Textbook Question
Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58. If {an} is a finite sequence whose last term is -83, how many terms does {an} contain?
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 9, Problem 53
Find f(x + h) − f(x)/h and simplify. f(x) = x4+7
Verified step by step guidance1
Start with the given function: \(f(x) = x^4 + 7\).
Find the expression for \(f(x + h)\) by substituting \(x + h\) into the function: \(f(x + h) = (x + h)^4 + 7\).
Expand the binomial \((x + h)^4\) using the binomial theorem or by repeated multiplication.
Form the difference quotient by subtracting \(f(x)\) from \(f(x + h)\) and then dividing by \(h\): \(\frac{f(x + h) - f(x)}{h} = \frac{(x + h)^4 + 7 - (x^4 + 7)}{h}\).
Simplify the numerator by canceling out like terms and then simplify the entire expression by factoring and reducing where possible.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Notation and Evaluation
Understanding function notation f(x) is essential to evaluate expressions like f(x + h). This involves substituting the input variable x with (x + h) in the function's formula and simplifying the resulting expression.
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Difference Quotient
The difference quotient, given by (f(x + h) - f(x)) / h, measures the average rate of change of the function over the interval from x to x + h. It is foundational for understanding derivatives and requires careful algebraic manipulation.
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Polynomial Expansion and Simplification
Expanding polynomials like (x + h)^4 using binomial expansion or other methods is necessary to simplify the difference quotient. Combining like terms and factoring where possible helps to reduce the expression to its simplest form.
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05:13Introduction to Polynomials
Related Practice
Textbook Question
If you toss a fair coin six times, what is the probability of getting all heads?
Textbook Question
Use the formula for nCr to solve Exercises 49–56. To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?
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Textbook Question
Use the formula for nCr to solve Exercises 49–56. You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?
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Textbook Question
In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+3+5+⋯+ (2n−1)
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Textbook Question
In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = 2n