Intro to Multivariable Calculus Test 1 Study Guide
Intro to Multivariable Calculus Test 1 Study Guide MATH 2204
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This 3 page Study Guide was uploaded by Jack Pullman on Wednesday September 21, 2016. The Study Guide belongs to MATH 2204 at Virginia Polytechnic Institute and State University taught by Peter Wapperom in Fall 2016. Since its upload, it has received 48 views. For similar materials see Introduction to Multivariable Calculus in Math at Virginia Polytechnic Institute and State University.
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Date Created: 09/21/16
Intro to Multivariable Calculus Test 1 Study Guide Wednesday, September 21, 2016 9:39 AM • Basic single variable calculus ○ Differentiationof basic lines ○ Product, quotient, and chain rule ○ Equations for 2D lines Lines: Parabolas: Hyperbolas: Ellipses: Trig functions: Logarithmic functions: ○ Limits: • Chapter 12: Vectors and the Geometryof Space ○ 12.1: Three Dimensional Coordinate Systems Coordinate axes: (x, y, z, etc….) Points: ( Coordinate planes: xy planes, xz plane, yz plane Projectionon coordinate plane: The shadow a vector would leave on a plane if a flashlight was shined over a vector at the plane Surfaces: An equation in x, y, and z that relates them in three dimensions Distance between points: d = Equation of a sphere: ○ 12.2: Vectors Definition of a vector:a line segment with a direction and length Notation: Scalar multiplication: c Vector addition: Vectors between 2 points: Length of a vector: Unit vector = vector with each componentdivided by vectorlength Newton's 2nd law: F = ma; vector componentsof tension ○ 12.3: The Dot Product Dot product definition: Geometricinterpretation: Angle and perpendicular vectors:when dot product is 0, angle is 90 degrees Vector projection of b on a: Scalar projection of b on a: Work: ○ 12.4: The Cross Product Geometricinterpretation:a vectororthogonal to the plane created by 2 given vectors Area of a parallelogram: Torque: ○ 12.5: Equations of Lines and Planes Vector equation of a line Parametricequation of a line: Parametricequation of a line: Parallel lines and parallel planes: if normal vectors are parallel, so are planes, if vectors are scalar multiples of each other, they are parallel Intersection of a line and a plane: a point at subbing parametric equations into plane equation Intersection of two planes: cross normal vectorsfor vector coefficientof t and add to point you found Distance from point to plane: create a vector between desired point and a point on the plane, scalar projection of that vectoron the normal vector ○ 12.6: Cylinders and Quadric Surfaces Sketching surfaces using cross sections: set one dimension to zero and treat as two dimensional to find the trace Cylinders and quadric surfaces: □ Ellipsoid: □ Cone: □ Cylinder: □ Hyperboloid of one sheet: □ Hyperboloid if two sheets: □ Elliptic paraboloid: □ Hyperbolic paraboloid: • Chapter 14: Partial Derivatives ○ 14.1: Functions of Several Variables Evaluating functions: several inputs, one output Domain:all points where function is defined Range: all possible outputs Graphing: in three dimensions is a surface Level curves: a curve along which ○ 14.2: Limits and Continuity Two path test for nonexistence of a limit: If you can get two different values for Limit L as it approaches a point from different paths, the limit Does Not Exist Limit exists: if you can't disprove it Cancel commonfactors: to simplify such that limit is a real number Polar coordinates:check for factors in terms of Function is continuous if: ○ 14.3: Partial Derivatives Notation of partial derivatives: Computationof partial derivatives:like taking a normal derivative in single variable calculus, but treat variables other than the one in respect to as constants Approximationof partial derivatives: Mixed derivativetheorem:if f is sufficiently smooth, Geometricinterpretation:plane made by two tangent lines to surface at a point with partials in x and y Implicit differentiation:take partial derivative of both sides and solve when function is not defined in terms of variable dependent on variable in question ○ 14.4: Tangent Planes and Linear Approximation Tangent planes: Linearization: Linear approximation:create linearization using easy values then plug in harder values Total differential: Total differential:
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