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In Chapter 5 we introduce the concept of the inner product, and through that we construct the concept of length and distance. The Least-Squares Problem aims to find the "closest" solution to what is technically an inconsistent system - one that has no solution. Most undergraduate math students are exposed to this problem when trying to find the line of best fit to a data set. However, from our vast experience with formulas, that they aren't all linear. This problem set deals with the problem of non-constant acceleration. Two researchers conduct an experiment with an electric car on a test track. While one is driving the car, the other will look at the odometer of the car at two-second intervals. The times are in seconds and the distances are in meters. The data set they create is: ((0,0), (2,8),(4,32), (6,72), (8,128), (10,192) They notice that the acceleration is not a constant value. They decide that a third-degree polynomial will be the best to describe the distance the car travels as a function of time. The task here is to determine the third-degree polynomial that fits this data set the best.
Exercises 1. Use a general third-degree polynomial and the data to construct six equations. Write the equations here: HO →>> => ⇒>> => Project=3 x (Seconds) y (meters) O 2 4 6 8 128 10 1.92 P(x) = ax³ + bxc² + cx+d O= a(0)³ + b(0)² + c(0) + d a (2)3 + b(2)+ ((2) +d a (4)3 + b(4)² + ((4) + d 32 72= a (6)3 + b(6)² + c(6) + d 128= a(8)³ + b (8)² + ((8) + d 192= a (10)³ + b(10)² + ((10) + d 32 72 4 4a + 2b + c 8 = 16a + 4b + c 12 36 a + (b + c 16 = 64a + 8b+ C 19.2= 100a+10b+c =
2. Construct the least-squares problem Ax=b . 3 A=L ΑΤΑ AT "1 AR=D 4 1.60 36 64 -100 77 2 4 4 2 6 8 10 +6₂ 4 F F 1 F 1 36 16 A B с 64 OOF OF ?=9 4 12 1.60 19-2
3. Construct the system of normal equations A¹ AX=A¹b. A¹A=L 3→ ATA AT 201 = 4 1.6 36 64 100 24 6 1 1 1 4 16 36 64 100 ATA= 2 4 6 8 10 1 1 1 1 1 15664 ATA= 1800 220 CS Scanned with CamScanner АТБ АТБ = 4 16 36 24 Co 1 FF Ab= 15664 1800 1800 200 220 30 64 100 8 3520 432 59.2 OF 1 1 JO TI 2.20 30 5 F ● 17 • 16 OF F 1800 220 200 30 55 36 OF OOF 30 85 4 2 8568 64 8 1 FX x₂ 4 12 16 19.2 Ex 거리게 되 1 F 1 1 A¹b=l 3520 432 59.2
4. Solve the system of normal equations. (I don't want you doing this by hand. Use a calculator or app.) x=L Ab= 15664 1800 220 3520 432 59.2 1800 200 30 220 30 5 x1 I₂ 15664x+1800 x₂ + 220 x 1.800 x1 + 200 x₂ + 30 x 3 2009 + 30 xe 45xg 2 3520 432 59.2 3590 432 59.2 15664x + 1800x₂ + 220x₂ = 3520 + 200x₂ +30 1800 220 x + 30x₂ = 432 + 5x3=59,2 carta Ordio Hea -0.1831, x₂ = 5.563 5. State the third-degree polynomial of best fit that relates the time (in seconds) to the distance traveled (in meters) by the car. 6. Use calculus to construct a polynomial that gives the velocity of the car as a function of time.