Calculus Revisited is a series of videos and related resources that covers the materials normally found in freshman- and sophomore-level introductory mathematics courses. Complex Variables, Differential Equations, and Linear Algebra is the third course in the series, consisting of 20 Videos, 3 Study Guides, and a set of Supplementary Notes. Students should have mastered the first two courses in the series (Single Variable Calculus and Multivariable Calculus) before taking this course. The series was first released in 1972, but equally valuable today for students who are learning these topics for the first time.

In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun. If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.

This brief article relates the legend of young Gauss and the summing of consecutive numbers. Readers are asked to apply the method and they are shown a general solution. A link to a printable page is provided as well as links to related topics.

This problem gives children an opportunity to explore an increasing pattern and then generalize the results with a rule. Students begin with a single counter, surround it by a ring of other counters and then each new ring is surrounded with more counters. Solvers record results as they replicate the pattern and make predictions about many counters there will be in any given ring. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.

In this investigation, students visualize and compare volumes in solids composed of unit cubes and look for patterns in the measurements. They work systematically to organize and analyze the results. Ideas for implementation, extension and support are included.

Count as you are carrying out daily routines.While counting, stop often to ask what number comes next.  Ask what number comes after or before a given number.

This activity provides students with an opportunity to recognize arithmetic sequences and at the same time reinforces identifying multiples. The interactivity displays five numbers and the student must discover the times table pattern and the numerical shift. On Levels 1 and 2, the first five numbers in the sequence are given and on Levels 3 and 4, the numbers given could be any five numbers in the sequence. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, ideas for extension and support.

This problem offers a simple context to begin an exploration of the properties of numbers and to make conjectures about those properties. Learners explore the sums of consecutive numbers and whether all positive numbers from 1-30 can be written as the sum of two or more consecutive numbers. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.

This problem gives children an opportunity to explore patterns in a practical context and to generalize the results with a rule. Students investigate how many blocks would be needed to build an up-and-down staircase with any number of steps up. An interactivity in the hints shows the blocks transformed into a square pattern. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.