As detailed on the syllabus, your assessment grade in this course will be determined by your proficiency on a variety of standards. (This is known as Standards Based Grading.)
Below is the list of standards that form the basis for this class, along with guiding questions that address each standard. You should put in the time to master each standard before taking the assessment. This involves reading the text, being present and asking questions in class, and completing the WebWork and book problems. If for some reason you do not master the knowledge the first time around, you will have the opportunity to re-assess your knowledge.
The Standards
Standard 1. 3-space. Can you draw the coordinate axes and coordinate planes? Are you able to place a point (x,y,z) in the correct location? Do you understand how to project a point onto a coordinate plane? Do you understand what an equation of the form x=a, y=b, or z=c corresponds to? Can you calculate the distance between two points? Between a point and a coordinate plane? A point and a coordinate axis? (Section 10.1)
Standard 2. Vectors. Can you explain how a vector is different from a scalar? How a vector is different from a coordinate triple? Can you find the components of a vector? Do you know the standard basis vectors? Can you add or subtract vectors? Can you calculate the scalar multiple of a vector? Do you understand the geometry of those operations? Can you determine when two vectors are parallel? Can you find the magnitude of a vector? Do you know the properties of vectors on page 552? Can you extend these ideas to n-dimensional vectors? (Sections 10.1 and 10.2)
Standard 3. Vector Products. Can you calculate the dot product of two vectors? Can you calculate the cross product of two vectors? Do you understand when the dot and cross products apply? Can you apply the properties of the dot and cross products (10.3.2 and 10.4.8)? Can you find the angle between two vectors? Can you use vector products to determine if two vectors are orthogonal? Can you use vector products to find the projection of one vector onto another? Do you understand the geometry of the cross product and scalar triple product? Can you use vector products to calculate work and torque? (Sections 10.3 and 10.4)
Standard 4. Mathematical Collaboration. Have you met with classmates and completed the writing prompts that are assigned at regular intervals in the class?
Standard 5. Vector Functions. (AKA Parametric Equations) Do you understand the definition of a vector function? Can you determine the domain and range of a vector function? Do you know what the initial point and terminal points are? Can you calculate them? Do you understand how to draw the graph of a vector function? Can you give the equation of a helix? Can you parametrize a circle? Can you use a computer to graph a vector function? (Sections 9.1 and 10.7)
Standard 6. Lines and Planes. Can you give the parametric equations of a line in 2D? 3D? (Using either vector equations or linear interpolations.) Can you give the equation of a specified plane in 3D? Can you use vector products and operations to explain the relationship between points, lines, and planes? (Sections 9.1 and 10.5)
Standard 7. Derivatives of Vector Functions. Can you calculate the limit of a vector function? Can you determine if a vector function is continuous at t=a? Can you determine if a vector function is smooth on an interval? Can you compute the derivative of a vector function? Can you apply differentiation rules, including when vector products and the chain rule are involved? Can you determine the tangent vector or tangent line to a vector function? (Sections 9.2, 10.7, 10.9)
Standard 8. Derivative Applications. Can you apply vector calculus to questions of position, velocity, speed, and acceleration? Do you understand what the Frenet (or TNB) frame is for a particle in space? Can you compute it? Do you understand the theory behind the curvature of a curve? Can you compute the curvature of a curve in space? (Sections 9.2, 10.7, 10.9)
Standard 9. Integrals and Arc Length. Do you know how to compute a definite or indefinite integral of a vector function? Can you compute the arc length of a vector function? Do you understand what it means to re-parametrize a curve? Can you re-parametrize a curve with repect to arc length? (Sections 9.2, 10.7, 10.8)
Standard 10. Functions of Multiple Variables. Do you understand the definition of a function of multiple (two, three, n) variables? Can you determine its domain and range? Can you use a computer to graph a function of multiple variables? Do you know what a level curve is? Do you know what a level surface is? Can you use a computer to graph a level curve or level surface? Can you reconstruct the graph of a function of two variables from its level curves? (Section 11.1)
Standard 11. Equations of Surfaces. Can you identify when an equation corresponds to a plane, sphere, ellipsoid, elliptic paraboloid, cone, hyperboloid of one sheet, hyperboloid of two sheets, or generalized cylinder? Do you understand what the traces / level curve of these surfaces are? Given an equation of such a surface, can you draw their traces / level curves? Can you use a computer to graph this type of surface? (Sections 10.1, 10.6, and 11.1)
Standard 12. Limits and Continuity. Can you write down the definition of the limit of a function of two variables? Do you understand what this definition means? Are you able to prove that a limit does not exist by finding two curves passing through a point along which the limit is different? Do you understand what it means for a function to be continuous at a point? Do you know families of functions that are continuous everywhere? Do you know at which points rational functions are continuous? (Section 11.2)
Standard 13. Definition and Interpretation of Partial Derivatives. Can you write down the definition of the partial derivative for functions of two or more variables? Do you understand mathematically a partial derivative represents? Do you understand conceptually what a partial derivative represents? Can you compute a single partial derivative of a function? Can you interpret your answer as a slope or rate of change? (Section 11.3)
Standard 14. Calculating Partial Derivatives and the Chain Rule. Can you compute multiple partial derivatives? Can you state Clairaut's Theorem? Are you able to determine whether Clairaut's Theorem applies and determine its consequences? Do you understand what a partial differential equation is? Do you understand the chain rule for functions of several variables? Can you write out the tree of variable dependencies and use it to write out a complete expression for what the chain rule says? Can you work through the steps to apply the chain rule? (Sections 11.3 and 11.5)
Standard 15. Applications of Partial Derivatives. Can you determine when a function is differentiable? Do you understand the concept of the tangent plane to a surface? Can you determine the equation of the tangent plane to a surface at a given point? Do you understand the concept of differentials? Can you write down the differential for a function of two or more variables? Can you apply tangent planes or differentials to give a linear approximation for a function value? Can you calculate the directional derivative of a function in a given unit direction? Can you interpret that quantity as a slope or rate of change? (Sections 11.4)
Standard 16. The Gradient Vector and Applications. Can you compute the gradient vector of a function of two (three? n?) variables? Do you understand how the gradient vector relates to the maximum value of the directional derivative and its maximal rate of change? Do you understand how a gradient vector is related to level curves (or level surfaces)? Do you understand how the gradient vector relates to the tangent plane to a level surface? Can you use the gradient vector to calculate the curve of steepest ascent or descent? (Section 11.6)
Standard 17. Optimization. Can you state the definition of a local (or global) maximum (or minimum)? Do you understand the difference between a maximum and a maximum value? Do you know the definition of a critical point? Can you compute the critical points of a function? Can you find the local extrema of a function? Can you state and apply the Second Derivatives Test? Do you understand what a closed set is? Do you understand what a bounded set is? Can you state and apply the Extreme Value Theorem? Can you find the global extrema of a function defined on a closed set? (Sections 11.7)
Standard 18. Riemann Sums and Integration over Rectangles. Can you state the Riemann Sum definition of the Double Integral? Do you understand it? Can you use a Riemann Sum to approximate the volume under a surface? Do you understand and can you apply Property 11 on page 713 (at the end of Section 12.2) to find upper and lower bounds for a double integral? Do you understand how to read a double integral? Can you compute a simple double integral over a rectangle? Do you understand and can you apply properties 12, 13, and 14 of double integrals stated on Page 704 (at the end of Section 12.1)? (Section 12.1)
Standard 19. Iterated Integrals. Do you understand when Fubini's Theorem applies? Can you compute more complex double integrals involving x and y? Can you compute a double integral over a non-rectangular region defined by two curves? Can you change the order of integration of a double integral? (Sections 12.1 and 12.2)
Standard 20. Applications of Integration. Can you apply double integration to calcuate the average value of a function? To find the volume of a region? To find the centroid of a region? To find the mass of an object? To find the electric charge on an object? To find the probability that an event occurs? (Sections 12.2 and 12.4)
Standard 21. Polar Coordinates and Regions. Do you understand what polar coordinates are? Can you identify and determine multiple polar coordinate pairs that represent the same point? Can you convert between cartesian and polar coordinates? Can you convert an equation involving r and θ into an equation involving x and y? Can you determine the polar equations for circles centered at the origin and lines through the origin? Can you determine the region in the plane given by a "polar rectangle" and can you write down the polar inequalities for a given polar rectangle? (Sections 9.3 and 12.3)
Standard 22. Polar Curves. Do you understand what a polar function r=f(θ) is? Can you graph a polar function r=f(θ) and polar inequalities f(θ)≤r≤g(θ) by hand? Do you understand the relationship between a cartesian plot of r as a function of θ and the polar plot of r=f(θ)? Can you graph a polar function and inequalities using Mathematica? Can you determine the intersection points of two polar curves? Can you determine the tangent line to a polar curve? Do you understand the formula for the arc length of a polar curve starting from the idea that L=∫dL? Can you set up the integral to calculate the arc length of a polar curve? (Sections 9.3 and 9.4)
Standard 23. Polar Integrals with applications. Do you understand the formula for area of a polar region starting from the idea that A=∫dA? Can you set up an integral to calculate the area of a region defined using polar inequalities? Can you determine when a given 2D domain of integration is better suited for cartesian integration or for polar integration? Can you convert a cartesian double integral into a polar integral? Can you apply polar integration to compute volume or other application similar to those in Standard 20? (Sections 9.4, 12.3, and 12.4)
Standard 24. Triple Integrals Do you understand what a triple integral is and is calculating? Can you compute a triple integral as an iterated integral? Can you use Mathematica to calculate a triple integral? Can you determine the bounds of a triple integral for a given three-dimensional domain (and vice versa)? Can you apply triple integration to calculate the volume or centroid of a region, find the average value of a function, find the mass of or electric charge on an object? (Section 12.5)
Standard 25. Cylindrical and Spherical Coordinates. Do you understand what cylindrical and spherical coordinates are? Can you convert between cartesian and cylindrical and spherical coordinates? Can you determine the cylindrical equations for cylinders, cones, and spheres centered at the origin and vertical planes through the origin? Can you determine the spherical equations for cylinders, cones, and spheres centered at the origin? Can you determine the region in the plane given by a "cylindrical box" or "spherical box" and can you write down the polar inequalities for a given polar rectangle? (Sections 12.6 and 12.7)
Standard 26. Cylindrical and Spherical Integration. Do you understand the formula for volume of a cylindrical or spherical region starting from the idea that V=∫dV? Can you set up an integral to calculate the volume of a region defined using cylindrical or spherical inequalities? Can you determine when a given 3D domain of integration is better suited for cartesian or cylindrical or spherical integration? Can you convert a cartesian triple integral into a cylindrical or spherical integral? Can you apply cylindrical or spherical integration to compute volume or other application similar to those in Standard 24? (Sections 12.6 and 12.7)
