Domanda del libro di testo
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?
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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 53
Find f(x + h) − f(x)/h and simplify. f(x) = x4+7
Guida verificata passo dopo passo1
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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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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Evaluating Composed Functions
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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3:49Product, Quotient, and Power Rules of Logs
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
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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
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