Students examine the foundational components of limits, derivatives, integrals, and series and apply this knowledge to real-world situations. Derivatives are used to find slopes of lines tangent to curves at specified points. Students learn specific rules of differentiation and explore real-world applications, including related rates and optimization. Students explore the graphs of functions and their first and second derivatives to reveal the functions’ characteristics. Functions increase in complexity to include logarithmic and exponential components. Integrals are explored as various methods of finding the area under a curve are examined and applied, and each method is supported graphically. Integration is used to revolve solids about an axis. At the conclusion of the course, students learn about series, including Taylor and Maclaurin series, as well as how to prove convergence or divergence using integral and p-series tests.
Tangents to curves
Derivative rules and notation
Concavity and extrema
Modeling and optimization Riemann sums and areas under curves
Definite and indefinite integration
Volumes of solids
Parametric and polar curves
Calculate and analyze functions, graphs, and limits.
Recognize and describe asymptotic behavior, continuity, rates of change and infinity on the coordinate plane.
Calculate, interpret, and analyze derivatives including those of trigonometric, exponential, logarithmic, and inverse functions.
Explore graphs of derivatives, concavity, inflection points, and extrema.
Model and solve optimization problems. Evaluate antiderivatives using formulas, using substitution techniques, and by parts.
Use sigma notation to represent, manipulate, and evaluate finite sums and other series including Taylor, Maclaurin, and power series.
Use integration to find areas under curves and areas between curves.
Calculate the volume of a solid using cross sections and washer methods.
Calculate arc length of parametric curves and apply parametric equations.