For the function $$ f(x)=5 x^{2}+5 x-5 $$, evaluate and fully simplify each of the following. $$ f(x+h)=\square $$ $$ \frac{f(x+h)-f(x)}{h}=\square $$

Question

UpStudy AI · Accepted Answer

Let's evaluate and simplify each part step by step for the function $$ f(x) = 5x^2 + 5x - 5 $$.

### 1. Evaluating $$ f(x + h) $$:

$$
\begin{align*}
f(x + h) &= 5(x + h)^2 + 5(x + h) - 5 \
&= 5(x^2 + 2xh + h^2) + 5x + 5h - 5 \
&= 5x^2 + 10xh + 5h^2 + 5x + 5h - 5
\end{align*}
$$

So,

$$
f(x + h) = 5x^2 + 10xh + 5h^2 + 5x + 5h - 5
$$

### 2. Evaluating $$ \frac{f(x + h) - f(x)}{h} $$:

First, find $$ f(x + h) - f(x) $$:

$$
\begin{align*}
f(x + h) - f(x) &= (5x^2 + 10xh + 5h^2 + 5x + 5h - 5) - (5x^2 + 5x - 5) \
&= 10xh + 5h^2 + 5h
\end{align*}
$$

Now, divide by $$ h $$:

$$
\frac{f(x + h) - f(x)}{h} = \frac{10xh + 5h^2 + 5h}{h} = 10x + 5h + 5
$$

### Final Answers:

$$
\boxed{
\begin{aligned}
f(x + h) &= 5x^2 + 10x h + 5h^2 + 5x + 5h - 5 \
\frac{f(x + h) - f(x)}{h} &= 10x + 5h + 5
\end{aligned}
}
$$
$$ f(x + h) = 5x^2 + 10xh + 5h^2 + 5x + 5h - 5 $$ $$ \frac{f(x + h) - f(x)}{h} = 10x + 5h + 5 $$

For the function , evaluate and fully simplify each of the following.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Calculus Questions