Find f(x + h) − f(x)/h and simplify. f(x) = x4+7

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 53

9장, 문제 53

Find f(x + h) − f(x)/h and simplify. f(x) = x⁴+7

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

15m

주요 개념

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

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.

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.

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.