⋅ 𝑏⃗ + 𝑎 ⋅ 𝑐

Symmetric equations are

Distance b\w two parallel planes is 𝐷 =

Two variable limits,:
𝑑𝑥
=
,
𝑑𝑥 2
𝛽
=
𝑑𝑥 2
𝑑𝑦 2
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑦
t−?
𝑟′ sin 𝜃+𝑟 cos 𝜃
= 𝑟′ cos 𝜃−𝑟 sin 𝜃
𝑑𝑥
𝑦
1
2
𝑏
∫𝑎 𝑟 2 𝑑𝜃
𝑑𝑟 2
𝑏
𝑅𝑖𝑠𝑒
⃗
𝑎⃗ ⋅𝑏
𝑎
𝑥−𝑥0
𝑎
=
𝑦−𝑦0
𝑏
=
𝑏
𝑐
𝑧−𝑧0
𝑐
|𝑑1 −𝑑2 |
|𝑛|
Direct substitution  Split the limit Algebraic(conjugate) u substitution Squeeze Polar . type of the function depends;
you should look at it and see which is best.
 Path testing for two variable limits is ONLY for checking EXISTANCE, NOT FOR CALCULATING THE VALUE of the limit.
 To check a path, change the function based on the path’s equation(substitute x,,,) calculate the limit of this new function.
 𝑧 is always considered as a function of 𝑥 and 𝑦 unless it is declared otherwise.

𝜕2 𝑧
𝜕𝑦𝜕𝑥
means first differentiate z with respect to x then with respect to y.
 Equation of tangent plane 𝑧 − 𝑧0 = 𝑓𝑥 ⋅ (𝑥 − 𝑥0 ) + 𝑓𝑦 ⋅ (𝑦 − 𝑦0 ),
 Equation of linearization near 𝑥0 and 𝑦0 is z from the tangent plane equation 𝑧 = 𝐿(𝑥, 𝑦) = 𝑧0 + 𝑓𝑥 ⋅ (𝑥 − 𝑥0 ) + 𝑓𝑦 ⋅ (𝑦 − 𝑦0 )
(𝑧0 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑧 𝑎𝑡 𝑥0 𝑎𝑛𝑑 𝑦0 , 𝑜𝑟 𝑓(𝑥0 , 𝑦0 ))

𝑓 is differentiable at (𝑎, 𝑏) if and only if 𝑓𝑥 and 𝑓𝑦 are continuous at (𝑎, 𝑏)

𝑑𝑧 = 𝜕𝑥 𝑑𝑥 + 𝜕𝑦 𝑑𝑦

Partial derivative of multiple variables 𝜕𝑡 = 𝜕𝑥 ⋅ 𝑑𝑡 + 𝜕𝑦 ⋅ 𝑑𝑡 , and it is better to draw a tree to know

𝑑𝑦

∇𝑓 (𝑥, 𝑦, 𝑧) =< 𝑓𝑥 , 𝑓𝑦 , 𝑓𝑧 >

𝐷𝑢 𝑓 (𝑥0 , 𝑦0 ) = ∇𝑓 (𝑥0 , 𝑦0 ) ⋅ 𝑢 = |∇𝑓(𝑥0 , 𝑦0 )| cos 𝜃, 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒𝑡𝑎 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑙𝑒 𝑏\𝑤 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑎𝑛𝑑 ∇𝑓

In Directional derivative, 𝑢 must be a unit vector, if you were given a ordinary vector, you must convert it to its unit

The value of maximum directional derivative is |∇𝑓(𝑥, 𝑦, 𝑧)| and the minimum −|∇𝑓(𝑥, 𝑦, 𝑧)|

The maximum directional derivative occurs at many points, at points where 𝑢 has the same direction as ∇𝑓

The gradient vector is the normal of the tangent plane at some point

Extreme values ≡ Local max or min

Critical points occur at either when {𝑓𝑥 = 𝑓𝑦 = 0} or {𝑓𝑥 𝑜𝑟 𝑓𝑦 𝐷𝑁𝐸}

𝜕𝑧
𝜕𝑧
𝜕𝑧
𝑑𝑥
−𝐹
𝜕𝑧
𝜕𝑧
𝑑𝑥
𝜕𝑧
𝑑𝑦
𝐹
= 𝐹 𝑥 , 𝑖𝑓 𝐹 (𝑥, 𝑦) = 0, 𝑎𝑛𝑑 𝜕𝑥 = − 𝐹𝑥 , 𝑜𝑟 𝑦, 𝑖𝑓 𝐹 (𝑥, 𝑦, 𝑧) = 0
𝑦
𝑧