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Real Analysis –
mth440
(3 credits)
This course focuses on the functions of real variables. Students investigate the topology of the real line and plane, and use these concepts to prove limit and convergence theorems about sequences, series, and functions. Students also examine continuity, differentiation, and the Riemann integral.
This undergraduate-level course is 5
weeks.
To enroll, speak with an Enrollment Advisor.
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Sequences, Series, and Limits
- Determine limits of functions.
- Prove statements about infinite series.
- Prove statements about sequences and subsequences.
- Determine convergence and sums of series of real numbers.
- Determine convergence and limits of sequences of real numbers.
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Preliminaries and Real Numbers
- Prove statements about intervals.
- Prove statements using the Completeness and Supremum Properties of real numbers.
- Find subsets of R defined by specified conditions.
- Prove statements about the sets R of real numbers and Q of rational numbers.
- Prove statements about finite and infinite sets.
- Use the Principle of Mathematical Induction to prove statements about the set of natural numbers.
- Prove statements about sets and functions.
- Find the results of operations on finite sets.
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The Riemann Integral
- Use the Trapezoidal Rule and Simpson's Rule to find approximations to integrals.
- Find integrals and derivatives, using the Fundamental Theorems of Calculus.
- Determine whether or not a function is integrable.
- Compute the value of a Riemann sum.
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Continuous Functions
- Prove statements about continuous functions.
- Prove statements about monotone and inverse functions.
- Determine whether or not a function is uniformly continuous.
- Prove statements about continuous functions on intervals.
- Prove continuity using rules for combinations of continuous functions.
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Differentiation
- Use Taylor's Theorem to develop polynomial approximations to functions.
- Use L'Hospital's Rules to find limits of functions.
- Use the Mean Value Theorem to prove the existence of relative extrema.
- Determine whether or not a given function is differentiable.
- The University of Phoenix reserves the right to modify courses.
- While widely available, not all programs are available in all locations or in both online and on-campus formats. Please check with a University Enrollment Advisor.
- Transferability of credit is at the discretion of the receiving institution. It is the student’s responsibility to confirm whether or not credits earned at University of Phoenix will be accepted by another institution of the student’s choice.
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