Calculus Index Cards

# Calculus Index Cards (scroll down to see all the cards)

 Here's one of the most common student questions: ‘What do I need to know?’ Answer: the ‘Index Card Method’

When I teach, every concept becomes an index card (4" x 6", blank both sides—it's easier to write math without lines). The cards guide class discussions. Students hand-copy the cards (adding additional information, as needed)—by course end, they have a stack that embodies every key idea. Cards can be mixed up so learning isn't order-dependent; ones that are well understood are taken out to focus on those that remain. Students hold the whole course in the palm of their hand.

I used this method for decades of teaching. Early on, the classroom blackboard became ‘one side of a card’: as I lectured, the card emerged on the board, and students took notes. Later on, students got photo-copies of the cards, or they were posted on the web. Many-a-student has told me (years later) that they still have their stack of cards!

Below are snapshots of the Calculus index cards. They're in my handwriting. You can purchase the entire set of jpg images for \$2.00. It's a zip file; you'll need to be able to unzip it on your computer. The purchase of this file includes the rights for teachers to make as many copies (electronic or printed) as needed for their students.  There are 91 card images (jpg) in a zipped folder. The cards are numbered 1–89, with two ‘extended’ cards: 43abcd and 53abcd. Click on any card below to see the full-size image. Beneath the images is a numbered list of all the cards by title/concept. You can see the cards used in the context of a college course here. STUDENTS: These are a great resource, even if your teacher isn't using the ‘index card method’. Just be sure you write them up yourself. Yep, you could print, cut, tape onto cards—but you'd reap only about 10% of the potential benefit. \$2.00 If \$2.00 is a hardship, then email me, and I'll send you the zipped folder containing the index cards. I do not want to deny anyone these materials. When you buy, you'll receive an email with a download link. (So, check your email after checking out!) # Index Card Titles/Concepts In each entry, the first line is side (a), the second line is side (b). Two cards (43 and 53) have additional sides (c) and (d). 1. average rate of change; the tangent problem 2. position functions; the derivative of a function 3. intuitive derivative (notation); example 4. limit of a function; 3 cases when$\,\lim_{x\rightarrow a} f(x) = \ell\,$5. one-sided limits; the squeeze (pinching) theorem 6. continuity at a point; three ways a function can fail to be continuous at$\,c\,$7. key idea (for continuity at a point); example 8. an important type of continuity problem; relationship between differentiability and continuity 9. limit rules; techniques for evaluating limits (examples) 10. indeterminate forms; deciding if you have an ‘indeterminate form’ 11. the derivative of$\,f\,$at$\,a\,$; using the definition of derivative 12. an alternative approach (compare with 11b); another example: using the definition of derivative 13. the linearization of$\,f\,$at$\,a\,$; intuition for the linearization 14. local max/min (extreme values); global max/min 15. strictly increasing/decreasing functions; getting increasing/decreasing behavior from the derivative 16. concavity; getting concavity info from the second derivative 17. power rule for differentiation; examples: using the power rule 18. partial proof of the power rule (both sides) 19. derivative of a constant times a function; finding$\,\frac{d}{dx} (kx^n)\,$20. derivatives of sums and differences; proof 21. differentiating composite functions: motivation; differentiating composite functions: the idea 22. the chain rule (prime notation); the chain rule (Leibnitz notation) 23. Why is it called the ‘chain rule’? Using the chain rule to differentiate$\,\bigl( f(x) \bigr)^n\,$24. pattern for generalizing all the basic differentiation formulas; examples 25. the product rule for differentiation; proof of the product rule 26. When is$\,\lim_{h\rightarrow 0} f(x+h) = f(x)\,$? When is this true? As$\,x\rightarrow c\,$,$\,f(x)\rightarrow f(c)\,$27. the quotient rule for differentiation; proof of the quotient rule 28. the irrational number$\,\text{e}\,$; equivalent limit statements defining$\,\text{e}\,$29. evaluating a limit involving$\,\text{e}\,$; Why is$\,\lim_{n\rightarrow\infty} \left[ (1 + \frac 1n)^n \right]^{2k} = \text{e}^{2k}\,$? 30. derivatives of logarithms; proof that$\,\frac d{dx} (\ln x) = \frac 1x\,$31. more notation for derivatives; notation: evaluating a derivative at a point 32. differentiating$\,{\text{e}}^x\,$; differentiating$\,a^x\,$33. what is$\,\lim_{x\rightarrow 0} \frac{\sin x}{x}\,$? example 34. derivatives of sine and cosine; derivatives of tangent and cotangent 35. derivatives of secant and cosecant a useful observation—derivatives of co-functions 36. finding the derivative of an inverse function; two methods: finding the derivative of an inverse function 37. What does$\,(f^{-1})'(x) = \frac{1}{f'\bigl(f^{-1}(x)\bigr)}\,$really mean? summary 38. derivatives of the inverse trigonometric functions; a typical derivation 39. differentiating variable stuff to variable powers: the log trick; example 40. l'Hopital's rule; motivation, ‘$\,\frac 00\,$’ case 41. basic use of l'Hopital's rule; more advanced use of l'Hopital's rule 42. Where can a function change its sign? negating ‘and’ and ‘or’ sentences 43. (a) info given by$\,f\,$,$\,f'\,$,$\,f''\,$; (b) a basic sign analysis of a function; (c) implications (and equivalent sentences); (d) the contrapositive of an implication 44. Where can a function have a local max/min? If$\,f\,$has a local max/min at$\,c\,$, then ... 45. critical points, critical numbers; Careful!! A critical point does not have to be a max or min!! 46. Where can a function have an inflection point? If$\,f\,$has an inflection point at$\,c\,$, then$\,\bigl(f''(c) = 0\ \ \text{or}\ \ f''(c) \text{DNE}\,$. 47. an algorithm for thorough function analysis; example—thorough function analysis 48. Are critical points actually max or min? Two tests: 1st and 2nd derivative tests; the first derivative test 49. open/closed intervals; bounded subsets of$\,\Bbb R\,$50. review of absolute/global max/min; the Extreme Value Theorem 51. optimization problems (finding max/min); example 52. a first optimization problem; You try it! 53. (a) an optimization problem; (b) ... continued; (c) a useful observation; (d) using the observation 54. an optimization problem; ... continued 55. explicit (‘showing’) versus implicit (‘hidden’); implicit differentiation 56. related rate problems; falling ladder problem 57. Is the car speeding? ... continued 58. the Mean Value Theorem; the Mean Value Theorem (MVT) intuition 59. a Mean Value Theorem example; a Mean Value Theorem example 60. bounded functions; the definite integral of a function 61. a definite integral may or may not exist; notation for the definite integral 62. dummy variables in definite integrals; simple definite integral examples 63. properties of the definite integral; linearity of the integral 64. definite integral example; definite integral example 65. How might we estimate areas beneath a curve? general notation for the area problem 66. summation notation; left and right estimates using summation notation 67. precise definition of a definite integral; Riemann sums 68. estimating a definite integral; finding a definition integral using the definition (for a very simple function) 69. the Fundamental Theorem, Part I; proof 70. antiderivatives; ‘undoing’ differentiation; antidifferentiation 71. Every differentiation formula gives an antidifferentiation formula! some antiderivatives you should know 72. examples: using the Fundamental Theorem; more examples 73. A very useful formula!! integrating a rate of change gives total change 74. connection between area and antiderivatives; making it more precise... 75. making it more precise, continued (both sides) 76. summary: the Fundamental Theorem of Calculus, Existence of an Antiderivative; the Evaluation Theorem 77. examples: derivatives involving integrals; a more complicated derivative involving an integral 78. indefinite integrals (general antiderivatives); more antiderivative formulas 79. finding a particular antiderivative; example 80. generalizing the formula for the derivative of$\,\ln x\,$; the antiderivatives of$\,\frac 1x\,\$
81. substitution;
format for substitution problems
82. a substitution example;
same problem, more compact
83. substitution with definite integrals;
2nd approach: merely indicate that limits are different
84. integration by parts;
basic strategy for using the parts formula
85. format for integration by parts;
using parts with a definite integral
86. periodically differentiable functions;
using parts with products of polynomials and periodically differentiable functions
87. inverse functions with simpler derivatives;
using parts for inverse functions
88. using parts with products of periodically differentiable functions;
... but it doesn't always work
89. analyzing falling objects;
example: integrating to find velocity and position information