CheckIt! problems bank aligned to a set of learning standards from Schlicker et al.'s Active Calculus: Vector textbook ( http://activecalculus.org/vector ).
Public version of the bank is available here: http://matthematician.github.io/vecit/ .
Find a quirk or error, or have a suggestion? Open an Issue in this Github repo.
VecIt! is a project of @matthematician, @tienchih, and @StevenClontz . For more information about the CheckIt! platform see http://checkit.clontz.org .
The learning standards below are arranged into core standards, with slugs ending in numerals (e.g., D4) and supplementary standards, with slugs ending in letters (e.g., VF). Core standards are intended to align with the fundamental, threshold knowledge of vector calculus while supplementary standards provide opportunities to extend core skills into deeper and broader applications.
N1 Surfaces and Graphs
Use traces and contours to describe surfaces in
$\mathbb{R}^3$ that are graphs of two-variable functions and level sets of three-variable functions, including quadric surfaces.
N2 Vector Arithmetic
Perform and geometrically interpret sums, differences, scalar multiples, and magnitudes of vectors in
$\mathbb{R}^n$ .
N3 Vector Geometry
Choose among, and use, dot products and cross products of vectors to compute quantities of geometric interest, including angles and areas.
N4 Parametrized Lines
Determine parametric equations for a line in
$\mathbb{R}^3$ defined either by a point and a direction vector, or by an intersection of two planes.
N5 Planes
Determine a level set equation for a plane in
$\mathbb{R}^3$ defined either by a point and a normal vector, or by three given non-collinear points.
NA Parametrized Curves
Sketch and parameterize portions of oriented curves, including those defined by intersections of surfaces and those which involve circular motions.
NB Derivatives and Curves
Use derivatives to determine, and distinguish among, a parameterized curve’s tangent, speed, velocity, and acceleration.
NC Parametrized Surfaces
Find a two-variable parametrization for portions of surfaces defined as graphs of two-variable functions and level sets of three-variable functions.
DA Multivariable Limits
Use continuity and polar-coordinate methods to calculate limits of multivariable functions, and use multiple-approach methods to demonstrate when such a limit does not exist.
D1 Partial Derivative Computation
Compute the partial derivatives of a given multivariable function, up to second order.
DB Partial Derivative Interpretation
Carefully interpret the first- and second-order partial derivatives of a multivariable function as slopes and concavities of its graph’s traces.
D2 Tangent Planes
Find equations for tangent planes to surfaces expressed as graphs of two-variable functions and as level sets of three-variable functions.
DC Multivariable Chain Rule
Identify all “routes” of composition when two or more multivariable functions are composed, and use the multivariable chain rule to compute such a function’s partial derivatives.
D3 The Gradient of
Compute the gradient vector of a two-variable function, give its geometric interpretation with respect both to the contours of the function and to its graph, and use it to calculate directional derivatives in a given direction.
DD The Gradient of
Compute the gradient vector of a three-variable function, give its geometric interpretation with respect to its level surfaces, and use it to calculate directional derivatives in a given direction.
D4 Critical Points of Multivariable Functions
Use its gradient vector to locate critical points of a multivariable function, and its second-order partial derivatives to classify critical points as local maximum, local minimum, saddle points, or degenerate.
DE Optimization and Extreme Values of Multivariable Functions
Use both critical points and a parametrization of its boundary to find absolute maximum and minimum values of a multivariable function on a closed and bounded domain.
DF Constrained Optimization
Use Lagrange multipliers to determine absolute maximum and minimum values of a multivariable function subject to one or more constraint equations.
S1 The "Tape-Measure Method"
Describe a given region in
$\mathbb{R}^2$ using coordinate inequalities, in at least two distinct ways.
S2 Double Integrals
Set up and evaluate double integrals over general regions, using iterated integrals and “tape-measured” bounds.
SA Double Integrals in Polar Coordinates
Use polar coordinates to transform the integrand, the bounds, and the area element of a given double integral.
SB Surface Area
Set up double integrals to compute surface areas of portions of parametrized surfaces.
SC Triple Integrals
Set up and evaluate triple integrals over general regions, using iterated integrals and “tape-measured” bounds.
SD1 Change of Variables
Use a given change of variables to rewrite a multiple integral.
SD2 Triple Integrals in Cylindrical and Spherical Coordinates
Use cylindrical and spherical coordinates to transform the integrand, the bounds, and the volume element of a given triple integral.
SE Applications of Multiple Integrals
Use double and triple integrals to solve applied problems involving mass, density, and probability.
V: Use, understand, and connect the various types of derivatives and integrals associated with a vector field.
V1 Vector Fields
Sketch and identify vector fields in two and three dimensions, including gradient vector fields of two- and three-variable functions.
VA Line Integral Heuristics
Predict from a sketch whether a given line integral of a vector field along an oriented path is positive, negative, or zero.
V2 Line Integral Computation
Directly compute the line integral of a vector field over a given oriented path.
V3 Gradient Vector Fields
Classify a given vector field as conservative and/or gradient; and use “partial integration” to determine a scalar potential function for a given gradient vector field.
VB Path-Independent Line Integrals
Identify when the Fundamental Theorem of Calculus for Line Integrals may be applied, and use the Theorem to compute line integrals using a scalar potential function.
VC Curl of a Vector Field
Compute the scalar curl of a two-dimensional vector field and the vector curl of a three-dimensional vector field. Use notions of circulation to predict whether the scalar curl of a given two-dimensional vector field is positive, negative, or zero.
V4 Green's Theorem
Identify when Green’s Theorem may be applied to find the total circulation of a vector field along a closed path, and use the Theorem to carry out this computation using a double integral.
VD Divergence of a Vector Field
Compute the divergence of a vector field, and use notions of flux to predict whether the divergence of a given vector field is positive, negative, or zero.
VE Flux Integrals
Directly compute the flux integral of a vector field through a given oriented surface.
VF Stokes' and Divergence Theorems
Identify when a line or flux integral in
$\mathbb{R}^3$ may be evaluated more simply through the application of Stokes' Theorem or the Divergence Theorem, and use the appropriate Theorem to convert it into a different integral form.