The difference quotient of a function $f(x)$ is given by the formula:

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

For the function $f(x) = 2x^2 - 5x$, let's compute the difference quotient step-by-step:

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

$f(x + h) = 2(x + h)^2 - 5(x + h)$ Expanding this, we have: $= 2(x^2 + 2xh + h^2) - 5(x + h)$ $= 2x^2 + 4xh + 2h^2 - 5x - 5h$

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

$f(x + h) - f(x) = (2x^2 + 4xh + 2h^2 - 5x - 5h) - (2x^2 - 5x)$ $= 4xh + 2h^2 - 5h$

Step 3: Divide by $h$

$\frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 - 5h}{h}$ $= 4x + 2h - 5$

Final Answer

The difference quotient is: $4x + 2h - 5$

Would you like more details or have any questions?

5 Related Questions

1. What is the geometric interpretation of the difference quotient?
2. How does the difference quotient lead to the derivative as $h \to 0$?
3. How would the difference quotient change if $f(x) = 3x^3 - 4x + 1$?
4. What is the significance of the term $2h$ in the difference quotient result?
5. How would you apply the difference quotient to linear functions?

Tip

The difference quotient represents the average rate of change of the function over the interval from $x$ to $x + h$. As $h$ approaches zero, it becomes the instantaneous rate of change, which is the derivative.