Multivariable Calculus Cheat Sheet

Multivariable Calculus Cheat Sheet By The WeSolveThem.com Team Simplicity is the ultimate sophistication. Leonardo da Vinci

Table of Contents 3D Calculus ......................................................................................................................................... 6 Magnitude ...................................................................................................................................................................................................... 6 Unit Vectors .................................................................................................................................................................................................. 6 Dot/Cross Product ..................................................................................................................................................................................... 6 Dot ..................................................................................................................................................................................................................... 6 Properties ....................................................................................................................................................................................................... 6 Cross .................................................................................................................................................................................................................. 6 Properties ....................................................................................................................................................................................................... 7 Angles Between Vectors .......................................................................................................................................................................... 7 Projections .................................................................................................................................................................................................... 7 Areas/Volume .............................................................................................................................................................................................. 7 Triangle ........................................................................................................................................................................................................... 7 Parallelogram ............................................................................................................................................................................................... 7 Parallelepiped ............................................................................................................................................................................................... 7 Line ................................................................................................................................................................................................................... 7 Line from tip to tip ...................................................................................................................................................................................... 7 Equation of a Plane .................................................................................................................................................................................... 8 Vector Functions ........................................................................................................................................................................................ 8 Limit .................................................................................................................................................................................................................. 8 Derivative ....................................................................................................................................................................................................... 8 Definite Integral .......................................................................................................................................................................................... 8 Indefinite Integral ....................................................................................................................................................................................... 8 Differentiation Rules ................................................................................................................................................................................. 8 Arc length ...................................................................................................................................................................................................... 8 Tangents ......................................................................................................................................................................................................... 9 Unit Tangent Vector .................................................................................................................................................................................. 9 Curvature 1 .................................................................................................................................................................................................... 9 Curvature 2 (vector function) ................................................................................................................................................................ 9 Curvature 3 (single variable) ............................................................................................................................................................... 9 Curvature 4 (parametric) ........................................................................................................................................................................ 9 Normal Vector .............................................................................................................................................................................................. 9 Binormal Vector .......................................................................................................................................................................................... 9 Tangential and Normal Components (acceleration) ............................................................................................................... 10 Physics Notations .................................................................................................................................................................................... 10 Position ......................................................................................................................................................................................................... 10 Velocity ......................................................................................................................................................................................................... 10 Speed .............................................................................................................................................................................................................. 10 Acceleration ................................................................................................................................................................................................ 10 Curvature ..................................................................................................................................................................................................... 10 Tangential Component (acceleration) ........................................................................................................................................... 10 Normal Component (acceleration) .................................................................................................................................................. 10 Acceleration ................................................................................................................................................................................................ 10 Note: .............................................................................................................................................................................................................. 10 Dot Product of Velocity and Acceleration ..................................................................................................................................... 10 Tangential Acceleration ........................................................................................................................................................................ 10 Normal Acceleration ............................................................................................................................................................................... 10 Frenet-Serret Formulas ......................................................................................................................................................................... 11 Partial Derivatives ............................................................................................................................ 11 Mixed Partial ............................................................................................................................................................................................. 11

Equation of a Plane ................................................................................................................................................................................. 11 Normal Vector ........................................................................................................................................................................................... 11 Distance/Vector Between Points ..................................................................................................................................................... 11 Vector from two points .......................................................................................................................................................................... 11 Tangent Plane ........................................................................................................................................................................................... 11 Equation of a sphere .............................................................................................................................................................................. 12 Chain Rule .................................................................................................................................................................................................. 12 Gradient ∇f ................................................................................................................................................................................................. 12 Directional Derivative ........................................................................................................................................................................... 12 Differentials ............................................................................................................................................................................................... 12 Implicit Differentiation ......................................................................................................................................................................... 12

Extrema ........................................................................................................................................... 13 Lagrange Multipliers .............................................................................................................................................................................. 13 Two Constraints ........................................................................................................................................................................................ 14 Multiple Integrals ............................................................................................................................. 14 Double .......................................................................................................................................................................................................... 14 Average Value ........................................................................................................................................................................................... 14 Type I ............................................................................................................................................................................................................ 14 Type II .......................................................................................................................................................................................................... 14 Polar .............................................................................................................................................................................................................. 15 Type III ......................................................................................................................................................................................................... 15 Moments & Center of Mass ................................................................................................................................................................. 15 Moments ....................................................................................................................................................................................................... 15 Center of mass ........................................................................................................................................................................................... 15 Moment of Inertia .................................................................................................................................................................................... 15 Surface Area .............................................................................................................................................................................................. 16 Triple Integrals ......................................................................................................................................................................................... 16 Moments & Center of Mass ................................................................................................................................................................. 16 Moments ....................................................................................................................................................................................................... 16 Center of Mass ........................................................................................................................................................................................... 17 Moments of Inertia .................................................................................................................................................................................. 17 Cylindrical Coordinates ........................................................................................................................................................................ 17 Spherical Coordinates ........................................................................................................................................................................... 17 Change of Variables ................................................................................................................................................................................ 17 2D Jacobian ................................................................................................................................................................................................. 17 3D Jacobian ................................................................................................................................................................................................. 18 Line Integrals .................................................................................................................................... 18 General ......................................................................................................................................................................................................... 18 Smooth .......................................................................................................................................................................................................... 18 Not Smooth ................................................................................................................................................................................................. 18 x, y Derivatives ......................................................................................................................................................................................... 18 Vector form ................................................................................................................................................................................................ 19 Respect to z ................................................................................................................................................................................................ 19 Multiple Functions P, Q, R .................................................................................................................................................................... 19 Work ............................................................................................................................................................................................................. 19 Gradient Line Integral ........................................................................................................................................................................... 20 Conservative Vector Field ................................................................................................................................................................... 20 Green’s Theorem ..................................................................................................................................................................................... 20 Curl ∇ ............................................................................................................................................................................................................ 20 Divergence ................................................................................................................................................................................................. 20 Stokes Theorem ....................................................................................................................................................................................... 20

Divergence Theorem ............................................................................................................................................................................. 21

PreCalculus Review .......................................................................................................................... 21 Arithmetic ................................................................................................................................................................................................... 21 Exponential ................................................................................................................................................................................................ 21 Radicals ....................................................................................................................................................................................................... 21 Fractions ..................................................................................................................................................................................................... 21 Logarithmic ................................................................................................................................................................................................ 22 Other Formulas/Equations ................................................................................................................................................................. 22 Areas ............................................................................................................................................................................................................. 24 Surface Areas ............................................................................................................................................................................................ 24 Volumes ....................................................................................................................................................................................................... 24 Domain Restrictions .............................................................................................................................................................................. 24 Right Triangle ........................................................................................................................................................................................... 25 Reciprocal Identities ............................................................................................................................................................................... 25 Double Angle Formulas ........................................................................................................................................................................ 26 Half Angle Formulas ............................................................................................................................................................................... 26 Sum and Difference Formulas ............................................................................................................................................................ 26 Product to Sum Formulas ..................................................................................................................................................................... 26 Sum to Product Formulas ..................................................................................................................................................................... 26 Unit Circle ........................................................................................................................................ 27 Pre-CALC III Reference ...................................................................................................................... 28 Derivative Rules (prime notations) ................................................................................................................................................ 28 Derivative of a Constant ........................................................................................................................................................................ 28 Power Rule .................................................................................................................................................................................................. 28 Constant Multiple Rule .......................................................................................................................................................................... 28 Product Rule ............................................................................................................................................................................................... 28 Quotient Rule ............................................................................................................................................................................................. 28 Chain Rule ................................................................................................................................................................................................... 28 Exponential and Logarithmic ............................................................................................................................................................. 28 exp{u} ............................................................................................................................................................................................................ 28 Natural Log ................................................................................................................................................................................................. 28 Base Log ....................................................................................................................................................................................................... 28 Exponential ................................................................................................................................................................................................. 28 Inverse Function Derivative ............................................................................................................................................................... 28 Trig Derivatives ....................................................................................................................................................................................... 29 Standard ...................................................................................................................................................................................................... 29 Inverse ........................................................................................................................................................................................................... 29 Common Derivatives ............................................................................................................................................................................. 29 Operator ....................................................................................................................................................................................................... 29 Prime ............................................................................................................................................................................................................. 30 Implicit Differentiation ..................................................................................................................... 30 Tangent Line .............................................................................................................................................................................................. 30 Related Rates ............................................................................................................................................................................................ 31 Hyperbolic Functions ........................................................................................................................ 31 Notation ....................................................................................................................................................................................................... 31 Identities ..................................................................................................................................................................................................... 31 Derivatives ................................................................................................................................................................................................. 32 Standard ...................................................................................................................................................................................................... 32 Inverse ........................................................................................................................................................................................................... 32

Antiderivatives & Integration ........................................................................................................... 32 Basic Rules ................................................................................................................................................................................................. 32 Riemann Sum for Area Approximation ......................................................................................................................................... 33 Area Approximation Rules .................................................................................................................................................................. 33 Midpoint Rule ............................................................................................................................................................................................. 33 Trapezoid Rule .......................................................................................................................................................................................... 33 The Integral Notation ∫ ................................................................................................................... 34 Definite Integral Properties ................................................................................................................................................................ 34 Fundamental Theorems ....................................................................................................................................................................... 34 Limit Definition of a Definite Integral ............................................................................................................................................ 35 Differential Equation (1st order) ....................................................................................................................................................... 35 Common Integrals ................................................................................................................................................................................... 35 Definite Integral Rules .......................................................................................................................................................................... 36 Substitution ................................................................................................................................................................................................ 36 Integration by Parts ................................................................................................................................................................................ 36 Trig Substitution ...................................................................................................................................................................................... 36 Trig Identity ............................................................................................................................................................................................... 36 Partial Fractions ...................................................................................................................................................................................... 36

3D Calculus Magnitude 𝑣 = 𝐯 = 𝑣! , 𝑣! , 𝑣! Unit Vectors 𝑣=

𝑣 𝑣

⇒

𝑣 = 𝐯 =

≡ 𝐯≡𝐮=

𝑣!! + 𝑣!! + 𝑣!!

𝐯 𝐯

𝚤 ≡ 𝐢 𝚥 ≡ 𝐣 𝑘 ≡ 𝐤 𝚤 = 1, 0, 0 𝚥 = 0, 1, 0 𝑘 = 1, 0, 0 Note: 𝑣 = 𝑣! , 𝑣! , 𝑣! = 𝑣! 1, 0, 0 + 𝑣! 0, 1, 0 + 𝑣! 0, 0, 1 = 𝑣! 𝚤 + 𝑣! 𝚥 + 𝑣! 𝑘 = 𝑣! 𝐢 + 𝑣! 𝐣 + 𝑣! 𝐤 Dot/Cross Product Dot 𝑎 ⋅ 𝑏 = 𝐚 ⋅ 𝐛 = 𝑎! , 𝑎! , 𝑎! ⋅ 𝑏! , 𝑏! , 𝑏! = 𝑎! 𝑏! + 𝑎! 𝑏! + 𝑎! 𝑏! Properties 𝐚 ⋅ 𝐚 = 𝐚 ! 𝐚 ⋅ 𝐛 + 𝐜 = 𝐚 ⋅ 𝐛 + 𝐚 ⋅ 𝐜

𝐚 ⋅ 𝐛 = 𝐛 ⋅ 𝐚 k𝐚 ⋅ 𝐛 = k 𝐚 ⋅ 𝐛 = 𝐚 ⋅ k𝐛

Cross 𝑎×𝑏 = 𝐚×𝐛 = 𝑎! , 𝑎! , 𝑎! × 𝑏! , 𝑏! , 𝑏! 𝚤 𝚥 𝑘 = 𝑎! 𝑎! 𝑎! 𝑏! 𝑏! 𝑏! 𝑎! 𝑎! 𝑎! 𝑎! 𝑎! 𝑎! = 𝑏 𝑏 𝚤 − 𝑏 𝑏 𝚥 + 𝑏 𝑏 𝑘 ! ! ! ! ! ! = 𝑎! 𝑏! − 𝑏! 𝑎! 𝚤 − 𝑎! 𝑏! − 𝑏! 𝑎! 𝚥 + 𝑎! 𝑏! − 𝑏! 𝑎! 𝑘

Properties 𝐚×𝐛 = −𝐛×𝐚 k𝐚 ×𝐛 = k 𝐚×𝐛 = 𝐚× k𝐛 𝐚 ⋅ 𝐛×𝐜 = 𝐚×𝐛 ⋅ 𝐜 𝐚 + 𝐛 ×𝐜 = 𝐚×𝐜 + 𝐛×𝐜 𝐚× 𝐛 + 𝐜 = 𝐚×𝐛 + 𝐚×𝐜 𝐚× 𝐛×𝐜 = 𝐚 ⋅ 𝐜 𝐛 − 𝐚 ⋅ 𝐛 𝐜 Angles Between Vectors 𝑎 ⋅ 𝑏 = 𝑎 𝑏 cos 𝜃 𝑎×𝑏 = 𝑎 𝑏 sin 𝜃 𝑎⋅𝑏 𝑎×𝑏 ⇒ 𝜃 = arccos ⇒ 𝜃 = arcsin 𝑎 𝑏 𝑎 𝑏 Projections Scalar Vector 𝑎⋅𝑏 𝑎⋅𝑏 comp! 𝑏 = proj𝐚 𝑏 = 𝑎 𝑎 𝑎! Areas/Volume Triangle Parallelogram Parallelepiped 1 𝐴 = 𝑎×𝑏 𝑉 = 𝑎 ⋅ 𝑏×𝑐 𝐴 = 𝑎×𝑏 2 Line ℒ 𝑡 = 𝑃! + 𝑡𝑣 𝑣 = 𝑃! 𝑃! = 𝑃! − 𝑃! = 𝑥! , 𝑦! , 𝑧! + 𝑡 𝑎, 𝑏, 𝑐 = 𝑥, 𝑦, 𝑧 − 𝑥! , 𝑦! , 𝑧! = 𝑥! + 𝑎𝑡, 𝑦! + 𝑏𝑡, 𝑧! + 𝑐𝑡 = 𝑥 − 𝑥! , 𝑦 − 𝑦! , 𝑧 − 𝑧! = 𝑥! , 𝑦! , 𝑧! + 𝑡 𝑥 − 𝑥! , 𝑦 − 𝑦! , 𝑧 − 𝑧! = 𝑎, 𝑏, 𝑐 Line from tip to tip A line segment from the tips two vectors beginning from the origin to 𝑣! → 𝑣! is ℒ 𝑡 = 1 − 𝑡 𝑣! + 𝑡𝑣! , 𝑡 ∈ 0, 1

Equation of a Plane 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 ⇒ 𝑛 = 𝑎, 𝑏, 𝑐 ⊥ surface 𝑛 is perpendicular to the surface 𝑣 is in the plane, 𝑃! = 𝑥! , 𝑦! , 𝑧! (point in plane) 𝑛 ⊥ 𝑣 ⇒ 𝑛 ⋅ 𝑣 = 𝑎, 𝑏, 𝑐 ⋅ 𝑥 − 𝑥! , 𝑦 − 𝑦! , 𝑧 − 𝑧! = 𝑎 𝑥 − 𝑥! + 𝑏 𝑦 − 𝑦! + 𝑐 𝑧 − 𝑧! = 0 Vector Functions 𝑟 𝑡 = 𝑟! 𝑡 , 𝑟! 𝑡 , 𝑟! 𝑡 = 𝑓 𝑡 , 𝑔 𝑡 , ℎ 𝑡 Limit

lim 𝑟 𝑡 = lim 𝑓 𝑡 , lim 𝑔 𝑡 , lim ℎ 𝑡

!→!

!→!

!→!

!→!

𝑑𝑟 = 𝑓 ! 𝑡 , 𝑔! 𝑡 , ℎ! 𝑡 𝑑𝑡

Derivative

!!

Definite Integral

𝑟 𝑡 𝑑𝑡 =

!!

Indefinite Integral

!! !!

𝑟 𝑡 𝑑𝑡 =

!!

𝑟! 𝑡 𝑑𝑡 𝚤 +

𝑟! 𝑡 𝑑𝑡 𝚤 +

!!

𝑟! 𝑡 𝑑𝑡 𝚥 +

𝑟! 𝑡 𝑑𝑡 𝚥 +

Differentiation Rules Note: 𝑣 𝑡 , 𝑢 𝑡 , 𝑓 𝑡 Function dot Vector

Vector cross Vector

𝑑 𝑓 𝑡 ⋅𝑢 𝑡 𝑑𝑡

𝑑 𝑢 𝑡 ×𝑣 𝑡 𝑑𝑡

=𝑢 𝑡

𝑑𝑓 𝑑𝑢 +𝑓 𝑡 𝑑𝑡 𝑑𝑡

Vector dot Vector 𝑑 𝑢 𝑡 ⋅𝑣 𝑡 𝑑𝑡

=

!! !!

𝑟! 𝑡 𝑑𝑡 𝑘

𝑟! 𝑡 𝑑𝑡 𝑘 + 𝐶

𝑑𝑢 𝑑𝑣 ×𝑣 𝑡 + 𝑢 𝑡 × 𝑑𝑡 𝑑𝑡

Chain Rule

=𝑣 𝑡 ⋅

𝑑𝑢 𝑑𝑣 +𝑢 𝑡 ⋅ 𝑑𝑡 𝑑𝑡

!

!

𝑑 𝑢 𝑓 𝑡 𝑑𝑡

= 𝑢! 𝑓 𝑡 𝑓 ! 𝑡

Arc length 𝐿=

!! !!

𝑑𝑟! 𝑑𝑡

+

𝑑𝑟! 𝑑𝑡

+

𝑑𝑟! 𝑑𝑡

!

𝑑𝑡 =

!! !!

𝑓! 𝑡

!

+ 𝑔! 𝑡

!

+ ℎ! 𝑡

!

𝑑𝑡 =

!! !!

𝑑𝑟 𝑑𝑡 𝑑𝑡

Tangents Unit Tangent Vector

𝐓 𝑡 =

𝐫! 𝑡 , 𝐫! 𝑡

𝐫! 𝑡

=

𝑑𝑠 𝑑𝑡

Curvature 1

𝑑𝐓 𝑑𝐓 𝑑𝐓 𝑑𝑡 𝐓! 𝑡 𝑑𝑡 𝜅(𝑡) = = = = ! 𝑑𝑠 𝑑𝑠 𝑑𝑡 𝑑𝑠 𝐫 𝑡 𝑑𝑡

Curvature 2 (vector function)

𝐫 ! 𝑡 ×𝐫 !! 𝑡 𝜅(𝑡) = 𝐫! 𝑡 !

Curvature 3 (single variable)

𝜅(𝑥) =

𝑓 !! 𝑥

! ! !

1 + 𝑓! 𝑥 Curvature 4 (parametric)

𝜅 𝑡 =

𝑥 ! 𝑡 𝑦 !! 𝑡 − 𝑦 ! 𝑡 𝑥 !! 𝑡 𝑥! 𝑡

Normal Vector Binormal Vector

𝐍 𝑡 =

!

+ 𝑦! 𝑡

𝐓! 𝑡 𝐓! 𝑡

𝐁 𝑡 = 𝐓 𝑡 ×𝐍 𝑡

! ! !



Tangential and Normal Components (acceleration) Physics Notations Position 𝑟 𝑡 ≡𝐫 𝑡 Velocity

𝑣 𝑡 = 𝑟! 𝑡 =

𝑑𝑟 𝑑𝐫 = = 𝐫! 𝑡 𝑑𝑡 𝑑𝑡

= 𝑟! 𝑡

Speed

𝑣= 𝑣 𝑡

Acceleration

𝑎 𝑡 = 𝑣 ! 𝑡 = 𝑟 !! 𝑡

𝐓 𝑡 =

𝐫! 𝑡 𝐫! 𝑡

=

𝑣 𝑡 𝑣 𝑡

𝑣 = 𝑣

𝑣 = 𝑣𝐓 ⇒

𝑑𝑣 = 𝑎 = 𝑣 ! 𝐓 + 𝑣𝐓′ 𝑑𝑡

Curvature

𝜅=

𝐓! 𝐓! = ⇒ 𝜅𝑣 = 𝐓′ 𝐫! 𝑣

Tangential Component (acceleration)

𝑎! =

𝑑 ! 𝑑𝑣 𝑟 = = 𝑣 !, 𝑑𝑡 𝑑𝑡

𝑣 = 𝑣 = 𝑟′ ≡ 𝐫′

Normal Component (acceleration)

𝑎! = 𝜅𝑣 !

Acceleration

𝒂 = 𝑣 ! 𝐓 + 𝜅𝑣 ! 𝐍 = 𝑎! 𝐓 + 𝑎! 𝐍

Note:

𝐓 ⋅ 𝐓 = 1 ∧ 𝐓 ⋅ 𝐍 = 0

Dot Product of Velocity and Acceleration 𝑣 ⋅ 𝑎 = 𝑣𝐓 ⋅ 𝑣 ! 𝐓 + 𝜅𝑣 ! 𝐍 = 𝑣𝑣 ! 𝐓 ⋅ 𝐓 + 𝜅𝑣 ! 𝐓 ⋅ 𝐍 = 𝑣𝑣′ Tangential Acceleration Normal Acceleration

𝑎! = 𝑣 ! =

𝑣 ⋅ 𝑎 𝐫 ! 𝑡 ⋅ 𝐫 !! 𝑡 = 𝑣 𝐫! 𝑡

𝑎! = 𝜅𝑣 ! =

𝐫 ! 𝑡 × 𝐫 !! 𝑡 𝐫! 𝑡

Frenet-Serret Formulas 𝑑𝐓 = 𝜅𝐍 𝑑𝑠

𝑑𝐍 = −𝜅𝐓 + 𝜏𝐁 𝑑𝑠

𝑑𝐁 = −𝜏𝐍 𝑑𝑥

Partial Derivatives Given a multivariable function e.g. 𝑓 𝑥, 𝑦, 𝑧 , then a partial derivative is the derivative with respect to a variable where the other variables are treating as constants i.e. do not implicitly differentiate. 𝜕𝑓 𝜕𝑓 𝜕𝑓 = 𝑓! = 𝑓! 𝑥, 𝑦, 𝑧 = 𝑓! = 𝑓! 𝑥, 𝑦, 𝑧 = 𝑓! = 𝑓! 𝑥, 𝑦, 𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕!𝑓 𝜕!𝑓 𝜕!𝑓 = 𝑓 = 𝑓 = 𝑓!! !! !! 𝜕𝑥 ! 𝜕𝑦 ! 𝜕𝑧 ! Mixed Partial 𝜕!𝑓 𝜕!𝑓 = 𝑓!" , = 𝑓!" 𝜕𝑥𝜕𝑦 𝜕𝑦𝜕𝑥 Equation of a Plane 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 Normal Vector The normal vector 𝑛 = 𝑎, 𝑏, 𝑐 , is extracted from the equation of a plane, and the normal vector is perpendicular to the surface. Distance/Vector Between Points Vector from two points 𝑃! 𝑎, 𝑏, 𝑐 ∧ 𝑃! 𝑑, 𝑒, 𝑓 ⇒ 𝑃! 𝑃! = 𝑃! − 𝑃! = 𝑑 − 𝑎, 𝑒 − 𝑏, 𝑓 − 𝑐 𝑃! 𝑃! = 𝑑 − 𝑎 ! + 𝑒 − 𝑏 ! + 𝑓 − 𝑐 ! Tangent Plane 𝑧 − 𝑧! = 𝑓! 𝑥! , 𝑦! 𝑥 − 𝑥! + 𝑓! 𝑥! , 𝑦! 𝑦 − 𝑦!

Equation of a sphere 𝑥 − ℎ ! + 𝑦 − 𝑘 ! + 𝑧 − 𝑙 ! = 𝑟 ! , center: ℎ, 𝑘, 𝑙 radius: 𝑟 Chain Rule 𝑑𝑧 𝜕𝑧 𝑑𝑥 𝜕𝑧 𝑑𝑦 = + , 𝑥 =𝑥 𝑡 ∧𝑦 =𝑦 𝑡 𝑑𝑡 𝜕𝑥 𝑑𝑡 𝜕𝑦 𝑑𝑡 𝜕𝑧 𝜕𝑧 𝜕𝑥 𝜕𝑧 𝜕𝑦 𝜕𝑧 𝜕𝑧 𝜕𝑥 𝜕𝑧 𝜕𝑦 = + , = + , 𝑥 = 𝑥 𝑠, 𝑡 ∧ 𝑦 = 𝑦 𝑠, 𝑡 𝜕𝑠 𝜕𝑥 𝜕𝑠 𝜕𝑦 𝜕𝑠 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑦 𝜕𝑡 Gradient 𝜵𝒇 The symbol 𝛻 is called nabla or del; 𝜕 is called partial or del. It would be appropriate to use “del” as del is for partial derivatives just as nabla is. The gradient of 𝑓 is noted as 𝛻𝑓, and is equal the vector function of partials i.e. 𝜕𝑓 𝜕𝑓 𝜕𝑓 𝛻𝑓 = 𝐢+ 𝐣+ 𝐤 𝜕𝑥 𝜕𝑦 𝜕𝑧 Directional Derivative Given 𝑓 𝑥, 𝑦, 𝑧 , 𝑣 = 𝑣! , 𝑣! , 𝑣! , and 𝑃 𝑥! , 𝑦! , 𝑧! 𝑣 𝐷𝐮 𝑓 ≡ 𝛻𝑓 𝑥! , 𝑦! , 𝑧! ⋅ 𝐮 ≡ 𝛻𝑓 𝑥! , 𝑦! , 𝑧! ⋅ 𝑣 1 𝐷𝐮 𝑓 = 𝑓 𝑥 , 𝑦 , 𝑧 , 𝑓 𝑥 , 𝑦 , 𝑧 , 𝑓 𝑥 , 𝑦 , 𝑧 ⋅ 𝑣! , 𝑣! , 𝑣! 𝑣 ! ! ! ! ! ! ! ! ! ! ! ! Differentials 𝑑𝑓 = 𝑓! 𝑥, 𝑦 𝛥𝑥 + 𝑓! 𝑥, 𝑦 𝛥𝑦 + 𝑓! 𝑥, 𝑦 𝛥𝑧 Implicit Differentiation 𝜕𝐹 𝜕𝐹 𝜕𝑧 𝜕𝑧 𝜕𝑦 = − 𝜕𝑥 ∧ =− 𝜕𝐹 𝜕𝐹 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑧

Extrema Given a three-dimensional function 𝑓, we can find the extrema by using partial derivatives, and derivative tests. Process: Set 𝑓! = 0 Set 𝑓! = 0 Solve for 𝑥, 𝑦 = 𝑐! , 𝑐! (critical point) Evaluate 𝑓!! 𝑐! , 𝑐! 𝑓!! 𝑓!" 𝑓!" True: 𝑓!" = 𝑓!! 𝑓!! 𝑓!" ! 𝐷= = 𝑓!! 𝑓!! − 𝑓!" 𝑓!" 𝑓!! Local Min: 𝐷 > 0 and 𝑓!! 𝑐! , 𝑐! > 0 Local Max: 𝐷 > 0 and 𝑓!! 𝑐! , 𝑐! < 0 Saddle: 𝐷 < 0 Lagrange Multipliers These are like puzzles i.e. the set up is pretty straight forward, but you may need to make multiple attempts to find the right pattern. 2D Given 𝑓 𝑥, 𝑦 (function) and 𝑔 𝑥, 𝑦 = 𝑘 (constraint) then 𝛻𝑓 𝑥, 𝑦 = 𝜆𝛻𝑔 𝑥, 𝑦 Solve the following system: 𝑓! = 𝜆𝑔! 𝑓! = 𝜆𝑔! 𝑔 𝑥, 𝑦 = 𝑘 3D Given 𝑓 𝑥, 𝑦, 𝑧 (function) and 𝑔 𝑥, 𝑦, 𝑧 = 𝑘 (constraint) then 𝛻𝑓 𝑥, 𝑦, 𝑧 = 𝜆𝛻𝑔 𝑥, 𝑦, 𝑧 Solve the following system: 𝑓! = 𝜆𝑔! 𝑓! = 𝜆𝑔! 𝑓! = 𝜆𝑔! 𝑔 𝑥, 𝑦, 𝑧 = 𝑘 Once you find all possible values, then you simply plug them into 𝑓, and see which is largest/smallest. These are then your max/min.

Two Constraints

𝛻𝑓 𝑥, 𝑦, 𝑧 = 𝜆𝛻𝑔 𝑥, 𝑦, 𝑧 + 𝜇𝛻ℎ 𝑥, 𝑦, 𝑧

𝑓! = 𝜆𝑔! + 𝜇ℎ!

𝑓! = 𝜆𝑔! + 𝜇ℎ!

𝑓! = 𝜆𝑔! + 𝜇ℎ!

𝑔 𝑥, 𝑦, 𝑧 = 𝑘!

ℎ 𝑥, 𝑦, 𝑧 = 𝑘!

Multiple Integrals Double !

!

𝑓 𝑥, 𝑦 𝑑𝑦𝑑𝑥 ≡ !

𝑓 𝑥, 𝑦 𝑑𝐴,

!

𝑅=

Note 1:

!

!

!

!

!

!

!

Note 2: 𝑓 𝑥, 𝑦 = 𝑓 𝑥 𝑔 𝑦

𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 ⇔ 𝑎 ≤ 𝑥 ≤ 𝑏 ∧ 𝑐 ≤ 𝑦 ≤ 𝑑 !

!

!

!

𝑓 𝑥, 𝑦 𝑑𝑦𝑑𝑥 = Average Value

!

!

!

𝑓 𝑥 𝑔 𝑦 𝑑𝑦𝑑𝑥 = !

!

1 1 𝑑−𝑐 𝑏−𝑎 Type I

!

𝑓 𝑥, 𝑦 𝑑𝐴 = !

!

!

𝑓 𝑥, 𝑦 𝑑𝐴 = !



≡ 𝑅 = 𝑎 , 𝑏 × 𝑐, 𝑑

!

𝑓 𝑥, 𝑦 𝑑𝑦𝑑𝑥 =

Type II

𝑥, 𝑦 𝑥 ∈ 𝑎 , 𝑏 , 𝑦 ∈ 𝑐, 𝑑

!

!

!! !

!

!

!! !

𝑓 𝑥 𝑑𝑥 !

!

𝑓 𝑥, 𝑦 𝑑𝑦𝑑𝑥 !

!

𝑓 𝑥, 𝑦 𝑑𝑦𝑑𝑥,

𝐷=

𝑥, 𝑦 𝑥 ∈ 𝑎 , 𝑏 , 𝑦 ∈ 𝑔! 𝑥 , 𝑔! 𝑥

𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦,

𝐷=

𝑥, 𝑦 𝑥 ∈ ℎ! 𝑦 , ℎ! 𝑦 , 𝑦 ∈ 𝑐, 𝑑

!! !

!! !

!

𝑔 𝑦 𝑑𝑦



Polar 𝑟! = 𝑥! + 𝑦! 𝑓 𝑥, 𝑦 𝑑𝐴 = !

𝑥 = 𝑟 cos 𝜃 !!

!!

!!

𝑦 = 𝑟 sin 𝜃

𝑟𝑓 𝑟 cos 𝜃 , 𝑟 sin 𝜃 𝑑𝑟𝑑𝜃,

𝑅=

!!

𝑟, 𝜃 𝑟 ∈ 𝑟! , 𝑟! , 𝜃 ∈ 𝜃! , 𝜃!

Note: Do not forget the extra 𝑟 multiplied by 𝑓

Type III 𝑓 is continuous on a polar region !!

𝑓 𝑥, 𝑦 𝑑𝐴 = !



!!

𝑅=

!! !

𝑓 𝑟 cos 𝜃 , 𝑟 sin 𝜃 𝑟 𝑑𝑟𝑑𝜃,

!! !

𝑟, 𝜃 𝑟 ∈ 𝑔! 𝜃 , 𝑔! 𝜃 , 𝜃 ∈ 𝜃! , 𝜃!

Moments & Center of Mass Moments 𝑀! 𝑦𝜌 𝑥, 𝑦 𝑑𝐴 !

𝑀!

𝑥𝜌 𝑥, 𝑦 𝑑𝐴 !

Center of mass 𝑀! 𝑥= 𝑚 𝑦=

𝑀! 𝑚

1 𝑚

!

1 𝑚

!

Moment of Inertia 𝐼!

𝑦𝜌 𝑥, 𝑦 𝑑𝐴,

𝑚=

𝑥𝜌 𝑥, 𝑦 𝑑𝐴,

𝑚=

𝑦 ! 𝜌 𝑥, 𝑦 𝑑𝐴 !

𝐼!

𝑥 ! 𝜌 𝑥, 𝑦 𝑑𝐴 !

𝐼! (about origin)

𝑥 ! + 𝑦 ! 𝜌 𝑥, 𝑦 𝑑𝐴 !



𝜌 𝑥, 𝑦 𝑑𝐴 !

𝜌 𝑥, 𝑦 𝑑𝐴 !

Surface Area 𝑧 = 𝑓 𝑥, 𝑦 , 𝑥, 𝑦 ∈ 𝐷, and 𝑓! , 𝑓! are continuous 𝐴! =

𝜕𝑧 𝜕𝑥

1+ !

!

+

𝜕𝑧 𝜕𝑦

!

𝑑𝐴 = !

1 + 𝑓! 𝑥, 𝑦

Triple Integrals 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉,

𝑅=

𝑥, 𝑦, 𝑧

!

𝑥! , 𝑥! × 𝑦! , 𝑦! × 𝑧! , 𝑧!

Type IV: !! !,!

𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = !

Type V:

!

!!

𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = Type VI:

!

!!

!!

𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = !

Moments & Center of Mass Moments 𝑀!" 𝑧𝜌 𝑥, 𝑦, 𝑧 𝑑𝑉

!!

!

𝑀!"

𝑥𝜌 𝑥, 𝑦, 𝑧 𝑑𝑉 !

𝑀!"

𝑦𝜌 𝑥, 𝑦, 𝑧 𝑑𝑉 !



+ 𝑓! 𝑥, 𝑦

!! !!

!! !!

!! !!

𝑓 𝑥, 𝑦, 𝑧 𝑑𝑧 𝑑𝐴

!! !,!

!! ! !! !

!! ! !! !

≡

!

!! !,!

𝑓 𝑥, 𝑦, 𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥

!! !,!

!! !,! !! !,!

𝑓 𝑥, 𝑦, 𝑧 𝑑𝑧 𝑑𝑥 𝑑𝑦

!

𝑑𝐴

𝑓 𝑥, 𝑦, 𝑧 𝑑𝑥𝑑𝑦𝑑𝑧

Center of Mass The centroid of 𝐸 is the center of mass (𝑥, 𝑦, 𝑧) for constant density. 𝑚= 𝑥=

𝑀!" 𝑚

𝑦=

Moments of Inertia 𝐼! =

𝜌 𝑥, 𝑦, 𝑧 𝑑𝑉 !

𝑦 ! + 𝑧 ! 𝜌 𝑥, 𝑦, 𝑧 𝑑𝑉

𝐼! =

!

Cylindrical Coordinates 𝑦 𝑟! = 𝑥! + 𝑦! tan 𝜃 = 𝑥 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = !

!

𝑧=

𝑥 ! + 𝑧 ! 𝜌 𝑥, 𝑦, 𝑧 𝑑𝑉 𝐼! =

𝑥 = 𝑟 cos 𝜃 !! !!

Note: Do not forget the extra 𝑟

𝑀!" 𝑚

!! ! !! !

!! ! !"# !,! !"# !

𝑀!" 𝑚

𝑥 ! + 𝑦 ! 𝜌 𝑥, 𝑦, 𝑧 𝑑𝑉 !

𝑦 = 𝑟 sin 𝜃

𝑟𝑓 𝑟 cos 𝜃 , 𝑟 sin 𝜃 , 𝑧 𝑑𝑧𝑑𝑟𝑑𝜃

!! ! !"# !,! !"# !

Spherical Coordinates 𝑥 = 𝜌 sin 𝜙 cos 𝜃 𝑦 = 𝜌 sin 𝜙 sin 𝜃 𝑧 = 𝜌 cos 𝜙 𝐸 = 𝜌, 𝜃, 𝜙 𝜌 ∈ 𝜌! , 𝜌! , 𝜃 ∈ 𝜃! , 𝜃! , 𝜙 ∈ 𝜙! , 𝜙! 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = !

!!

Change of Variables 2D Jacobian



!!

𝜕𝑥 𝜕 𝑥, 𝑦 = 𝜕𝑢 𝜕𝑦 𝜕 𝑢, 𝑣 𝜕𝑢

!!

!!

!!



𝜌! = 𝑥 ! + 𝑦 ! + 𝑧 !

𝑓 𝜌 sin 𝜙 cos 𝜃 , 𝜌 sin 𝜙 sin 𝜃 , 𝜌 cos 𝜙 𝜌! sin 𝜙 𝑑𝜌𝑑𝜃𝑑𝜙

!!

𝜕𝑥 𝜕𝑣 = 𝜕𝑥 𝜕𝑦 − 𝜕𝑥 𝜕𝑦 , 𝜕𝑦 𝜕𝑢 𝜕𝑣 𝜕𝑣 𝜕𝑢 𝜕𝑣

𝑓 𝑥, 𝑦 𝑑𝐴 = !

𝑧 = 𝑧

𝑓 𝑥 𝑢, 𝑣 , 𝑦 𝑢, 𝑣 !

𝑥 = 𝑥 𝑢, 𝑣 ∧ 𝑦 = 𝑦 𝑢, 𝑣

abs

𝜕 𝑥, 𝑦 𝜕 𝑢, 𝑣

𝑑𝑢 𝑑𝑣

Note: Do not confuse the determinant with the absolute value i.e. 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕 𝑥, 𝑦 𝜕𝑢 𝜕𝑣 ≠ = abs 𝜕𝑢 𝜕𝑣 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕 𝑢, 𝑣 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣 3D Jacobian 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕𝑤 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝐽= , 𝑥 = 𝑥 𝑢, 𝑣, 𝑤 ∧ 𝑦 = 𝑦 𝑢, 𝑣, 𝑤 ∧ 𝑧 = 𝑧 𝑢, 𝑣, 𝑤 𝜕𝑢 𝜕𝑣 𝜕𝑤 𝜕𝑧 𝜕𝑧 𝜕𝑧 𝜕𝑢 𝜕𝑣 𝜕𝑤 𝜕 𝑥, 𝑦 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 ⇒ 𝑑𝑉 = 𝐽 𝑑𝑢 𝑑𝑣 𝑑𝑤, 𝐽= 𝜕 𝑢, 𝑣 !

Line Integrals General Smooth 𝑓 𝑥, 𝑦 𝑑𝑠 = !

Not Smooth

!!

𝑓 𝑥, 𝑦 𝑑𝑠 = !

𝑓 𝑥, 𝑦 𝑑𝑠 + !!

𝒙, 𝒚 Derivatives Respect to 𝑥

!

+ 𝑦! 𝑡

𝑓 𝑥, 𝑦 𝑑𝑠 + ⋯ !!

!

𝑑𝑡

Respect to 𝑦

𝑓 𝑥, 𝑦 𝑑𝑦 !

𝑓 𝑥, 𝑦 𝑑𝑠 !!

!!

𝑓 𝑥, 𝑦 𝑑𝑥 !

Note: Changing direction of 𝑥, 𝑦

𝑥! 𝑡

𝑓 𝑥 𝑡 ,𝑦 𝑡

!!

!! !!

𝑓 𝑥 𝑡 , 𝑦 𝑡 𝑥 ! 𝑡 𝑑𝑡 𝑓 𝑥 𝑡 , 𝑦 𝑡 𝑦 ! 𝑡 𝑑𝑡

!!

𝑓 𝑥, 𝑦 𝑑𝑠 = − !!

Arc length

𝑓 𝑥, 𝑦 𝑑𝑠 = !!

𝑓 𝑥, 𝑦 𝑑𝑠 !

𝑓 𝑥, 𝑦 𝑑𝑠 !

Vector form 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑠 = !

!!

∵ 𝐫 𝑡 = 𝑥 𝑡 ,𝑦 𝑡 ,𝑧 𝑡 ∴

!!

!

+ 𝑦! 𝑡

𝑓 𝑥 𝑡 ,𝑦 𝑡 ,𝑧 𝑡

𝑥! 𝑡

∧ 𝐫 𝑡 !

𝑥! 𝑡

=

+ 𝑦! 𝑡

!

!

+ 𝑧! 𝑡

+ 𝑦! 𝑡 !

!!

Multiple Functions 𝑷, 𝑸, 𝑹

!! !!

𝑊=

𝑊= 𝑊=

Case VI

+ 𝑧! 𝑡 𝑓 𝐫 𝑡

!

𝐫 𝑡

𝑑𝑡

𝐅 𝑥, 𝑦, 𝑧 ⋅ 𝐓 𝑥, 𝑦, 𝑧 𝑑𝑠 𝐅 ⋅ 𝐓 𝑑𝑠 ! !!

𝐅 𝐫 𝑡

⋅

!! !!

𝐅 𝐫 𝑡

𝐫! 𝑡 𝐫! 𝑡

𝐫 ! 𝑡 𝑑𝑡

⋅ 𝐫 ! 𝑡 𝑑𝑡

!!

𝑊=

𝑔! 𝑡 + 𝑔! 𝑡 + 𝑔! ! 𝑑𝑡

!

𝑊=

Case V

𝑑𝑡

!!

!

Case IV

!

𝑓 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 𝑧 ! 𝑡 𝑑𝑡

𝑃 𝑥, 𝑦, 𝑧 𝑑𝑥 + 𝑄 𝑥, 𝑦, 𝑧 𝑑𝑦 + 𝑅 𝑥, 𝑦, 𝑧 𝑑𝑧 =

Case III

+ 𝑧! 𝑡

!!

Respect to 𝒛

Case II

! !!

𝑑𝑡 =

!!

Work Case I

!

!!



𝑥! 𝑡

𝑓 𝑥 𝑡 ,𝑦 𝑡 ,𝑧 𝑡

𝐅 𝐫 𝑡

⋅ 𝑑𝐫

!

𝑊=

𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧 , !

𝐅 = 𝑃, 𝑄, 𝑅

Gradient Line Integral Case I: Fundamental Theorem

!

Case II Case III Case IV

!

𝛻𝑓 ⋅ 𝑑𝐫 = 𝑓 𝐫 𝑡!

𝛻𝑓 ⋅ 𝑑𝐫 = 𝑓 x! , y! , 𝑧! − 𝑓 x! , 𝑦! , 𝑧!

!!

𝛻𝑓 𝐫 𝑡

!!

!!

⋅ 𝐫 ! 𝑡 𝑑𝑡 =

!!

!!

Conservative Vector Field

− 𝑓 𝐫 𝑡!

!!

𝑑 𝑓 𝐫 𝑡 𝑑𝑡 = 𝑓 𝐫 𝑡! 𝑑𝑡

𝛻𝑓 = 𝐅 x, y = 𝑃 x, y , 𝑄 x, y Green’s Theorem 𝑃𝑑𝑥 + 𝑄𝑑𝑦 = !

!

∧

𝜕𝑓 𝑑𝑥 𝜕𝑓 𝑑𝑦 𝜕𝑓 𝑑𝑧 + + 𝑑𝑡 𝜕𝑥 𝑑𝑡 𝜕𝑦 𝑑𝑡 𝜕𝑧 𝑑𝑡 − 𝑓 𝐫 𝑡!

𝜕𝑃 𝜕𝑄 = 𝜕𝑦 𝜕𝑥

𝜕𝑄 𝜕𝑃 − 𝑑𝐴 𝜕𝑥 𝜕𝑦

Curl 𝜵 Note: gradient of 𝑓 is 𝛻𝑓, and curl/divergence of 𝑓 is 𝛻×𝛻𝑓 and 𝛻 ⋅ 𝛻𝑓, where 𝛻 (nabla) is referred to as del. 𝜕 𝜕 𝜕 𝛻= 𝐢+ 𝐣 + 𝐤 ≡ 𝜕! , 𝜕! , 𝜕! 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝐢 𝐣 𝐤 𝜕𝑓 𝜕𝑓 𝜕𝑓 ! ! ! 𝛻×𝛻𝑓 = 𝛻×𝐅 = 𝜕! 𝜕! 𝜕! , 𝜕! , 𝜕! , 𝜕! ≡ 𝐢+ 𝐣+ 𝐤 𝜕𝑥 𝜕𝑦 𝜕 ! ! ! 𝜕! 𝜕! 𝜕! Conservative if curl 𝐅 = 0 Divergence ! ! ! 𝛻 ⋅ 𝛻𝑓 = 𝛻 ⋅ 𝐅 = 𝜕! , 𝜕! , 𝜕! ⋅ 𝜕! , 𝜕! , 𝜕! Stokes Theorem 𝛻𝑓 ⋅ 𝑑𝐫 = !

𝐅 ⋅ 𝑑𝐫 = !

∇×𝐅 ⋅ 𝑑𝐒 = !

curl 𝐅 ⋅ 𝑑𝐒 !

Divergence Theorem 𝐅 ⋅ 𝑑𝐒 = !



𝛻 ⋅ 𝛻𝑓 𝒅𝑽 = !

𝛻 ⋅ 𝐅 𝑑𝑉 = !

div 𝐅 𝑑𝑉 !

PreCalculus Review Arithmetic

𝑎 𝑏 = 𝑎 𝑐 𝑏𝑐

𝑎𝑏 ± 𝑎𝑐 = 𝑎 𝑏 ± 𝑐 = 𝑏 ± 𝑐 𝑎 𝑎−𝑏 𝑏−𝑎 = 𝑐−𝑑 𝑑−𝑐

𝑎𝑏 + 𝑎𝑐 = 𝑏 + 𝑐, 𝑎 ≠ 0 𝑎

𝑎 𝑎 𝑐 𝑎𝑐 = ∙ = 𝑏 1 𝑏 𝑏 𝑐 Exponential 𝑎! = 𝑎

𝑎=

𝑎=

!

!"

𝑎=

𝑎 𝑏

!

=

𝑎!! = 𝑎! 𝑏!

!

𝑎 = 𝑎!"

!

𝑎!

=

! 𝑎!

Fractions 𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 ± = 𝑏 𝑑 𝑏𝑑

𝑎 𝑏

!

!

!!

1 𝑎! =

𝑎 𝑏 = 𝑎 ∙ 𝑑 = 𝑎𝑑 𝑐 𝑏 𝑐 𝑏𝑐 𝑑

1 = 𝑎! 𝑎!!

𝑏! 𝑎!

𝑎!

𝑎! = 𝑎, 𝑛 𝑖𝑠 𝑜𝑑𝑑

𝑎!

=

! 𝑎!

𝑏 𝑎𝑏 = 𝑐 𝑐

𝑎

𝑎±𝑏 𝑎 𝑏 = ± 𝑐 𝑐 𝑐

𝑎! = 1

𝑎! = 𝑎!!! 𝑎! Radicals ! !

𝑎 𝑐 𝑎𝑑 ± 𝑏𝑐 ± = 𝑏 𝑑 𝑏𝑑

! !

𝑎! 𝑎! = 𝑎!!! ! !

𝑎!

= 𝑎!

!

!

!

= 𝑎! !

𝑎! = 𝑎 , 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎 = 𝑏

!

𝑎

!

𝑏

!

=

𝑎! ! 𝑏!

𝑎 = 𝑏

𝑔 𝑥 ℎ 𝑥 𝑔 𝑥 𝑟 𝑥 ± 𝑓 𝑥 ℎ 𝑥 ± = 𝑓 𝑥 𝑟 𝑥 𝑓 𝑥 𝑟 𝑥

! !



Logarithmic ln 𝑏 = log ! 𝑏 ln 𝑎 log ! 1 = 0

𝑦 = log ! 𝑥 ⇔ 𝑥 = 𝑏 !

𝑒 ≈ 2.72

log ! 𝑎 = 1

log ! 𝑎! = 𝑢

log ! 𝑢 = ln 𝑢

log ! 𝑢! = 𝑏 log ! 𝑢

log ! 𝑢𝑣 = log ! 𝑢 + log ! 𝑣

log !



𝑢 = log ! 𝑢 − log ! 𝑣 𝑣

log ! 𝑏 = !

𝑣 = ln 𝑢 ⇒ 𝑢 = 𝑒

!

𝑣=𝑒

⇒ 𝑢 = ln 𝑣

ln 𝑢! = 𝑏 ln 𝑢

1 𝑛!

ln 𝑒 ! = 𝑢 ⇒ 𝑒 !" ! = 𝑢

ln 1 = 0 ln 𝑢𝑣 = ln 𝑢 + ln 𝑣

Other Formulas/Equations Quadratic Formula

ln

𝑢 = ln 𝑢 − ln 𝑣 𝑣



𝑎𝑥 ! + 𝑏𝑥 + 𝑐 = 0

⇒

𝑥=

Discriminant

𝑒= !!!

ln 𝑎 = undefined, 𝑎 ≤ 0 ln 𝑒 ! = 1 ⇒ 𝑒 !" ! = 1

!

ln 𝑏 ln 𝑎

−𝑏 ± 𝑏 ! − 4𝑎𝑐 2𝑎

Two Real Solutions 𝑏 ! − 4𝑎𝑐 > 0 Repeated Solution 𝑏 ! − 4𝑎𝑐 = 0 Complex Solution 𝑥 = 𝛼 ± 𝛽𝑖 if 𝑏 ! − 4𝑎𝑐 < 0

Complete the Square

!

𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐

⇒

𝑏 𝑦=𝑎 𝑥+ 2𝑎

!

𝑏! +𝑐− 4𝑎

Other Formulas Distance Formula 𝐷=

Midpoint Formula !

𝑥 − 𝑥!

+ 𝑦 − 𝑦! !

𝑀=

Equation of a Line 𝑠𝑙𝑜𝑝𝑒 = 𝑚 =

𝑥 + 𝑥! 𝑦 + 𝑦! , 2 2

𝑦 = 𝑚𝑥 + 𝑏 𝑦! − 𝑦! = 𝑚 𝑥! − 𝑥! 𝐴𝑥 + 𝐵𝑦 = 𝐶 𝑦 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐 𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘 ! 𝑥 − ℎ + 𝑦 − 𝑘 ! = 𝑟!

𝑦! − 𝑦! 𝑥! − 𝑥!

Equation of Parabola Vertex: ℎ, 𝑘 Equation of Circle Center: ℎ, 𝑘 Radius: 𝑟 Equation of Ellipse

𝑥−ℎ 𝑎!

!

+

Right Point: ℎ + 𝑎, 𝑘

𝑦−𝑘 𝑏!

!

= 1

Left Point: ℎ − 𝑎, 𝑘 Top Point: ℎ, 𝑘 + 𝑏

Bottom Point: ℎ, 𝑘 − 𝑏 Equation of Hyperbola Center: ℎ, 𝑘 ! Slope: ± !

𝑥−ℎ 𝑎!

!

𝑦−𝑘 − 𝑏!

!

= 1

!

Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ + 𝑎, 𝑘 , ℎ − 𝑎, 𝑘 ! 𝑦−𝑘 𝑥−ℎ − 𝑎! 𝑏!

Equation of Hyperbola Center: ℎ, 𝑘 ! Slope: ± !

!

= 1

!

Asymptotes: 𝑦 = ± ! 𝑥 − ℎ + 𝑘 Vertices: ℎ, 𝑘 + 𝑏 , ℎ, 𝑘 − 𝑏



Areas Square: 𝐴 = 𝐿! = 𝑊 ! Rectangle: 𝐴 = 𝐿 ∙ 𝑊 Circle: 𝐴 = 𝜋 ∙ 𝑟 ! ! ! Ellipse: 𝐴 = 𝜋 ∙ 𝑎𝑏 Triangle: 𝐴 = ! 𝑏 ∙ ℎ Trapezoid: 𝐴 = ! 𝑎 + 𝑏 ∙ ℎ !" Parallelogram: 𝑏 ∙ ℎ Rhombus: 𝐴 = ! , 𝑝 and 𝑞 are the diagonals Surface Areas Cube: 𝐴! = 6𝐿! = 6𝑊 ! Box: 𝐴! = 2(𝐿𝑊 + 𝑊𝐻 + 𝐻𝐿) Sphere: 𝐴! = 4𝜋𝑟 ! Cone: 𝐴! = 𝜋𝑟 𝑟 + ℎ! + 𝑟 ! Cylinder: 2𝜋𝑟ℎ + 2𝜋𝑟 ! Volumes ! Cube: 𝑉 = 𝐿! = 𝑊 ! Box: 𝑉 = 𝐿 ∙ 𝑊 ∙ 𝐻 Sphere: 𝑉 = ! 𝜋 ∙ 𝑟 ! ! ! Cone: 𝑉 = ! 𝜋 ∙ 𝑟 ! ℎ Ellipsoid: 𝑉 = ! 𝜋 ∙ 𝑎𝑏𝑐, 𝑎, 𝑏, 𝑐 are the radii Domain Restrictions 𝑢 𝑦= , 𝑣≠0 𝑦 = 𝑢, 𝑢≥0 𝑦 = ln 𝑢 , 𝑢 > 0 𝑣 ! 𝑦 = 𝑎! , none 𝑦 = 𝑢 none if 𝑛 is odd, 𝑢 ≥ 0 if 𝑛 is even

Right Triangle

𝑥! + 𝑦! = 𝑟! ⇔ cos 𝛼 = tan 𝛼 = sin 𝛼 =

𝑟=

𝑥! + 𝑦!

𝑥 𝑟

𝑦 cos 𝛽 = 𝑟

𝑦 𝑥

𝑥 tan 𝛽 = 𝑦

𝑦 𝑟

𝑥 sin 𝛽 = 𝑟

𝑥 = 𝑟 cos 𝛼 𝑦 = 𝑟 sin 𝛼 𝑦 𝑦 𝛼 = arctan = tan!! 𝑥 𝑥 Reciprocal Identities 1 sin 𝜃 = csc 𝜃 1 csc 𝜃 = sec 𝜃 sin 𝜃 tan 𝜃 = cos 𝜃

𝑦 = 𝑟 cos 𝛽 𝑥 = 𝑟 sin 𝛽 𝛽 = arctan

𝑥 𝑥 = tan!! 𝑦 𝑦

csc 𝜃 =

1 sin 𝜃

tan 𝜃 =

1 cot 𝜃

sec 𝜃 =

1 cos 𝜃

cot 𝜃 =

1 tan 𝜃

cot 𝜃 =

cos 𝜃 sin 𝜃

Double Angle Formulas sin 2𝜃 = 2 sin 𝜃 cos 𝜃 cos 2𝜃 = 1 − 2 sin! 𝜃 cos 2𝜃 = cos ! 𝜃 − sin! 𝜃 cos 2𝜃 = 2 cos ! 𝜃 − 1 ! !"# ! cos 2𝜃 = 1 − 2 sin! 𝜃 tan 2𝜃 = !!!"#! !Officia Half Angle Formulas 1 1 1 − cos(2𝜃) sin! 𝜃 = 1 − cos 2𝜃 cos ! 𝜃 = 1 + 𝑐𝑜𝑠 2𝜃 tan! 𝜃 = 2 2 1 + cos(2𝜃) Sum and Difference Formulas sin 𝛼 ± 𝛽 = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽 cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽 tan 𝛼 ± tan 𝛽 tan 𝛼 ± 𝛽 = 1 ∓ tan 𝛼 𝑡𝑎𝑛𝛽 Product to Sum Formulas 1 1 sin 𝛼 sin 𝛽 = [cos 𝛼 − 𝛽 − cos(𝛼 + 𝛽)] cos 𝛼 cos 𝛽 = [cos 𝛼 − 𝛽 + cos(𝛼 + 𝛽)] 2 2 1 1 sin 𝛼 cos 𝛽 = [sin 𝛼 + 𝛽 + sin 𝛼 − 𝛽 ] cos 𝛼 sin 𝛽 = sin 𝛼 + 𝛽 − sin 𝛼 − 𝛽 2 2 Sum to Product Formulas 𝛼+𝛽 𝛼−𝛽 𝛼+𝛽 𝛼−𝛽 sin 𝛼 + sin 𝛽 = 2 sin cos sin 𝛼 − sin 𝛽 = 2 cos sin 2 2 2 2 𝛼+𝛽 𝛼−𝛽 𝛼+𝛽 𝛼−𝛽 cos 𝛼 + cos 𝛽 = 2 cos cos cos 𝛼 − cos 𝛽 = −2 sin sin 2 2 2 2

Unit Circle





Pre-CALC III Reference Derivative Rules (prime notations) Derivative of a Constant 𝑐 ! = 0 Power Rule 𝑥 ! ′ = 𝑛𝑥 !!! Constant Multiple Rule 𝑐𝑢 ! = 𝑐𝑢′ Product Rule 𝑢𝑣 ! = 𝑢𝑣 ! + 𝑣𝑢′ 𝑢 ! 𝑣𝑢! − 𝑢𝑣′ Quotient Rule = 𝑣 𝑣! Chain Rule



[𝑢 𝑣 ]′ = 𝑢! 𝑣 ∙ 𝑣′

Exponential and Logarithmic

𝑑 ! 𝑒 𝑑𝑥

exp{u}

Note: log ! 𝑎 ≡

!" ! !" !



Exponential

Inverse Function Derivative 𝑑 !! 𝑓 𝑥 𝑑𝑥

!

= 𝑒!

!

∙ 𝑓! 𝑥

𝑑 𝑓! 𝑥 ln 𝑓 𝑥 = 𝑑𝑥 𝑓 𝑥 𝑑 1 𝑓! 𝑥 log ! 𝑓 𝑥 = ⋅ 𝑑𝑥 ln 𝑏 𝑓 𝑥 𝑑 !! 𝑎 = 𝑎 ! ! 𝑓 ! 𝑥 ln 𝑎 𝑑𝑥

Natural Log

Base Log

Operator

!

=

1 𝑓!

𝑓 !!

𝑎

𝑒!

!

ln 𝑢

Prime = 𝑒 ! ⋅ 𝑢′

!

𝑢! = 𝑢

log ! 𝑢 ! = 𝑎!

!

= 𝑎! 𝑢! ln 𝑎

𝑓 !! 𝑎 = 𝑏 ⇔ 𝑓 𝑏 = 𝑎

,

1 𝑢! ⋅ ln 𝑏 𝑢

Trig Derivatives Standard sin 𝑢 ! = cos 𝑢 ∙ 𝑢! csc 𝑢

!

= − csc 𝑢 cot 𝑢 ∙ 𝑢!

Inverse sin!! 𝑢 csc !! 𝑢

!

!

𝑢′

=

=−

1 − 𝑢!

!

cos 𝑢



𝑢′

𝑢 𝑢! − 1

Common Derivatives Operator 𝑑 𝑑𝑦 𝑦= 𝑑𝑥 𝑑𝑥 𝑑 ! 𝑒 = 𝑒! 𝑑𝑥

sec 𝑢

!

= − sin 𝑢 ∙ 𝑢!

tan 𝑢

= sec 𝑢 tan 𝑢 ∙ 𝑢!

cos !! 𝑢

!

sec !! 𝑢

!

=− =

𝑢′ 1 − 𝑢!



𝑢′

𝑢 𝑢! − 1

cot 𝑢

!

!

= sec ! 𝑢 ∙ 𝑢!

= − csc ! 𝑢 ∙ 𝑢′

tan!! 𝑢 cot !! 𝑢

!

!

=

𝑢′ 1 + 𝑢!

=−

𝑢′ 1 + 𝑢!

𝑑 𝑓! 𝑥 ln 𝑓 𝑥 = 𝑑𝑥 𝑓 𝑥

𝑑 ! 𝑥 = 𝑛𝑥 !!! 𝑑𝑥 𝑑 !! 𝑒 = 𝑒! ! 𝑓! 𝑥 𝑑𝑥 𝑑 ! 𝑎 = 𝑎 ! ln 𝑎 𝑑𝑥

𝑑 ! 𝑑𝑦 𝑦 = 𝑛𝑦 !!! 𝑑𝑥 𝑑𝑥 𝑑 1 ln 𝑥 = 𝑑𝑥 𝑥 𝑑 !! 𝑎 = 𝑎 ! ! 𝑓 ! 𝑥 ln 𝑎 𝑑𝑥

𝑑 sin 𝑥 = cos 𝑥 𝑑𝑥 𝑑 (sec 𝑥) = sec 𝑥 tan 𝑥 𝑑𝑥 𝑑 1 sin!! 𝑥 = 𝑑𝑥 1 − 𝑥!

𝑑 csc 𝑥 = −csc 𝑥 cot 𝑥 𝑑𝑥 𝑑 tan 𝑥 = sec ! 𝑥 𝑑𝑥 𝑑 −1 csc !! 𝑥 = 𝑑𝑥 𝑥 𝑥! − 1

𝑑 cos 𝑥 = − sin 𝑥 𝑑𝑥 𝑑 cot 𝑥 = − csc ! 𝑥 𝑑𝑥 𝑑 −1 cos !! 𝑥 = 𝑑𝑥 1 − 𝑥!

𝑑 sec !! 𝑥 = 𝑑𝑥 𝑥

𝑑 1 tan!! 𝑥 = 𝑑𝑥 1 + 𝑥!

𝑑 −1 cot !! 𝑥 = 𝑑𝑥 1 + 𝑥!

𝑑 csch 𝑥 = − csch 𝑥 coth 𝑥 𝑑𝑥 𝑑 tanh 𝑥 = sech! 𝑥 𝑑𝑥

𝑑 cosh 𝑥 = sinh 𝑥 𝑑𝑥 𝑑 coth 𝑥 = − csch! 𝑥 𝑑𝑥

1

𝑥! − 1

𝑑 sinh 𝑥 = cosh 𝑥 𝑑𝑥 𝑑 sech 𝑥 = − sech 𝑥 tanh 𝑥 𝑑𝑥





Prime 𝑒 ! ! = 𝑢! 𝑒 !

ln 𝑢 ! =

𝑢! 𝑢

𝑎! ! = 𝑢! 𝑎! ln 𝑎

sin 𝑢 ! = 𝑢! cos 𝑢

cos 𝑢 ! = −𝑢! sin 𝑢

tan 𝑢 ! = 𝑢! sec ! 𝑢

csc 𝑢 ! = −𝑢! csc 𝑢 cot 𝑢

sec 𝑢 ! = 𝑢! sec 𝑢 tan 𝑢

cot 𝑢 ! = −𝑢! csc ! 𝑢

𝑢!

arcsin 𝑢 ! = !

arccsc 𝑢 =

1 − 𝑢! −𝑢!

arccos 𝑢 ! =



!

𝑢 𝑢! − 1

arcsec 𝑢 =

−𝑢! 1 − 𝑢! 𝑢!



𝑢 𝑢! − 1

arctan 𝑢 ! =

𝑢! 1 + 𝑢!

−𝑢! arccot 𝑢 = 1 + 𝑢! !



Implicit Differentiation

𝑑 𝒚 𝑑𝒙

𝑑𝑦 = 𝑦! 𝑑𝑥

Always pay attention to the variables 𝑑 ! 𝑦 𝑑𝑥

𝑑 𝑦 = 2𝑦𝑦′ 𝑑𝑥 𝑑 ! 𝑑𝑦 𝑦 = 𝑛𝑦 !!! ≡ 𝑛𝑦 !!! 𝑦′ 𝑑𝑥 𝑑𝑥 𝑑 𝑑𝑦 𝑑𝑥 𝑥𝑦 = 𝑥 +𝑦 ≡ 𝑥𝑦 ! + 𝑦 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑥 𝑦 𝑑𝑥 − 𝑥 𝑑𝑥 𝑦 − 𝑥𝑦′ 𝑑 𝑥 = ≡ 𝑑𝑥 𝑦 𝑦! 𝑦! 2 𝑦

Chain/Power Rule Chain/Product Chain/Quotient

!!!

Logarithmic

𝑑 𝑦! ln 𝑦 = 𝑑𝑥 𝑦

Exponential

𝑑 ! 𝑎 = 𝑦 ! 𝑎 ! ln 𝑎 𝑑𝑥 𝑑 ! 𝑒 = 𝑦!𝑒! 𝑑𝑥 𝑑 𝑑𝑦 sin 𝑦 = cos 𝑦 ⋅ = cos 𝑦 ⋅ 𝑦′ 𝑑𝑥 𝑑𝑥

Euler’s Number Trigonometric Tangent Line

𝑓 𝑥, 𝑦 = 0,

𝑃 𝑎, 𝑏

⇒

𝑦! = 𝑓 ! 𝑎, 𝑏 𝑥 − 𝑎 + 𝑏

Related Rates The idea for related rates, in general, is to find the equation that relates geometrically to the question, implicitly differentiate it, and then plug in the given variables and solve for the unknown. Here are a few examples i.e. just use the equation/formula that mimics the object in question. Right triangle 𝑎! + 𝑏 ! = 𝑐 ! ⇒ 𝑎𝑎! 𝑡 + 𝑏𝑏 ! 𝑡 = 𝑐𝑐 ! 𝑡 𝑑𝐴 = 2𝜋𝑟𝑟 ! 𝑟 𝑡 𝑑𝑡 4 𝑑𝑟 𝑉 = 𝜋𝑟 ! ⇒ 𝑉 ! 𝑡 = 4𝜋𝑟 ! 3 𝑑𝑡

Circle

𝐴 = 𝜋𝑟 ! ⇒

Sphere

Hyperbolic Functions Notation 𝑒 ! − 𝑒 !! sinh 𝑥 = 2 2 sech 𝑥 = ! 𝑒 + 𝑒 !!

csch 𝑥 =

2 ! 𝑒 + 𝑒 !!

𝑒 ! + 𝑒 !! cosh 𝑥 = 2

𝑒 ! − 𝑒 !! 𝑒 ! + 𝑒 !! 𝑒 ! + 𝑒 !! coth 𝑥 = ! 𝑒 − 𝑒 !! tanh 𝑥 =

Identities



sinh −𝑥 = − sinh 𝑥 cosh −𝑥 = cosh 𝑥 ! ! cosh 𝑥 − sinh 𝑥 = 1 1 − tanh! 𝑥 = sech! 𝑥 sinh 𝑥 + 𝑦 = sinh 𝑥 cosh 𝑦 + cosh 𝑥 sinh 𝑦 cosh 𝑥 + 𝑦 = cosh 𝑥 cosh 𝑦 + sinh 𝑥 sinh 𝑦 !! ! sinh 𝑥 = ln 𝑥 + 𝑥 + 1 , −∞ ≤ 𝑥 ≤ ∞ cosh!! 𝑥 = ln 𝑥 + 𝑥 ! − 1 , 𝑥 ≥ 1 1 1+𝑥 tanh!! 𝑥 = ln , −1 < 𝑥 < 1 2 1−𝑥

Derivatives Standard sinh 𝑢 ! = 𝑢′ cosh 𝑢 csch 𝑢 ! = −𝑢! csch 𝑢 coth 𝑢 Inverse sinh

!!

𝑢!

!

𝑢 =

1 + 𝑢!

csch!! 𝑢 ! = −

cosh 𝑢 ! = 𝑢! sinh 𝑢

tanh 𝑢 ! = 𝑢! sech! 𝑢

sech 𝑢 ! = −𝑢! sech 𝑢 tanh 𝑢

coth 𝑢 ! = −𝑢! csch! 𝑢

!!



cosh

𝑢! 𝑢 1 + 𝑢!



!

𝑢 =

𝑢!

!!

tanh

𝑢! − 1

sech!! 𝑢 ! = −

𝑢! 𝑢 1 − 𝑢!

𝑢! 𝑢 = 1 − 𝑢! !

coth!! 𝑢 ! =



𝑢! 1 − 𝑢!



Antiderivatives & Integration Basic Rules Power Rule for antiderivatives Exponential Natural Log (case 1) Natural Log (case 2) Natural Log (case 3) Euler’s Number (case 1) Euler’s Number (case 2)

1 𝑥 ! + 𝐶 ⇔ 𝑛 ≠ −1 𝑛+1 𝑎! ⇒ 𝑦= + 𝐶 ln 𝑎

𝑦! = 𝑥! ⇒ 𝑦 = 𝑦! = 𝑎!

1 ⇒ 𝑦 = ln 𝑥 + 𝐶 𝑥 1 1 𝑦! = ⇒ 𝑦 = ln 𝑎𝑥 + 𝑏 + 𝐶 𝑎𝑥 + 𝑏 𝑎 𝑦! =

𝑦! =

𝑢! 𝑥 ⇒ 𝑦 = ln 𝑢 𝑥 + 𝐶 𝑢 𝑥 1 !" 𝑒 + 𝐶 𝑎 1 ⇒ 𝑦 = 𝑒 !"!! + 𝐶 𝑎

𝑦 ! = 𝑒 !" ⇒ 𝑦 = 𝑦 ! = 𝑒 !"!!

Euler’s Number (case 3)

𝑦 ! = 𝑢! 𝑥 𝑒 !

Anti-Chain-Rule Substitution Method

𝑦 ! = 𝑓 ! 𝑔 𝑥 𝑔! 𝑥 ⇒ 𝑦 = 𝑓 𝑔 𝑥



!

⇒ 𝑦 = 𝑒!

!

+ 𝐶 + 𝐶

Riemann Sum for Area Approximation !

𝑓 𝑥!∗ 𝛥𝑥 ,

𝐴 ≈ lim

!→∞



!!!

!

𝛥𝑥 =

𝑏−𝑎 , 𝑛

𝑥! = 𝑎 + 𝑖 ∙ 𝛥𝑥 !

𝑐 = 𝑐𝑛

𝑖=

!!!

!!!

!

!

!

𝑐𝑓 𝑥! = 𝑐 !!!

𝑖! =

𝑓 𝑥! !!!

!!!

!

!

𝑓 𝑥! ± 𝑔 𝑥! !!!

𝑛 𝑛+1 2

=

!

𝑓 𝑥! ± !!!

!

𝑛 𝑛+1 𝑖 = 2 !

𝑔 𝑥! !!!

𝑛 𝑛 + 1 2𝑛 + 1 6

!!!

Area Approximation Rules Midpoint Rule ! !

𝑏−𝑎 𝑥! + 𝑥! 𝑥! + 𝑥! 𝑓 𝑥 𝑑𝑥 ≈ 𝑓 +𝑓 +⋯ 𝑛 2 2

Trapezoid Rule

!

𝑓 𝑥 𝑑𝑥 ≈

!

𝑏−𝑎 𝑓 𝑥! + 2𝑓 𝑥! + 2𝑓 𝑥! + ⋯ + 2𝑓 𝑥!!! + 𝑓 𝑥! 2𝑛

!



The Integral Notation ∫

!

lim

!→∞

!

𝑓(𝑥!∗ ) 𝛥𝑥 !!!

Definite Integral Properties

≡

𝑓(𝑥) 𝑑𝑥 !



!

!

𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎

𝑐 𝑑𝑥 = 𝑐 𝑏 − 𝑎

!

!

!

!

𝑓 𝑥 𝑑𝑥 = 0

!

𝑐𝑓 𝑥 𝑑𝑥 = 𝑐

!

𝑓 𝑥 𝑑𝑥

!

!

!

!

𝑓 𝑥 𝑑𝑥 = 0

!

𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥 =

!!

!

⇔ 𝑓 −𝑥 = −𝑓 𝑥 ! !!

!

𝑓 𝑥

𝒌

𝑓 𝑥 𝑑𝑥 =

!

!

⇔ 𝑓 −𝑥 = 𝑓 𝑥

𝑔 𝑥 𝑑𝑥 !

!

𝑓 𝑥 𝑑𝑥 + !

𝑓 𝑥 𝑑𝑥 𝒌

even

NOTE: 𝑓 𝑥 ⋅ 𝑔 𝑥 𝑑𝑥 ≠

!

odd

!

𝑓 𝑥 𝑑𝑥 = 2

!

𝑓 𝑥 𝑑𝑥 ±

𝑔 𝑥 𝑑𝑥 ⋅

!

𝑓 𝑥 𝑑𝑥

!

𝑓 𝑥 𝑑𝑥 = − !

𝑓 𝑥 𝑑𝑥 !

Fundamental Theorems Let 𝑓 𝑥 = 𝑢 and 𝑔 𝑥 = 𝑣 for the following: !

𝑖)

𝑦=

𝑓 𝑡 𝑑𝑡 ⇒ !

𝑓 𝑡 𝑑𝑡 ⇒ !

𝑓 𝑡 𝑑𝑡 ⇒ !

𝑦 ! = 𝑓 𝑣 ∙ 𝑣 ! − 𝑓 𝑎 ∙ 𝑎! = 𝑓 𝑣 ∙ 𝑣 ! − 0 = 𝑓 𝑣 ∙ 𝑣 !

!

𝑦=

𝑦 ! = 𝑓 𝑣 ∙ 𝑣 ! − 𝑓 𝑢 ∙ 𝑢′

!

𝑦=



𝑦 ! = 𝑓 𝑏 ∙ 𝑏 ! − 𝑓 𝑢 ∙ 𝑢! = 0 − 𝑓 𝑢 ∙ 𝑢! = −𝑓 𝑢 ∙ 𝑢′

Limit Definition of a Definite Integral !

𝑖𝑖)

𝑓(𝑥!∗ ) 𝛥𝑥 =

lim

!→∞

!!!

𝑏−𝑎 𝛥𝑥 = , 𝑛



!

𝑓(𝑥) 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎 !

𝑥! = 𝑎 + 𝑖 ∙ 𝛥𝑥

Differential Equation (1st order) 𝑑𝑦 𝑦! = 𝑓! 𝑥 ⇒ = 𝑓 ! 𝑥 ⇒ 𝑑𝑦 = 𝑓 ! 𝑥 𝑑𝑥 ⇒ 𝑑𝑦 = 𝑓 ! 𝑥 𝑑𝑥 𝑑𝑥 ⇒ 𝑦 + 𝑐! = 𝑓 𝑥 + 𝑐! ⇒ 𝑦 = 𝑓 𝑥 + 𝑐! − 𝑐! = 𝑓 𝑥 + 𝑐! ≡ 𝑓 𝑥 + 𝐶 Common Integrals 1 𝑑𝑥 = 𝑥 + 𝐶 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶 𝑥 𝑑𝑥 = 𝑥 ! + 𝐶 2 1 𝑥 ! 𝑑𝑥 = 𝑥 ! + 𝐶 3

𝑥 ! 𝑑𝑥 =

1 𝑥 !!! + 𝐶 𝑛+1

1 𝑑𝑥 = ln |𝑥| + 𝐶 𝑥

⇔ 𝑛 ≠ −1 1 !" 𝑒 + 𝐶 𝑎

𝑒 ! 𝑑𝑥 = 𝑒 ! + 𝐶

𝑒 !" 𝑑𝑥 =

1 𝑑𝑥 = ln 𝑥 + 1 + 𝐶 𝑥+1

1 1 𝑑𝑥 = ln 𝑎𝑥 + 𝑏 + 𝐶 𝑎𝑥 + 𝑏 𝑎

𝑒 ! 𝑢′ 𝑑𝑢 = 𝑒 ! + 𝐶

𝑢! 𝑑𝑢 = ln 𝑢 + 𝐶 𝑢

1 !"!! 𝑒 + 𝐶 𝑎

𝑓 𝑢 𝑢′ 𝑑𝑢 = 𝐹 𝑢 + 𝐶 !

𝑓 𝑥 =𝐹 𝑏 −𝐹 𝑎 !

𝑢! cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶

𝑢! sin 𝑢 𝑑𝑢 = − cos 𝑢 + 𝐶

𝑢! sec ! 𝑢 𝑑𝑢 = tan 𝑢 + 𝐶

𝑢! csc 𝑢 sec 𝑢 𝑑𝑢 = − csc 𝑢 + 𝐶

𝑢! sec 𝑢 tan 𝑢 𝑑𝑢 = sec 𝑢 + 𝐶

𝑢! csc ! 𝑢 𝑑𝑢 = − cot 𝑢 + 𝐶

𝑢! 1 − 𝑢!

𝑑𝑢 = arcsin 𝑢 + 𝐶

−𝑢! 1 − 𝑢!

𝑑𝑢 = arccos 𝑢 + 𝐶



𝑒 !"!! 𝑑𝑥 =

𝑢! 𝑑𝑢 = arctan 𝑢 + 𝐶 1 + 𝑢!

Definite Integral Rules Substitution

!

Integration by Parts

𝑓 𝑔 𝑥 𝑔! 𝑥 𝑑𝑥 =

! !

! !

𝑓 𝑢 𝑑𝑢 ! !

𝑓 𝑥 𝑔! 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑔 𝑥

!

Let 𝑢=𝑓 𝑥 𝑑𝑢 = 𝑓 ! 𝑥 𝑑𝑥 Then !

𝑢 𝑑𝑣 = 𝑢𝑣

Trig Substitution 𝑎! − 𝑥 !

!

! !

! !

!

−

𝑔 𝑥 𝑓 ! 𝑥 𝑑𝑥

!

𝑑𝑣 = 𝑔! 𝑥 𝑑𝑥 𝑣=𝑔 𝑥

!

−

𝑣 𝑑𝑢 !

𝑎! + 𝑥 !

𝑥 ! − 𝑎!

1 − sin! 𝜃 = cos ! 𝜃

1 + tan! 𝜃 = sec ! 𝜃

sec ! 𝜃 − 1 = tan! 𝜃

𝑥 = 𝑎 sin 𝜃 𝜋 𝜋 𝜃∈ − , 2 2

𝑥 = 𝑎 tan 𝜃 𝜋 𝜋 𝜃∈ − , 2 2

𝑥 = 𝑎 sec 𝜃

Trig Identity tan 𝑥 𝑑𝑥 =

sin 𝑥 𝑑𝑥 = − cos 𝑥

𝜃 ∈ 0,

1 ⋅ − sin 𝑥 𝑑𝑥, cos 𝑥

𝑑 ln 𝑢 𝑥 𝑑𝑥

= − ln cos 𝑥 + 𝐶 = ln Partial Fractions 𝑝 𝑥 𝐴 𝐵 = + 𝑥 𝑥+1 𝑥 𝑥+1 𝑝 𝑥 𝐴 𝐵𝑥 + 𝐶 = + ! ! 𝑥 𝑥 +1 𝑥 𝑥 +1

=

𝜋 3𝜋 ∨ 𝜃 ∈ 𝜋, 2 2

1 𝑑𝑢 𝑢 𝑑𝑥

1 + 𝐶 = ln sec 𝑥 + 𝐶 cos 𝑥

𝑝 𝑥 𝑥! 𝑥 + 1 𝑝 𝑥 𝑥 𝑥! + 1

!

=

𝐴 𝐵 𝐶 + !+ 𝑥 𝑥 𝑥+1

=

𝐴 𝐵𝑥 + 𝐶 𝐷𝑥 + 𝐸 + ! + ! 𝑥 𝑥 +1 𝑥 +1 !

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