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Retaining the Concepts. If f(x) = 4x2 - 5x - 2, find [f(x + h) - f(x)]/h, h ≠ 0
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 9, Problema 92
How many four-digit odd numbers less than 6000 can be formed using the digits 2, 4, 6, 7, 8, and 9?
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Identify the constraints for the four-digit number: it must be less than 6000, so the first digit can only be 2 or 4 (since 6, 7, 8, 9 would make the number 6000 or greater).
Determine the possible digits for the last digit since the number must be odd. From the given digits (2, 4, 6, 7, 8, 9), the odd digits are 7 and 9, so the last digit can be either 7 or 9.
For the first digit, we have 2 choices (2 or 4). For the last digit, we have 2 choices (7 or 9).
For the second and third digits, since there are no restrictions mentioned, each can be any of the 6 given digits (2, 4, 6, 7, 8, 9), so each has 6 choices.
Calculate the total number of such four-digit odd numbers by multiplying the number of choices for each digit: number of choices for the first digit × number of choices for the second digit × number of choices for the third digit × number of choices for the last digit.
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Place Value and Number Formation
Understanding place value is essential to form four-digit numbers. The first digit determines the thousands place and restricts the number's size (less than 6000 means the first digit must be 2, 4, or 5 if available). Each digit's position affects the overall number's value and constraints.
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4:47
The Number e
Odd Number Criteria
An odd number ends with an odd digit. To form odd numbers, the units (last) digit must be chosen from the set of odd digits available. Identifying which digits are odd (in this case, 7 and 9) is crucial to ensure the number formed is odd.
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4:47The Number e
Counting Principles and Permutations
Counting the total number of valid numbers involves applying the multiplication principle. For each digit position, count the number of possible choices considering the restrictions, then multiply these counts to find the total number of combinations.
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4:04Fundamental Counting Principle
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Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3(n-1) to find the seventh term of the sequence 11, 33, 99, 297,...
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