The materials provided here are for an advanced version of calculus. Eventually I'll add more so it can serve as a two semester sequence in calculus. It does not really conform to any sequence of courses at Occidental at this point. I will post separate materials for Math 110 at Occidental eventually.
The novelty of this course is it covers topics from multivariable calculus that are essential for economics. Many economics students do not take multivariable calculus, so this course prepares them for studying economics.
The flipped class videos should be watched before the in class recordings. Group work is provided and should be worked before moving on to the next section.
Velocity and Tangent Lines: We begin motivating the idea of a derivative by considering the problem of estimating the velocity of an object by using its position function and using smaller and smaller time intervals to calculate average velocity. We then repeat the procedure to estimate the slope of the tangent line by taking slopes of secant lines over smaller and smaller intervals.
Velocity and Tangent Lines group work: Link
2. Limits, Asymptotes, and Continuity: In this section we introduce limits via numerical approximations, graphical approximations, and limit laws. We also cover horizontal and vertical asymptotes as well as the notion of continuity. Epsilon-delta definitions and proofs are not included as this is meant for a general calculus class serving majors and non-majors.
2. Limits, Asymptotes, and Continuity group work: Link
3. The Derivative: In this section we introduce the derivative. We start with motivating examples via practical applications. We work towards the formal definition as a limit. The section focuses on understanding the concept of the derivative and what it is telling us in different situations.
3. The Derivative group work: Link
4. The Second Derivative and Concavity: In this section we introduce the second derivative and concavity. Practical applications are given to show how these concepts apply in real world settings.
4. The Second Derivative and Concavity group work: Link
5. Some Basic Derivatives: In this section we introduce the first shortcuts for derivatives including polynomials, the exponential function, and sine and cosine.
5. Some Basic Derivatives group work: Link
6. The Product and Quotient Rules: We introduce the product and quotient rules in this section.
6. The Product and Quotient Rules group work: Link
7. The Chain Rule: In this section we introduce the chain rule.
7. The Chain Rule group work: Link
8. Hyperbolic Functions: We introduce hyperbolic functions, establish some basic properties, and see a couple of examples of how they are used.
8. Hyperbolic Functions group work: Link
9. Linear approximation and differentials: We introduce the notion of linear approximation and see several examples of how it can be used. We then introduce differentials and look at applications.
9. Linear Approximation and Differentials group work: Link
10. Taylor Polynomials: We introduce Taylor polynomials as well as applications.
10. Taylor Polynomials group work: Link
11. The Mean Value Theorem: This theorem is only covered in a flipped class video.
12. Maximum and Minimum Values: Local maximum and minimum values are introduced as well as the Extreme Value Theorem.
12. Maximum and Minimum Values group work: Link
13. Optimization: Various applied problems in optimization are solved.
13. Optimization group work: Link
14. Newton's Method: The method is developed and then a demonstration is given using Mathematica. (See below for the Mathematica worksheet.)
14. Newton's Method group work: Link
15. Functions of Several Variables: We introduce functions of multiple variables, focusing mainly on functions of two variables. Contour diagrams are also included.
15. Functions of Multiple Variables group work: Link
16. Partial Derivatives: Partial derivatives and applications are covered.
16. Partial Derivatives group work: Link
17. Tangent Plane Approximations and Differentials: In this section we expand the notion of tangent line approximations to that of tangent planes. We also cover differentials of functions of multiple variables and applications.
17. Tangent Plane Approximations and Differentials group work: Link
18. Chain Rule in Multiple Variables: We cover the chain rule in multiple variables and show how this leads to implicit differentiation.
18. Chain Rule in Multiple Variables group work: Link
19. Implicit Differentiation and Related Rates: We approach implicit differentiation and related rates through the chain rule of multiple variables. This avoids the unmotivated "just trust me this works" approach normally used in calculus 1.
19. Implicit Differentiation and Related Rates group work: Link
20. Optimization of Functions with Two Variables: We generalize our earlier optimization results of one variable to two variables. This section can be omitted as it is encountered again in multivariable calculus and the economics students really only need Lagrange Multipliers.
21. Constrained Optimization and Lagrange Multipliers: We give a treatment of Lagrange multipliers that does not rely on any knowledge of vectors or the gradient of a function.
21. Constrained Optimization and Lagrange Multipliers group work: Link