MATH 170 • Midterm • Calculus for Scientists and Engineers II

Distance Between Points 1

Cosine Theorem in 3D & Euclidean N-Space

Vectors: Arithmetic, Parallelism and Unit Vector

Angles between Vectors: Dot Product

Vector Projection

Cross Product of Vectors

Applications of Cross Product

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Plane Equations

Line Equations 1

Line Equations 2

Intersection of Planes

Intersection of Planes and Lines

Distance of a Point to Lines & Planes

Graphs of Surfaces

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Domain of Multivariable Functions

Level Curves

Level Curves and Level Surfaces

Limits of Multivariable Functions 1

Limits of Multivariable Functions 2

Limits of Multivariable Functions 3

The Squeeze Theorem

Continuity of Multivariable Functions

Limits with Polar Coordinates

Limit Definition of Partial Derivative

Partial Differentiation Rules 1

Partial Differentiation Rules 2

Higher Order Partial Differentiation

Implicit Differentiation I

Implicit Differentiation II

Implicit Differentiation III

Implicit Higher Differentiation

Chain Rule 1

Chain Rule 2

Chain Rule 3

Equation of a Tangent Plane 1

Equation of a Tangent Plane 2

Horizontal Tangent Plane

Normal Vector to a Multivariable Function

Limit of Multivariable Functions 1

Limit of Multivariable Functions 2

Limit of Multivariable Functions 3

Limit of Multivariable Functions 4

Limit of Multivariable Functions 5

Limit of Multivariable Functions 6

Limit of Multivariable Functions 7

Limit of Multivariable Functions 8

Limit of Multivariable Functions 9

Limit of Multivariable Functions 10

Continuity 1

Continuity 2

Definition of Partial Derivative 1

Definition of Partial Derivative 2

Partial Derivative 1

Partial Derivative 2

Partial Derivative 3

Partial Derivative 4

Partial Derivative 5

Partial Derivative 6

Higher Order Differentiation 1

Higher Order Differentiation 2

Higher Order Differentiation 3

Higher Order Differentiation 4

Higher Order Differentiation 5

Tangent Plane 1

Tangent Plane 2

Tangent Plane 3

Tangent Plane 4

Tangent Plane and Normal Line 1

Tangent Plane and Normal Line 2

Chain Rule 1

Chain Rule 2

Chain Rule 3

Chain Rule 4

Chain Rule 5

Chain Rule 6

Chain Rule and Tangent Plane

Gradient Vector

Directional Derivative

Gradient and Directional Derivative

Extreme Values

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Extreme Values on Closed Regions 1

Extreme Values on Closed Regions 2

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Lagrange Multipliers I

Lagrange Multipliers II

Optimization with Lagrange Multipliers 1

Optimization with Lagrange Multipliers 2

Lagrange Multipliers with Two Constraints

Gradient and Tangent Plane 1

Gradient and Tangent Plane 2

Gradient and Directional Derivative 1

Gradient and Directional Derivative 2

Gradient and Directional Derivative 3

Gradient and Directional Derivative 4

Gradient and Directional Derivative 5

Gradient and Directional Derivative 6

Extreme Values on Closed Region 1

Extreme Values on Closed Region 2

Extreme Values on Closed Region 3

Extreme Values on Closed Region 4

Extreme Values on Closed Region 5

Extreme Values on Closed Region 6

Lagrange Multipliers 1

Lagrange Multipliers 2

Lagrange Multipliers 3

Lagrange Multipliers 4

Lagrange Multipliers 5

Lagrange Multipliers 6

Lagrange Multiplier 7

Intuition: Double Integrals

Volume Calculation with Double Integration

Area Calculation with Double Integral

Sketching the Area of Integration 1

Sketching the Area of Integration 2

Sketching the Area of Integration 3

Reversing the Order of Integration 1

Reversing the Order of Integration 2

Reversing the Order of Integration 3

Cartesian to Polar Coordinates 1

Cartesian to Polar Coordinates 2

Polar to Cartesian Coordinates

Double Integrals in Polar Coordinates 1

Double Integrals in Polar Coordinates 2

Double Integrals in Polar Coordinates 3

Double Integrals in Polar Coordinates 4

Introduction to Triple Integrals

Triple Integrals 2

Triple Integrals 3

Important Functions and Their Graphs

Triple Integrals 4

Triple Integrals 5

Triple Integrals 6

Reversing the Order of Triple Integrals

Reversing the Order of Integration 1

Reversing the Order of Integration 2

Reversing the Order of Integration 3

Double Integral in Polar Coordinate 1

Double Integral in Polar Coordinate 2

Double Integral in Polar Coordinate 3

Double Integral in Polar Coordinate 4

Double Integral in Polar Coordinate 5

Double Integral in Polar Coordinate 6

Double Integral in Polar Coordinate 7

Double Integral in Polar Coordinate 8

Double Integral in Polar Coordinate 9

Double Integral in Polar Coordinate 10

Limits of Multivariable Functions

Limits of Multivariable Functions

Directional Derivatives

Critical Points (Max-Min and Saddle Points)

Lagrange Multipliers

Double Integral: Reversing the Order of Integration

Limits of Multivariable Functions

Limits of Multivariable Functions

Tangent Planes and Normal Lines

Extreme Values over Closed Regions

Double Integrals in Polar Coordinates

Limit of Multivariable Functions

Limit of Multivariable Functions

Limit of Multivariable Functions

Reversing the Order of Integration

Extreme Values

Directional Derivative

Double Integrals in Poolar Coordinates

Change of Variable: The Jacobian

Plane Equations

Limits of Multivariable Functions

Limits of Multivariable Functions

Directional Derivatives

The Chain Rule

Double Integrals in Cartesian Coordinates

Extreme Values over Closed Regions

Reversing the order of Integration

Double Integrals in Poolar Coordinates

Graphs and Surfaces in 3D

Domain of Multivariable Functions

The Limit of Multivariable Functions

The Limit of Multivariable Functions

Higher Order Partial Derivative

The Chain Rule

Critical Points (Max-Min and Saddle Points)

Extreme Values on Closed Region

The Lagrange Multipliers

Reversing the order of Integration

Double Integrals in Cartesian Coordinates

Double Integrals in Cartesian Coordinates

Double Integrals in Polar Coordinates