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Combined Math Solutions

The document presents a sales analysis of hybrid vehicles, detailing a regression line equation and estimating vehicle sales for 2022 with a percentage relative error of 4.25%. It also includes logarithmic and calculus problems, providing the domain of a function, its derivative, and solutions to specific logarithmic equations and inequalities. Key findings include the defined domain of a function and the conclusion that a certain inequality holds for all real numbers.

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ramadansajed53
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4 views2 pages

Combined Math Solutions

The document presents a sales analysis of hybrid vehicles, detailing a regression line equation and estimating vehicle sales for 2022 with a percentage relative error of 4.25%. It also includes logarithmic and calculus problems, providing the domain of a function, its derivative, and solutions to specific logarithmic equations and inequalities. Key findings include the defined domain of a function and the conclusion that a certain inequality holds for all real numbers.

Uploaded by

ramadansajed53
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Combined Math Solutions (Hybrid Vehicle & Logarithmic Problems)

Part 1: Hybrid Vehicles Sales Analysis

Given data: x = [1, 2, 3, 4, 5, 6], y = [18, 32, 65, 84, 105, 123]

Regression Line Equation: y = 21.80x + -5.13

Center of Gravity G = (3.50, 71.17)

Correlation Coefficient r = 0.9953

Estimate for 2022 (x = 7): y = 147.47 thousand vehicles

Actual (15% increase from 2021): y = 141.45 thousand vehicles

Percentage Relative Error = 4.25%

Part 2: Logarithmic and Calculus Problems

1) Domain of f(x) = x / (1 - ln(x))

To be defined, ln(x) must exist => x > 0

Also, denominator != 0 => 1 - ln(x) != 0 => ln(x) != 1 => x != e

Final Domain: (0, e) U (e, infinity)

2) Derivative of f(x) = ln(2x - 1) - ln(2x + 1)

Using d/dx ln(u) = u'/u:

f'(x) = (2)/(2x - 1) - (2)/(2x + 1)

3) Solve ln(x + e) = 1

x + e = e^1 = e => x = 0

4) Solve inequality e^x + 4 > 1

e^x > -3 which is always true since e^x > 0


So solution set is all real numbers R

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