Breaking Free from Traditional Calculus Textbooks with Mathematica
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1 Breaking Free from Traditional Calculus Textbooks with Mathematica Matt Cathey Joseph Spivey Department of Mathematics Wofford College International Conference on Technology in Collegiate Mathematics, 2016
2 Problems with Our Old Calculus Sequence Short semesters Taken as a sequence, the courses were good, but few students take whole sequence. South Carolina scholarships
3 Old Sequence Served Very Few Students Calculus I: no integrals, lots of theory Calculus II: overlapped heavily with AP Calculus Students entering Wofford in who took Calculus Took one half of sequence at Wofford 189 Took whole sequence at Wofford 62
4 Old Sequence Served Very Few Students Calculus I: no integrals, lots of theory Calculus II: overlapped heavily with AP Calculus Students entering Wofford in who took Calculus Took one half of sequence at Wofford 189 Took whole sequence at Wofford 62
5 Calculus I: Old Version Review of precalculus Limits (intuitive definition) Limits (rigorous definition) Using Limit Laws Proving Limit Laws Continuity (intuitive) Continuity (rigorous) Continuity and limits Limits at infinity The derivative as a limit The derivative function Basic derivative rules Product/Quotient Rules Trig/Exp derivatives Chain Rule Implicit differentiation Derivatives of inverse functions Related Rates Differentials Optimization Mean Value Theorem l Hôpital s Rule Newton s Method Antidifferentiation Riemann sums Definite integration FTC
6 Calculus II: Old Version Review of antidifferentiation Review of definite integrals Review of Fundamental Theorem Integration by substitution Areas of planar regions Volumes Work Average value of a function Integration by parts Integrals of trig functions Trigonometric substitutions Partial fractions Numerical integration Improper integration Other topics
7 Calculus I: Changing... Review of precalculus Limits (intuitive definition) Limits (rigorous definition) Using Limit Laws Proving Limit Laws Continuity (intuitive) Continuity (rigorous) Continuity and limits Limits at infinity The derivative as a limit The derivative function Basic derivative rules Product/Quotient Rules Trig/Exp derivatives Chain Rule Implicit differentiation Derivatives of inverse functions Related Rates Differentials Optimization Mean Value Theorem l Hôpital s Rule Newton s Method Antidifferentiation Riemann sums Definite integration FTC
8 Calculus I: Changing... Review of precalculus Limits (intuitive definition) Limits (rigorous definition) Using Limit Laws Proving Limit Laws Continuity (intuitive) Continuity (rigorous) Continuity and limits Limits at infinity The derivative as a limit The derivative function Basic derivative rules Product/Quotient Rules Trig/Exp derivatives Chain Rule Implicit differentiation Derivatives of inverse functions Related Rates Differentials Optimization Mean Value Theorem l Hôpital s Rule Newton s Method Antidifferentiation Riemann sums Definite integration FTC
9 Calculus II: Changing... Review of antidifferentiation Review of definite integrals Review of Fundamental Theorem Integration by substitution Areas of planar regions Volumes Work Average value of a function Integration by parts Integrals of trig functions Trigonometric substitutions Partial fractions Numerical integration Improper integration Other topics
10 Calculus II: Changing... Review of antidifferentiation Review of definite integrals Review of Fundamental Theorem Integration by substitution Areas of planar regions Volumes Work Average value of a function Integration by parts Integrals of trig functions Trigonometric substitutions Partial fractions Numerical integration Improper integration Other topics
11 Calculus I: New version Historical context Limits (intuitive definition) Using Limit Laws Continuity (intuitive) Continuity and limits Limits at infinity Derivative as a limit Derivative function Basic derivative rules Antidifferentiation Intro to DEs Trig/Exp derivatives Antiderivs. of trig/exp Product/Quotient rules Int. by parts Differentials Chain rule Int. by substitution Implicit diff. Derivatives of inverses Related rates Riemann sums Definite integration FTC Areas of planar regions Numerical integration Optimization Separable DEs
12 Calculus I: New version Historical context Limits (intuitive definition) Using Limit Laws Continuity (intuitive) Continuity and limits Limits at infinity Derivative as a limit Derivative function Basic derivative rules Antidifferentiation Intro to DEs Trig/Exp derivatives Antiderivs. of trig/exp Product/Quotient rules Int. by parts Differentials Chain rule Int. by substitution Implicit diff. Derivatives of inverses Related rates Riemann sums Definite integration FTC Areas of planar regions Numerical integration Optimization Separable DEs
13 Calculus II: New version Rigor in mathematics Limits (rigorous definition) Proving Limit Laws Continuity (rigorous) Proof of MVT Proof of FTC Proofs of derivative rules l Hôpital s rule Volumes Work Average value Trigonometric subs. Partial fractions Improper integration Sequences Series Convergence tests Power series Taylor series Applications
14 Reasons for Creating Our Own Textbook Digital textbook Textbook market WebAssign Interactive figures
15 Demonstration of the Textbook
16 Feedback Students rated half-chapters of Calculus I on a 1 10 scale; averages ranged from 6.6 to 8.2. Sample Comments From First Half of Textbook: I personally hate reading stuff electronically and I found reading a math textbook particularly difficult. While I was carefully reading it, it felt like the professor was actually speaking to me, a human aspect that I can relate to. Making it easier to remember.
17 Sample Comments, Continued Like I said in the previous response I am a visual learner therefore the graphs helped me understand the formula a TON! If I didn t understand what the formula was saying I just scrolled down and watch the graph move based on how I changed it. None were particularly hard to understand. Once or twice I had to return to the explanation of the concept to actually see what was happening.
18 Link for More Information A draft of the textbook is available at digitalcommons.wofford.edu/facultybooks/1/. Materials from this presentation are available at presentation/. For questions or comments, you can reach Dr. Cathey at catheyme@wofford.edu and Dr. Spivey at spiveyja@wofford.edu. Thank you for coming!
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