Retaining the Concepts. If f(x) = 4x2 - 5x - 2, find [f(x + h) - f(x)]/h, h ≠ 0

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 96

9장, 문제 96

Retaining the Concepts. If f(x) = 4x² - 5x - 2, find [f(x + h) - f(x)]/h, h ≠ 0

검증된 단계별 안내

Start by writing the given function:

f (x) = 4x x^{2} - 5x - 2

Find the expression for

f(x + h)

by substituting

x + h

into the function:

f(x + h) = 4(x + h)^2 - 5(x + h) - 2

Expand the squared term and distribute the constants: expand

(x + h)^2

x^2 + 2xh + h^2

, then multiply by 4, and distribute -5 over

(x + h)

Write the difference quotient expression:

\(\frac{f(x + h) - f(x)}{h}\)

, substituting the expanded form of

f(x + h)

and the original

f(x)

Simplify the numerator by combining like terms and then factor out

h

from the numerator to cancel with the denominator, remembering that

h \(\neq\) 0

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Notation and Evaluation

Function notation, such as f(x), represents a rule that assigns each input x to an output. Evaluating f(x + h) means substituting x + h into the function in place of x, which is essential for understanding how the function changes with small increments.

Difference Quotient

The difference quotient [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 in calculus as it approximates the derivative and helps analyze how functions behave locally.

Polynomial Simplification

Simplifying expressions involving polynomials requires expanding terms, combining like terms, and factoring when possible. Mastery of these algebraic manipulations is necessary to simplify the difference quotient and express it in a reduced form.