The difference quotient is used to approximate the derivative of a function $f(x)$. It is given by the formula:

$\frac{f(x+h) - f(x)}{h}$

Step 1: Find $f(x+h)$

Given $f(x) = x^2 - x + 4$, we substitute $x+h$ into the function:

$f(x+h) = (x+h)^2 - (x+h) + 4$

Expanding each term:

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

Thus,

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

Step 2: Compute the Difference Quotient

Now, plug $f(x+h)$ and $f(x)$ into the difference quotient formula:

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

Distribute and simplify the numerator:

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

Step 3: Factor and Simplify

Now, factor out an $h$ from the numerator:

$\frac{2xh + h^2 - h}{h} = \frac{h(2x + h - 1)}{h}$

Cancel the $h$ from the numerator and denominator:

$= 2x + h - 1$

Final Answer:

The simplified difference quotient is:

$2x + h - 1$

Would you like a detailed explanation of any specific step?

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Here are five related questions to expand your understanding:

1. What happens to the difference quotient as $h$ approaches zero?
2. How would you find the derivative of $f(x) = x^2 - x + 4$ directly using differentiation rules?
3. Can you apply the difference quotient to a trigonometric function like $f(x) = \sin(x)$?
4. How does the difference quotient relate to the concept of a secant line on the graph of $f(x)$?
5. What role does the difference quotient play in understanding the slope of a curve?

Tip: The difference quotient is a fundamental concept for understanding how the derivative measures the rate of change of a function at a point.