Calculus

This badge demonstrates the earner's ability to calculate volume, lengths of plane curves, areas of surfaces of revolutions, work done by force, and the location of an object's center of mass using the definite integral.

This badge demonstrates the earner's ability to apply derivatives as a tool in order to find the optimal solutions to problems such as finding extreme values of functions, analyzing the shapes of graphs, solving equations numerically, recovering a function from its derivative, and applying the Mean Value Theorem.

This badge demonstrates the earner's ability to determine the derivative function including the rules used for finding derivative functions, tangents and the derivative point, derivatives as a rate of change, implicit differentiation, and linearization and differentials.

This badge demonstrates the earner's ability to solve first-order differential equations including slope fields, first-order linear equations, autonomous equations, systems of equations, and phase planes.

This badge demonstrates the earner's ability to determine what functions are, how they are pictured as graphs, how they are combined and transformed, and how they can be classified.

This badge demonstrates the earner's ability to compute functions using infinite series and sequences including the integral test, comparison tests, absolute convergence, alternating series and conditional convergence, power series, Taylor and Maclaurin Series, and the Binomial Series.

This badge demonstrates the earner's ability to calculate areas, volumes, and averages using the definite integral including finite sums, sigma notation, the definite integral, the Fundamental Theorem of Calculus, and the area between curves.

This badge demonstrates the earner's ability to solve a variety of applied problems using rigorous analytic problem solving approaches to integrals and transcendental functions including the natural logarithm function, exponential change and separable differential equations, and hyperbolic functions.

This badge demonstrates the earner's ability to use the theory of line and surface integrals to find the work done by a force in moving an object along a path, to find the mass of a curved wire with variable density, and to find the rate of flow of fluid across a surface, including the fundamental theorems of vector integral calculus, their mathematical consequences, their physical applications, and specifically the Fundamental Theorem of Calculus.

This badge demonstrates the earner's ability to develop limits to find the velocity of a moving object, and the tangent to a curve including intuitive and formal methods, the limit of a function and limit laws, one-sided limits, continuity, and infinity.

This badge demonstrates the earner's ability to determine the double and triple integrals for a function with multiple variables over a region in space including double and iterated integrals over rectangles and over general regions, area by double integration, double integrals in polar form, triple integrals in rectangular coordinates, moments and centers of mass, triple integrals in cylindrical and spherical coordinates, and substitutions in multiple integrals.

This badge demonstrates the earner's ability to determine the parametrization of plane curves using calculus with parametric curves including polar coordinates, graphing polar coordinate equations, areas and lengths in polar coordinates, and conics in polar coordinates.

This badge demonstrates the earner's ability to apply the derivatives from functions using several variable differentials including limits and continuity in higher dimensions, partial derivatives, directional derivatives and gradient vectors, tangent planes and differentials, extreme values and saddle points, and Lagrange multipliers.

This badge demonstrates the earner's ability to develop techniques used to find antiderivatives, also called indefinite integrals, to evaluate definite integrals for specialized classes of functions including integration by parts, trigonometric integrals, trigonometric substitutions, integration of rational functions by partial fractions, integral tables and computer algebra systems, numerical integration, and improper integrals.

This badge demonstrates the earner's ability to apply multivariable calculus to determine three-dimensional space including vectors, the dot product, the cross product, lines and planes in space, and cylinders and quadric surfaces.

This badge demonstrates the earner's ability to determine vector-valued functions of curves in space and their tangents including projectile motion, arc length in space, curvature of a plane curve, normal vectors of a curve, velocity, and acceleration.