STEP 1: 1st, 2nd, and 3rd sums
OBSERVE: for any X, nth sum is slope (of tangent) of n+1th sum
open Impact
clear()
let mutable y1, y2, y3 = 0.0, 0.0, 0.0
x <- 0.0
while x < 200. do
  dot (x,y1) green
  dot (x,y2) blue
  dot (x,y3) red
  x <- x + 1.
  y1 <- y1 + 0.05
  y2 <- y2 + y1
  y3 <- y3 + y2
 
STEP 2: Change initial value of 3rd sum
OBSERVE: Simply displaced 3rd sum.  (why?)
open Impact
clear()
let mutable y1, y2, y3 = 0.0, 0.0, 20.0
x <- 0.0
while x < 200. do
  dot (x,y1) green
  dot (x,y2) blue
  dot (x,y3) red
  x <- x + 1.
  y1 <- y1 + 0.05
  y2 <- y2 + y1
  y3 <- y3 + y2
 
STEP 3: Change initial value of 1st sum, hid 3rd sum (for simplicity)
OBSERVE: displaced 1st sum.  2nd sum has turning point
open Impact
clear()
let mutable y1, y2, y3 = -1.0, 0.0, 20.0
x <- 0.0
while x < 200. do
  dot (x,y1) green
  dot (x,y2) blue
//   dot (x,y3) red
  x <- x + 1.
  y1 <- y1 + 0.05
  y2 <- y2 + y1
  y3 <- y3 + y2
 
STEP 4: Changed initial value of 2nd sum
OBSERVE: displaced 2nd sum
open Impact
clear()
let mutable y1, y2, y3 = -1.0, 5.0, 20.0
x <- 0.0
while x < 200. do
  dot (x,y1) green
  dot (x,y2) blue
//  dot (x,y3) red
  x <- x + 1.
  y1 <- y1 + 0.05
  y2 <- y2 + y1
  y3 <- y3 + y2
 
STEP 5: Display 3rd sum
OBSERVE: multiple turning points at every zero of 2nd sum
open Impact
clear()
let mutable y1, y2, y3 = -1.0, 5.0, 20.0
x <- 0.0
while x < 200. do
  dot (x,y1) green
  dot (x,y2) blue
  dot (x,y3) red //uncomment
  x <- x + 1.
  y1 <- y1 + 0.05
  y2 <- y2 + y1
  y3 <- y3 + y2

STEP 6: Change initial values of 1st and 2nd sums
OBSERVE: Predict location of turning points of new 3rd sum
open Impact
clear()
let mutable y1, y2, y3 = -6.0, 250.0, 20.0
x <- 0.0
while x < 200. do
  dot (x,y1) green
  dot (x,y2) blue
  dot (x,y3) red
  x <- x + 1.
  y1 <- y1 + 0.05
  y2 <- y2 + y1
  y3 <- y3 + y2

STEP 7: Show 3rd sum (check answer previous question)
OBSERVE: 3rd sum shown in different scale
open Impact
clear()
let mutable y1, y2, y3 = -6.0, 250.0, 20.0
x <- 0.0
while x < 200. do
  dot (x,y1) green
  dot (x,y2) blue
  dot (x,y3/100.) red
  x <- x + 1.
  y1 <- y1 + 0.05
  y2 <- y2 + y1
  y3 <- y3 + y2