For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4.
$ƒ (x) = 1 + 2 x^{2}$

검증된 단계별 안내

Start by writing down the given function: $f(x) = 1 + 2x^2$.

To find $f(x+h)$, substitute every $x$ in the function with $(x+h)$, so write $f(x+h) = 1 + 2(x+h)^2$.

Next, expand the squared term $(x+h)^2$ using the formula $(a+b)^2 = a^2 + 2ab + b^2$, which gives $x^2 + 2xh + h^2$. Substitute this back into $f(x+h)$.

Calculate $f(x+h) - f(x)$ by subtracting the original function $f(x) = 1 + 2x^2$ from the expression you found for $f(x+h)$.

Finally, find the difference quotient by dividing the expression $f(x+h) - f(x)$ by $h$, which is $rac{f(x+h) - f(x)}{h}$. Simplify this expression as much as possible.

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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 ƒ(x), represents a rule that assigns each input x to an output. Evaluating ƒ(x+h) means substituting x+h into the function in place of x, which helps analyze how the function behaves when its input changes by h.

Difference of Function Values

The expression ƒ(x+h) - ƒ(x) calculates the change in the function's output as the input changes from x to x+h. This difference is fundamental in understanding how the function varies over an interval and is a stepping stone toward concepts like average rate of change.

Difference Quotient

The difference quotient, [ƒ(x+h) - ƒ(x)] / h, measures the average rate of change of the function over the interval from x to x+h. It is a key concept in calculus, representing the slope of the secant line, and is used to approximate derivatives.

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4.ƒ(x)=1+2x2ƒ(x)=1+2x^2