Calculus with Analytic Geometry II

Course Description

MAC2312 / MAC2312C – Calculus with Analytic Geometry II is a 4-credit lecture course in the Mathematics: Calculus and Pre-Calculus taxonomy of Florida's Statewide Course Numbering System (SCNS). The course is the second semester of the standard three-semester calculus sequence and continues the study of single-variable calculus from MAC2311. The course covers advanced techniques of integration, applications of integration (volumes, arc length, surface area, work), improper integrals, L'Hopital's Rule, infinite sequences and series (including convergence tests, power series, Taylor and Maclaurin series), and parametric, polar, and conic-section equations.

MAC2312 is part of Florida's state-mandated General Education Core in Mathematics, satisfying the Gen-Ed math requirement at every Florida public college and university. The course is offered at 49 Florida public institutions and transfers as equivalent across the state. The "C" suffix variant denotes integrated lecture and supplemental instruction. MAC2312 is the prerequisite for MAC2313 (Calculus III - Multivariable Calculus), MAP2302 (Differential Equations), and many upper-division engineering, mathematics, and physics courses.

Learning Outcomes

Required Outcomes

Upon successful completion of this course, students will be able to:

• Apply integration techniques, including integration by parts, trigonometric integrals, trigonometric substitution, partial fractions decomposition, and rationalizing substitutions.
• Evaluate improper integrals with infinite intervals or unbounded integrands; determine convergence or divergence.
• Apply L'Hopital's Rule to evaluate limits of indeterminate forms (0/0, ∞/∞, 0·∞, ∞−∞, 0⁰, 1^∞, ∞⁰).
• Apply integration to compute volumes of solids of revolution using disk, washer, and cylindrical shell methods.
• Apply integration to compute arc length and surface area for curves and surfaces of revolution.
• Apply integration to physical applications including work, hydrostatic force, center of mass, and moments.
• Solve separable differential equations and apply them to growth, decay, and mixing problems.
• Determine the convergence or divergence of infinite sequences and apply limit theorems to sequences.
• Determine the convergence or divergence of infinite series using appropriate tests (nth-term test, integral test, comparison tests, alternating series test, ratio test, root test).
• Determine intervals and radii of convergence for power series.
• Construct and apply Taylor and Maclaurin series, including using known series and operations (differentiation, integration, multiplication) to derive new series.
• Apply Taylor's Theorem to estimate the error of polynomial approximations.
• Work with parametric equations, including computing derivatives, slopes of tangent lines, and arc length parametrically.
• Work with polar coordinates, including converting between rectangular and polar forms, graphing polar curves, and computing area in polar coordinates.

Optional Outcomes

Depending on institutional emphasis, students may also:

• Analyze conic sections in detail (parabolas, ellipses, hyperbolas) including parametric and polar forms.
• Apply numerical integration methods (trapezoidal rule, Simpson's rule) including error estimation.
• Apply hyperbolic functions and inverse hyperbolic functions in integration and applications.
• Apply logistic differential equations to population dynamics.
• Use computational tools (MATLAB, Python, Wolfram Alpha) for series visualization and integration verification.
• Apply calculus concepts in economics, biology, or physical sciences through context-specific problems.

Major Topics

Required Topics

• Techniques of Integration: Integration by parts; trigonometric integrals (powers of sine, cosine, tangent, secant); trigonometric substitution; partial fractions; rationalizing substitutions; strategies for choosing integration techniques.
• Improper Integrals: Type I (infinite intervals); Type II (discontinuous integrands); convergence and divergence; comparison theorems for improper integrals.
• L'Hopital's Rule and Indeterminate Forms: Application to 0/0 and ∞/∞; algebraic manipulation for other indeterminate forms (0·∞, ∞−∞, 0⁰, 1^∞, ∞⁰).
• Applications of Integration — Volumes: Volumes by disks and washers; volumes by cylindrical shells; volumes by cross sections.
• Applications of Integration — Arc Length and Surface Area: Arc length of plane curves; surface area of solids of revolution.
• Applications of Integration — Physics: Work; hydrostatic pressure and force; center of mass; moments.
• Differential Equations: Separable differential equations; applications to exponential growth and decay, mixing problems, Newton's law of cooling, basic population models.
• Sequences: Definition; convergence and divergence; monotonic sequences; bounded sequences; theorems on sequences.
• Infinite Series: Definition and partial sums; geometric series; harmonic series; nth-term test for divergence; telescoping series.
• Convergence Tests for Series: Integral test; comparison and limit comparison tests; alternating series test; absolute and conditional convergence; ratio test; root test; strategy for testing series.
• Power Series: Radius and interval of convergence; differentiation and integration of power series; representations of functions as power series.
• Taylor and Maclaurin Series: Definition; common Maclaurin series (e^x, sin x, cos x, ln(1+x), 1/(1-x), arctan x); Taylor's Theorem and remainder; applications.
• Parametric Equations: Curves defined parametrically; eliminating the parameter; calculus with parametric curves (slopes, areas, arc length).
• Polar Coordinates: Polar coordinate system; conversion between polar and rectangular; graphing polar equations; calculus in polar coordinates (slopes, area enclosed by polar curves, arc length).

Optional Topics

• Conic Sections: Parabolas, ellipses, hyperbolas; parametric and polar representations; eccentricity.
• Numerical Integration: Midpoint rule, trapezoidal rule, Simpson's rule; error estimation.
• Hyperbolic Functions: Definitions, identities, derivatives, integrals; inverse hyperbolic functions.
• Logistic Differential Equation: Application to limited-growth population models.
• Approximation of Functions: Taylor polynomial approximation with error analysis; convergence acceleration.
• Computational Verification: Use of MATLAB, Mathematica, Wolfram Alpha, or Desmos for series visualization and computation verification.

Resources & Tools

• Standard Textbooks: Calculus: Early Transcendentals by Stewart (most widely adopted in Florida); Calculus by Larson and Edwards; Calculus: Single Variable by Hughes-Hallett, McCallum, et al.; OpenStax Calculus Volume 2 (free, openstax.org)
• Online Homework Platforms: WebAssign (Stewart); Pearson MyLab Math; Cengage MindTap; UF Xronos
• Required Calculator: Texas Instruments TI-84 Plus or TI-84 Plus CE; some institutions allow TI-89 or TI-Nspire CX CAS but prohibit CAS calculators on exams; some institutions prohibit graphing calculators entirely on exams.
• Free Online Tools: Desmos (desmos.com) — exceptional for visualizing parametric and polar curves; Wolfram Alpha for series convergence checks; Symbolab for step-by-step integration; GeoGebra
• Tutoring Resources: Free college tutoring centers; Khan Academy Calculus 2 (free); Paul's Online Math Notes (tutorial.math.lamar.edu); Professor Leonard, Krista King, 3Blue1Brown video channels on YouTube
• UF Resources: UF MAC2312 syllabus archive (syllabus.math.ufl.edu); UF Math Tutoring Center

Career Pathways

MAC2312 deepens the calculus foundation needed for STEM transfer pathways and supports continued progression through:

• Engineering Pathways – Required for transfer to all Florida public engineering programs; part of the engineering A.A. transfer pathway. Series and integration techniques are foundational to circuit analysis, signals and systems, structural analysis, and thermodynamics.
• Mathematics and Statistics Majors – Required for the calculus sequence (MAC2311, MAC2312, MAC2313); foundation for advanced calculus, analysis, differential equations, and complex variables.
• Physical Sciences – Required for physics (PHY2048/2049 and beyond), chemistry (PCHM 3411/3412 physical chemistry), and astronomy majors.
• Computer Science – Required at all Florida public universities for the Computer Science B.S.; foundation for algorithms analysis, machine learning, and computer graphics.
• Florida Industry Application – Calculus II concepts (especially series, Taylor expansions, and integration techniques) underpin signal processing, control systems, scientific computing, and machine learning applications in Florida's aerospace, semiconductor, finance, and tech industries.

Special Information

Gen-Ed Core Designation

MAC2312 is part of Florida's General Education Core Course Options in Mathematics, established by the Florida Department of Education and codified in Florida Statute 1007.25. All Florida public colleges and universities accept MAC2312 as fulfilling the Gen-Ed Mathematics core requirement. Students must earn a grade of C or better for the course to satisfy degree requirements.

Course Equivalence and Variations

MAC2312 is offered as both MAC2312 (lecture-only, 4 credits) and MAC2312C (with integrated supplemental instruction, 4-5 credits). The two forms are equivalent for transfer and Gen-Ed credit. UF and several other institutions offer an honors version (MAC3473 - Honors Calculus 2) for highly prepared students.

Workload and Time Expectations

MAC2312 is widely considered the most challenging course in the calculus sequence — "the wall" for many engineering students — primarily due to series. Most institutions expect 9-12 hours of weekly out-of-class work, including 4-6 hours on online homework. Series convergence tests and Taylor series in particular require substantial practice and conceptual understanding. DFW rates are historically elevated; success depends on consistent daily practice and active engagement with worked examples.

Foundation for Upper-Division Coursework

MAC2312 is the prerequisite for MAC2313 - Calculus III (Multivariable Calculus), MAP2302 - Differential Equations, and indirectly for upper-division engineering, physics, and computer science courses. Mastery of integration techniques, improper integrals, and series is essential for differential equations, real analysis, complex analysis, and advanced engineering mathematics.