We need to find and simplify the expression

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

for the function $f(x) = 2x^2 - 2x + 7$, where $h \neq 0$.

Step 1: Compute $f(a + h)$

Substitute $x = a + h$ into the function $f(x) = 2x^2 - 2x + 7$: $f(a + h) = 2(a + h)^2 - 2(a + h) + 7$ Expand $(a + h)^2$: $(a + h)^2 = a^2 + 2ah + h^2$ Now substitute back: $f(a + h) = 2(a^2 + 2ah + h^2) - 2(a + h) + 7$ Distribute the terms: $f(a + h) = 2a^2 + 4ah + 2h^2 - 2a - 2h + 7$

Step 2: Compute $f(a)$

Substitute $x = a$ into the function: $f(a) = 2a^2 - 2a + 7$

Step 3: Subtract $f(a)$ from $f(a + h)$

Now subtract $f(a)$ from $f(a + h)$: $f(a + h) - f(a) = (2a^2 + 4ah + 2h^2 - 2a - 2h + 7) - (2a^2 - 2a + 7)$ Simplify by canceling like terms: $f(a + h) - f(a) = 4ah + 2h^2 - 2h$

Step 4: Divide by $h$

Now divide by $h$: $\frac{f(a + h) - f(a)}{h} = \frac{4ah + 2h^2 - 2h}{h}$ Factor out $h$ from the numerator: $\frac{f(a + h) - f(a)}{h} = \frac{h(4a + 2h - 2)}{h}$ Since $h \neq 0$, we can cancel out $h$: $\frac{f(a + h) - f(a)}{h} = 4a + 2h - 2$

Final Answer:

The simplified expression is:

$\frac{f(a + h) - f(a)}{h} = 4a + 2h - 2$

Would you like further details or have any questions about the process?

Here are 5 questions related to the topic:

1. What is the significance of the expression $\frac{f(a + h) - f(a)}{h}$ in calculus?
2. How would the answer change if $f(x)$ had an additional linear term?
3. Can you explain the role of the limit as $h$ approaches 0 in finding the derivative?
4. How do we expand binomials like $(a + h)^2$?
5. What would the expression look like if $f(x)$ was a cubic function?

Tip: In calculus, the expression $\frac{f(a + h) - f(a)}{h}$ is key to understanding the concept of derivatives. It represents the average rate of change of the function over the interval from $a$ to $a + h$.