**Data Table:**| x | 1 | 2 | 3 | 4 | 5 | 6 ||---|-----|-----|-----|-----|-----|-----|| y | 831 | 1073| 1317| 1715| 2127| 2487|**Instructions:**1. **Exponential Regression:** Use exponential regression to find an exponential equation that best fits this data. Provide both $a$ and $b$ rounded to at least four decimal places. \[ y = \]2. **Linear Regression:** Use linear regression to find a linear equation that best fits this data. Provide both $m$ and $b$ rounded to at least two decimal places. \[ y = \]3. **Best Fit Equation:** Of these two, which equation best fits the data? - [ ] Linear - [ ] Exponential**Submit Question Button:**- [Submit Question]

Answered: y X y 1 2 831 1073 y = 4 3 1317 1715…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Data Table:**

| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|-----|-----|-----|-----|-----|-----|
| y | 831 | 1073| 1317| 1715| 2127| 2487|

**Instructions:**

1. **Exponential Regression:**
Use exponential regression to find an exponential equation that best fits this data. Provide both $a$ and $b$ rounded to at least four decimal places.
\[
y =
\]

2. **Linear Regression:**
Use linear regression to find a linear equation that best fits this data. Provide both $m$ and $b$ rounded to at least two decimal places.
\[
y =
\]

3. **Best Fit Equation:**
Of these two, which equation best fits the data?
- [ ] Linear
- [ ] Exponential

**Submit Question Button:**

- [Submit Question]

Expert Solution

Step 1

Sol:-

Exponential Regression:-

$T h e c u r v e t o b e f i t t e d i s y = a e^{b x} t a k i n g \log a r i t h m o n b o t h s i d e s, w e g e t \log_{10} (y) = \log_{10} (a) + b x \log_{10} (e) Y = A + B x w h e r e Y = \log_{10} (y), A = \log_{10} (a), B = b \log_{10} (e) w h i c h l i n e a r i n Y, x S o t h e c o r r e s p o n d i n g n o r m a l e q u a t i o n s a r e \sum Y = n A + B \sum x \sum x Y = A \sum x + B \sum x^{2}$

The values are calculated using the following table

x	y	Y=log10(y)	x²	x⋅Y
1	831	2.9196	1	2.9196
2	1073	3.0306	4	6.0612
3	1317	3.1196	9	9.3588
4	1715	3.2343	16	12.9371
5	2127	3.3278	25	16.6388
6	2487	3.3957	36	20.3741
---	---	---	---	---
∑x=21	∑y=9550	∑Y=19.0275	∑x²=91	∑x⋅Y=68.2895

Substituting these values in the normal equations

6A+21B=19.0275

21A+91B=68.2895

Solving these equation :-

6a+21b=19.0275

and 21a+91b=68.2895

∴21a+91b=68.29

6a+21b=19.0275→(1)

21a+91b=68.2895→(2)

equation(1)×7⇒42a+147b=133.1925

equation(2)×2⇒42a+182b=136.579

Subtracting ⇒-35b=-3.3865

⇒35b=3.3865

⇒b=0.096757

Putting b=0.096757 in equation (1), we have

6a+21(0.096757)=19.0275

⇒6a=19.0275-2.0319

⇒6a=16.9956

⇒a=2.8326

∴a=2.8326 and b=0.096757

we obtain A=2.8326,B=0.0968

$∴ a = a n t i \log_{10} (A) = a n t i \log_{10} (2.8326) = 680.1426 a n d b = \frac{B}{\log_{10} (e)} = \frac{0.0968}{0.4343} = 0.2228 N o w s u b s t i t u t i n g t h i s v a l u e s i n t h e e q u a t i o n i s y = a e^{b x}, w e g e t y = 680.1426 \cdot e^{0.2228 x}$