Calculus Course

1. Calculus Course

2. Baseline Rendering

3. Baseline vs. Progressive Rendering 25% 50% 100%

4. Current Garden Frank George

5. Current Garden Frank George

6. Current Garden Frank George

7. Area change = w + h + 1 Current Garden Current Garden

8. Three inputs, 3 changing perspectives to include

9. Scenario With 3 Parts Change Simplifies To A A A A + * * ^ B B B B / * + ^ C C C C A’s changes B’s changes C’s changes + + …

10. Three inputs, 3 changing perspectives to include

11. Scenario With 2 Parts Fuzzy Viewpoint A’s changes B’s changes + A + B …

12. Course Strategy/Content • Lesson 1: Intro • Lesson 2: Calculus In a Few Short Minutes

13. Lesson 3: Building A Garden • Go through this example end-to-end • Learning how to break things down step-by-step • Why? Business person. Want to see a record of each sale and expense, or just the bank balance at the end of the month? • Coach? Want to see the play-by-play summary, or just the final score? • If you’re making decisions… you want to see the changes that LED to a result, not just the final result! (You can always “replay” all the changes and get the final result anyway) • Seeing changes *evolve*, not just the final result • Want to give an example so you can discover calculus ON YOUR OWN, for yourself. It’ll click that much better. • Adding up the pieces… get a series of soil dumped your way. You know it’s making a square pattern! • Tick… tick. Every tick, they take a step. • Simple description • Ask two friends to help set up a garden • One walks North, one walks East • You’re not sure how big you want it. So you ring a bell. Each time it rings, they walk 1 foot more. • Clang! Way to small. Clang clang clang! It’s 4x4… still too small! Clang clang clang clang! Now it’s 8x8. Not bad. Clang! It’s 9x9. Decent, but a tad bigger would do. Clang! Now it’s 10x10. And your friends are glaring. • After 10 ticks, we have 10x10 square. Perimeter 40. Area 100. Fine. • But along the way • Did you notice… 4 feet of perimeter added for each tick • Did you notice… area went up 1, 3, 5, 7, 9, 11, 13, 15, 17, 19… and all the increases combine to the 100 square feet we now have! • Imagine we get a delivery of 10 square feet of mulch every day • In the beginning, we can just make our garden • At some point, we can no longer take a step forward. Once we’re at 5x5, the next step (to 6x6) will require 5*2 + 1 = 11 square feet. Ack. We have to “save up” a day. • And later • As long as 2*x + 1 < incoming rate. Don’t start abstracting too early. We can just notice these patterns. The problems that may arise. This is “calculus”. • Calculus is putting a name to these special patterns we noticed (the derivative – how it changed along the way) • Calculus is working backwards from the sequence of changes, to what pattern is being made (see 1, 3, 5, 7, 9… a square!) • Calculus is optimizing what you need (what if we need to stop after we *add* 10 or more square feet of area…) • Calculus is strategies for breaking a “final result” into a sequence of smaller steps

14. Lesson 4: Splitting a cake • Goal: Another scenario to examine step-by-step • You’re having a birthday part. About to cut the cake. • Friend comes in • And another • And another • How do you model the changes? How much cake are you losing with each additional person? • Inverse relationship (your share is 1/x). How is your share changing as x (the number of people) increases? • So… can you see how ¼ = 1 [starting] – ½ [one person] - 1/6 [another comes in] • And 1/5 = 1 – ½ - 1/6 – 1/12 [and another] • Neat! • How many people does it take to save $50/person? (1/20 the total?) • Calculus. Transaction-by-transcation. • You could manually start adding differences… but calculus gave you the pattern! • Where do you need to be to see the change you need? • Would have noticed that pattern on your own?

15. Lesson 5: Find The Perfect Rate • Let’s dive in: time to see how to make calculations on our own. Here’s the basic process • Come up with the change formula • Find change on a per-change basis • Make it perfect • How to • If x is our variable, then “dx” (aka delta x) is the amount it changes • If we have a 10x10 garden and go to 11x11, then x =10 and dx = 11 • Example: Building our Garden • Current area: x^2 (i.e., 10 x 10) • New area: (x + dx)^2 (i.e., 11x11) • Change: 2x.dx + dx^2 = 20*10*1 + 1^2 = 21 • Change on a “per dx’ basis: 2x + dx • What do we see? One part of it depends on our current value (x) but another part ONLY depends on the change we made! (dx^2) • Example: Measuring your speed. Imagine the speed measurement forced you to go a whole hour… ugh! We want to make it independent. • So we get the measurement, but then make it INDEPENDENT of the size of our change (i.e., a perfect change) • We let dx go to zero. • Example: Cutting our Cak • Current: 1/x • New: 1/(x + dx) • Change: -1/x(x+dx) • Make it perfect: -1/x^2 • Note • Limits and infinitesimals are the formal rules we use to “let dx turn to zero”.

16. Nested Systems

17. Open ended question • How can we account for the size of the change? • What does d/dx f^2 mean? • We have a system which is based on x somewhere along the line (f) • But we are going to square it • So we know the change in the outer system: 2f + 1 • And then we multiply by df/dx, how much it then depends on x • Look at the outer system’s changes • Then drill into the inner one • Most common mistake • Getting confused about when to include the other dx • 1/g^2 … dg/dx [which could be 1!!!]

18. Lesson 6: Exchanging Currency • Pretend you can get $3 in profit per square foot • Make a clap… how much more profit will you get? • We have a chain reaction • We clap, which makes area go up… • Which makes profit go up… • Which makes EUR go up • And later • Chain reaction. Change A (how much does it change?) which changes B (how much does it change?) which changes C (how much does it change?), on and on. • We have a 2x2 and we clap… • Which makes our area go up 5 feet (2x + 1) • Which makes our profit go up (3) • Which could be converted • Now, let’s say our profit was not just linear… another proportion? • Clap, which is 2x + 1 • And that is how long we run another clapper for (2x + 1) over there!

19. More Chain Rule Analogies • We want the “currency” of u, v, w, etc. • x changes… which goes up the chain • Or work top-to-bottom • The outer system is based on this inner one • When this inner system changes, we SCALE IT UP [here is the confusion… the inner system is changing BY A RATE… and we are scaling up that rate] • Can we have the inner system change by a set amount? x => x + dx? • Then the outer system changes by df • df/dx = 2x [rate it changes, on a per-dx basis] • df = 2x.dx [actual amount it changes]

20. Search online for chain rule analogies… have it really click • So, the key is realizing the rates are NOT like gears (which are fixed) • The conversion rate changes! • A to X = A to B, B to C, C to D… D to X • Each step can be non-linear • Not just a number! An entire conversion • How much are we going to get? What it’s based on? • Let’s say the “A to B” step is A = B^2 • Then changing B (dB) has the effect 2A * dA • So we drop in the 2A dA • Keep going! Now we can start to get really complex. Neat. Let’s solve some problems for real. • Giant applications of the chain rule.

21. Slicing A Cake Among Friends A B C Cake New person? Cut a slice from everyone D A B C

22. Lesson 5: How to find derivatives • Lesson 6: How to throw the unneeded things away

24. Concept Goals • Want you to think step-by-step • Not just final results • Learn how to make better estimates • Go from discrete (large, noticeable steps) to continuous (perfectly smooth, undetectable steps) • Learn how to calculate the actual changes • Derivative: see what the next step will be • Integral • Mechanical: accumulate previous steps to get a total • Artistic: Figure out what shape the total is forming

25. Gotcha! • Remember it’s • (1/g)’ = -1/g^2 * dg/dx • We need that dg! Have to jump into the rate. We’re saying… • The scaling factor is -1/g^2 times the rate we end up moving • Don’t forget about that extra dg. Every term needs that scaling factor.

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27. X-Ray Original Rings Slices Boards

28. Ring By Ring Timelapse 25% 50% 75% 100%

30. X-Ray

32. Shape

33. X-Ray View Step-by-step

34. X-Ray Shape Step-by-step

38. dr r p dp x dx

39. Shape Animation Spin

45. pi * r^2 / r = pi * r