We are tasked with determining the expression $\frac{g(x+h) - g(x)}{h}$, where $g(x)$ is given by:

$g(x) = -2x^2 + x + 6$

This expression is a difference quotient, which is used to find the slope of the secant line through two points on the graph of the function $g(x)$, and is closely related to the derivative of $g(x)$. Let's go through the steps to find the complete solution.

Step 1: Calculate $g(x + h)$

Substitute $x + h$ into the function $g(x)$ to find $g(x + h)$.

$g(x + h) = -2(x + h)^2 + (x + h) + 6$

First, expand $(x + h)^2$:

$(x + h)^2 = x^2 + 2xh + h^2$

Now, substitute this into the equation for $g(x + h)$:

$g(x + h) = -2(x^2 + 2xh + h^2) + (x + h) + 6$

Distribute the $-2$:

$g(x + h) = -2x^2 - 4xh - 2h^2 + x + h + 6$

So,

$g(x + h) = -2x^2 - 4xh - 2h^2 + x + h + 6$

Step 2: Compute $g(x + h) - g(x)$

Now, subtract $g(x)$ from $g(x + h)$. Recall that $g(x) = -2x^2 + x + 6$, so:

$g(x + h) - g(x) = \left(-2x^2 - 4xh - 2h^2 + x + h + 6\right) - \left(-2x^2 + x + 6\right)$

Simplify this expression:

$g(x + h) - g(x) = -2x^2 - 4xh - 2h^2 + x + h + 6 + 2x^2 - x - 6$

Now cancel out the like terms:

• The $-2x^2$ and $2x^2$ cancel.
• The $x$ and $-x$ cancel.
• The $6$ and $-6$ cancel.

We are left with:

$g(x + h) - g(x) = -4xh - 2h^2 + h$

Step 3: Divide by $h$

Now, divide the entire expression by $h$:

$\frac{g(x + h) - g(x)}{h} = \frac{-4xh - 2h^2 + h}{h}$

Simplify each term:

$\frac{g(x + h) - g(x)}{h} = \frac{-4xh}{h} + \frac{-2h^2}{h} + \frac{h}{h}$

This simplifies to:

$= -4x - 2h + 1$

Final Answer:

$\frac{g(x+h) - g(x)}{h} = -4x - 2h + 1$

This is the simplified form of the difference quotient.

Would you like more details or clarifications on any step?

Related Questions:

1. What is the geometric interpretation of the difference quotient?
2. How does the difference quotient relate to the derivative of $g(x)$?
3. What happens to $\frac{g(x+h) - g(x)}{h}$ as $h \to 0$?
4. Can you find the derivative $g'(x)$ from the result?
5. What is the significance of each term in the expression $-4x - 2h + 1$?

Tip:

The difference quotient becomes the derivative when $h \to 0$. This is a key concept in understanding rates of change and the slope of a function at a point.