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Calculus: Getting Started

MATH 2100, MATH 2200, MATH 3200, MATH 4050, MATH 4060

Calculus

Calculus
MATH 2100, 2200, 3200, 4050, 4060
Fall 2020

Functions, limits, and derivatives of algebraic functions. Introduction to derivatives of trigonometric functions, logarithmic functions; application of derivatives to physics problems; related rates and maximum/minimum problems, and definite and indefinite integrals with applications.
Riemann sums; the definite integral; method of integration; continuation of exponential, logarithmic functions, and inverse trigonometric functions. L'Hopital's rule and improper integrals.
Functions of several variables, surfaces, in three-space, vectors, techniques of partial differentiation and multiple integration with applications. Vector calculus topics will include the theorems of Green, Gauss and Strokes.
The course is an introduction to the theoretical treatment of the real numbers, sets, functions, sequences, limits and calculus. The course places an emphasis on reading and writing formal mathematical proofs. Topics include: the real number system, convergence of sequence and series, continuity, limits, functions of one real variable, and the theoretical foundations of differentiation and integration of functions of a single variable. 
‚ÄčThis course is a continuation of Advanced Calculus I providing an introduction to metric spaces and their topology. The course places an emphasis on extending results for real functions to multivariable functions. Topics include: metric spaces and topology, integration, differentiation, optimization and analysis in several variables.

Calculus I and II - MATH 2100 and 2200

Advanced Calculus I and II - MATH 4050 and 4060

  1. Describe the concept of limit of a function at a point and at infinity and the concept of continuity at a point and over an interval. Compute limits of sums, products, differences, and quotients of functions.
  2. Describe the relationship between line slope, rate change, and derivative and apply the definition of derivative to compute the derivative of a function.
  3. Compute derivatives of functions using the product, quotient, power and chain rules explicitly and implicitly.
  4. Use the second derivative to find intervals where a function is concave up/down.
  5. Describe how the integral came about and be able to compute areas under curves.
  1. Understand and apply differentiation rules to exponential and logarithmic functions as well as inverse trigonometric functions.
  2. Learn new techniques of integration by parts, integration by trigonometric substitution, and integration via a partial fraction decomposition and utilize them where appropriate.
  3. Apply the Fundamental Theorem of Calculus to solve a variety of applied problems such as the area between 2 curves, volumes of revolution, surface area, length of a curve, and problems involving the concept of work.
  4. Comprehend the fundamental concepts and differences between sequences and series of numbers and apply the appropriate tests to determine their convergence or divergence.
  1. Define and use vector operations in two and three dimensions.
  2. Demonstrate an understanding of partial derivatives and multiple integrals.</span
  3. Perform applications of partial derivatives and multiple integrals.
  4. Understand the fundamentals of vector space calculus including the Green, Gauss and Stokes theorems.
  1. Construct mathematically rigorous proofs of the real number line from the axioms for the field of real numbers.
  2. Utilize a rigorous approach to applying limits to problems on limits of sequences and functions.
  3. Apply the definition of derivative to prove elementary results of differentiation.
  4. Apply the definition of the Riemann integral to prove elementary properties of Riemann Integral and Fundamental Theorem of Calculus.
  1. Utilize the definition and topology of metric spaces and prove basic results of compactness.
  2. Apply the definition of derivatives in a multi-dimensional space to prove elementary results of differentiation.
  3. Demonstrate the pointwise and uniform convergence of infinite series of functions.
  4. Apply the integration in multi-variable spaces to prove basic results of line integrals.

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