1. Find and simplify the difference quotient for for $$ f(x)=\frac{-5 x^{2}}{2}-3 x $$. Show all work.

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UpStudy AI · Accepted Answer

**Step 1. Write the definition of the difference quotient**

The difference quotient is given by  
$$
\frac{f(x+h)-f(x)}{h}.
$$

**Step 2. Compute $$f(x+h)$$**

Given 
$$
f(x)=\frac{-5x^{2}}{2}-3x,
$$
we substitute $$x+h$$ for $$x$$ to get
$$
f(x+h)=\frac{-5(x+h)^{2}}{2}-3(x+h).
$$

**Step 3. Expand $$f(x+h)$$**

First, expand $$(x+h)^2$$:
$$
(x+h)^2=x^2+2xh+h^2.
$$
Thus,
$$
f(x+h)=\frac{-5(x^2+2xh+h^2)}{2}-3x-3h.
$$
Distributing the fraction gives
$$
f(x+h)=\frac{-5x^2}{2}-5xh-\frac{5h^2}{2}-3x-3h.
$$

**Step 4. Form the difference $$f(x+h)-f(x)$$**

Recall that
$$
f(x)=\frac{-5x^{2}}{2}-3x.
$$
Subtract $$f(x)$$ from $$f(x+h)$$:
$$
f(x+h)-f(x)=\left(\frac{-5x^2}{2}-5xh-\frac{5h^2}{2}-3x-3h\right)-\left(\frac{-5x^2}{2}-3x\right).
$$
Distribute the minus sign:
$$
f(x+h)-f(x)=\frac{-5x^2}{2}-5xh-\frac{5h^2}{2}-3x-3h+\frac{5x^2}{2}+3x.
$$
Notice that $$\frac{-5x^2}{2}+\frac{5x^2}{2}=0$$ and $$-3x+3x=0$$. Thus, we have
$$
f(x+h)-f(x)=-5xh-\frac{5h^2}{2}-3h.
$$

**Step 5. Factor and simplify the expression**

Factor out the common factor $$h$$:
$$
f(x+h)-f(x)=h\left(-5x-\frac{5h}{2}-3\right).
$$
Now, divide by $$h$$ to obtain the difference quotient:
$$
\frac{f(x+h)-f(x)}{h}=-5x-\frac{5h}{2}-3.
$$

**Final Answer**

$$
\boxed{-5x-\frac{5h}{2}-3}
$$
The difference quotient for $$ f(x) = \frac{-5x^{2}}{2} - 3x $$ is $$ -5x - \frac{5h}{2} - 3 $$.

1. Find and simplify the difference quotient for for \( f(x)=\frac{-5 x^{2}}{2}-3 x \). Show all work.

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