aa section 11-10
TRANSCRIPT
SECTION 11-10MODELING DATA WITH POLYNOMIALS
EXAMPLE 1
MATT MITARNOWSKI ROLLED A BALL DOWN AN INCLINED PLANE IN A PHYSICS LAB. HE
ACCURATELY MEASURED THE TOTAL DISTANCE TRAVELED BY THE BALL AS A FUNCTION OF TIME
AND OBTAINED THE FOLLOWING DATA:
Time(sec) 1 2 3 4 5 6 7 8
Distance (cm) 3 12 27 48 75 108 147 192
EXAMPLE 1
A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE?
Time(sec) 1 2 3 4 5 6 7 8
Distance (cm) 3 12 27 48 75 108 147 192
EXAMPLE 1
A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE?
Time(sec) 1 2 3 4 5 6 7 8
Distance (cm) 3 12 27 48 75 108 147 192
9 15 21 27 33 39 45
EXAMPLE 1
A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE?
Time(sec) 1 2 3 4 5 6 7 8
Distance (cm) 3 12 27 48 75 108 147 192
9 15 21 27 33 39 45
6 6 6 6 6 6
EXAMPLE 1
A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE?
Time(sec) 1 2 3 4 5 6 7 8
Distance (cm) 3 12 27 48 75 108 147 192
9 15 21 27 33 39 45
6 6 6 6 6 6
YES, THERE IS A QUADRATIC MODEL FOR THIS DATA
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a a = 3
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a a = 3
15 = 5(3)+ b
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a a = 3
15 = 5(3)+ b 15 = 15 + b
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a a = 3
15 = 5(3)+ b 15 = 15 + b
b = 0
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a a = 3
15 = 5(3)+ b 15 = 15 + b
b = 0
3 = 3 + 0 + c
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a a = 3
15 = 5(3)+ b 15 = 15 + b
b = 0
3 = 3 + 0 + c c = 0
EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
d = at 2 + bt + c d = distance, t = time
3 = a + b + c
12 = 4a + 2b + c
27 = 9a + 3b + c
27 = 9a + 3b + c
−12 = −4a − 2b − c
15 = 5a + b
12 = 4a + 2b + c
−3 = −a − b − c
9 = 3a + b
15 = 5a + b
−9 = −3a − b
6 = 2a a = 3
15 = 5(3)+ b 15 = 15 + b
b = 0
3 = 3 + 0 + c c = 0
d = 3t 2
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 2
FIT A POLYNOMIAL MODEL TO THE DATA.
X 1 2 3 4 5 6 7 8
Y 8 15 34 71 132 223 350 519
EXAMPLE 2
FIT A POLYNOMIAL MODEL TO THE DATA.
X 1 2 3 4 5 6 7 8
Y 8 15 34 71 132 223 350 519
7 19 37 61 91 127 169
EXAMPLE 2
FIT A POLYNOMIAL MODEL TO THE DATA.
X 1 2 3 4 5 6 7 8
Y 8 15 34 71 132 223 350 519
7 19 37 61 91 127 169
12 18 24 30 36 42
EXAMPLE 2
FIT A POLYNOMIAL MODEL TO THE DATA.
X 1 2 3 4 5 6 7 8
Y 8 15 34 71 132 223 350 519
7 19 37 61 91 127 169
12 18 24 30 36 42
6 6 6 6 6
EXAMPLE 2
FIT A POLYNOMIAL MODEL TO THE DATA.
X 1 2 3 4 5 6 7 8
Y 8 15 34 71 132 223 350 519
7 19 37 61 91 127 169
12 18 24 30 36 42
6 6 6 6 6
A CUBIC MODEL WILL FIT.
EXAMPLE 2
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
71 = 64a + 16b + 4c + d
−34 = −27a − 9b − 3c − d
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
71 = 64a + 16b + 4c + d
−34 = −27a − 9b − 3c − d
37 = 37a + 7b + c
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
71 = 64a + 16b + 4c + d
−34 = −27a − 9b − 3c − d
37 = 37a + 7b + c
34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
71 = 64a + 16b + 4c + d
−34 = −27a − 9b − 3c − d
37 = 37a + 7b + c
34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d
19 = 19a + 5b + c
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
71 = 64a + 16b + 4c + d
−34 = −27a − 9b − 3c − d
37 = 37a + 7b + c
34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d
19 = 19a + 5b + c 15 = 8a + 4b + 2c + d −8 = −a − b − c − d
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
71 = 64a + 16b + 4c + d
−34 = −27a − 9b − 3c − d
37 = 37a + 7b + c
34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d
19 = 19a + 5b + c 15 = 8a + 4b + 2c + d −8 = −a − b − c − d
7 = 7a + 3b + c
EXAMPLE 2
y = ax 3 + bx 2 + cx + d
8 = a + b +C + d
15 = 8a + 4b + 2c + d
34 = 27a + 9b + 3c + d
71 = 64a + 16b + 4c + d
71 = 64a + 16b + 4c + d
−34 = −27a − 9b − 3c − d
37 = 37a + 7b + c
34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d
19 = 19a + 5b + c 15 = 8a + 4b + 2c + d −8 = −a − b − c − d
7 = 7a + 3b + c
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
18 = 18 + 2b
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
18 = 18 + 2b b = 0
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
18 = 18 + 2b b = 0
7 = 7 + 0 + c
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
18 = 18 + 2b b = 0
7 = 7 + 0 + c
c = 0
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
18 = 18 + 2b b = 0
7 = 7 + 0 + c
c = 0
8 = 1 + 0 + 0 + d
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
18 = 18 + 2b b = 0
7 = 7 + 0 + c
c = 0
8 = 1 + 0 + 0 + d
d = 7
EXAMPLE 2
37 = 37a + 7b + c
19 = 19a + 5b + c
7 = 7a + 3b + c
37 = 37a + 7b + c
−19 = −19a − 5b − c
18 = 18a + 2b
19 = 19a + 5b + c
−7 = −7a − 3b − c
12 = 12a + 2b
18 = 18a + 2b −12 = −12a − 2b
6 = 6a a = 1
18 = 18 + 2b b = 0
7 = 7 + 0 + c
c = 0
8 = 1 + 0 + 0 + d
d = 7 y = x 3 + 7
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
HOMEWORK
P. 734 #1-17
“ACTION MAY NOT ALWAYS BRING HAPPINESS, BUT THERE IS NO HAPPINESS WITHOUT ACTION.”
- BENJAMIN DISRAELI