1. Intuition-First Learning
Concept | Key Analogy / Takeaway |
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Learning Strategy: Blurry, then refine | ![]() |
Music Analogy | In music: Appreciation (sounds good!), description (hum it), symbols (sheet music), performance (play it) In math: Appreciation (aha!), description (English), symbols (math), performance (calculate) |
Appreciation vs. Performance | We can enjoy listening to music even if we can't play it. We can think with calculus even if we can't (yet) compute with calculus. |
2. Appreciating Calculus
Concept | Key Analogy / Takeaway |
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Calculus in 1 minute | See the world with X-Ray and Time-Lapse vision.
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Calculus in 10 minutes | New viewpoints lead to insights:
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So what can I do with Calculus? | See patterns at a deeper level and make predictions we couldn't before. |
Example: Analyzing a Circle | Build with rings, slices, or boards; each has tradeoffs.
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Ring-by-ring View | Organic growth, increasing effort. ![]() |
Slice-by-slice View | Assembly-line, predictable progress ![]() |
Board-by-board View | Robotic, never retracing ![]() |
3d versions | ![]() |
How to think with calculus | For your situation:
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3. Describing Calculus (English → Math)
Concept | Key Analogy / Takeaway |
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Organize Our Descriptions | Direction of slices in orange. Arrange slices side-by-side for easy comparison.
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Term: Derivative | Creating a pattern of step-by-step slices along a path (i.e., rings, slices, boards, etc.). |
Term: Integral | Accumulating slices into a shape (what is being built up as we go?) |
Notation Details | Derivatives only require the direction we move when taking slices: frac(d)(dr) means slice along r's direction.
Integrals require direction we glue together (dr), where we start/stop (0 to r), and the size of each slice (2 π r dr:
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Computer Calculation | Wolfram Alpha can compute integrals/derivatives when asked:
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Note about Abstraction | Begin thinking about general patterns (x, x^2), not just specific shapes (a line, a square). |
Lines | Lines (f(x) = ax) change by a steady a each time, like building a fence. |
Squares | Squares (f(x) = x2) change by 2x + 1:
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4. Learning To Measure Change
Concept | Key Analogy / Takeaway |
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Infinite processes can point to a result | Analogy: fly going back and forth. Count its paths ("infinite"), or just the time traveled. Pixellated letters point to the smooth whole. |
Analogy: Measuring Heart Rate | Get on treadmill, hook up wires, run. The measurement is your heart rate under stress. Must then remove impact of wires. |
The formal derivative | Find change, then assume change had zero effect:
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Example | 2x + dx overset(dx = 0) Longrightarrow 2x, so frac(d)(dx) x2 = 2x, with errors artifacts removed |
Fundamental Theorem of Calculus | The shortcut to computing the integral is finding a pattern that made the changes we're seeing. |
Integrals are Reverse Engineered | See patterns of steps and ask: what shape could have made this?
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5. The Core Rules Of Calculus
Concept | Key Analogy / Takeaway |
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Multiplication Rule
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Grow a garden on two sides; ignore the corner.
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Simple Division
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Splitting cake, new person enters (from halves to thirds).
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Insight: Each perspective makes a contribution | With 3 variables you have 3 perspectives to add:
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Power Rule
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Each side has a point of view; I change, others are the same.
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Rule Summary | ![]() |
6. Finding Archimedes' Formulas
Concept | Key Analogy / Takeaway |
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Find the circle/sphere formulas | X-Ray and Time-Lapse a single ring into the other shapes.
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Circumference to Area | ![]() ![]() |
Area to Volume | ![]() ![]() ![]() |
Volume to Surface Area | ![]() ![]() ![]() |
Historical Note | Archimedes had a calculus mindset: he re-arranged discs, cylinders, cones, etc. to make "easy to measure" slices. The techniques you already know would make him tear up. |