Let's solve the questions based on the function provided:

The function is:

$f(x) = x^2 + 2x + 1$

a. $f(a)$

To find $f(a)$, substitute $a$ for $x$ in the function $f(x)$:

$f(a) = a^2 + 2a + 1$

b. $f(a+h)$

To find $f(a+h)$, substitute $a + h$ for $x$ in the function:

$f(a+h) = (a+h)^2 + 2(a+h) + 1$

Now expand this expression:

$(a+h)^2 = a^2 + 2ah + h^2$ $2(a+h) = 2a + 2h$

Thus:

$f(a+h) = a^2 + 2ah + h^2 + 2a + 2h + 1$

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

Now, calculate the difference quotient:

$\frac{f(a+h) - f(a)}{h} = \frac{(a^2 + 2ah + h^2 + 2a + 2h + 1) - (a^2 + 2a + 1)}{h}$

Simplify the expression in the numerator:

$= a^2 + 2ah + h^2 + 2a + 2h + 1 - a^2 - 2a - 1$

After cancellation, we are left with:

$= 2ah + h^2 + 2h$

Thus, the expression becomes:

$\frac{2ah + h^2 + 2h}{h}$

Now, factor out $h$ from the numerator:

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

Cancel the $h$:

$= 2a + h + 2$

So:

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

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore:

1. What is the meaning of the difference quotient in calculus?
2. How would you find the derivative of $f(x) = x^2 + 2x + 1$ using limits?
3. How does the expression $\frac{f(a+h) - f(a)}{h}$ lead to the derivative when $h \to 0$?
4. Can you apply the same method to find the difference quotient for a cubic function?
5. How does the difference quotient represent the slope of the tangent line at a point?

Tip: The difference quotient is an important step toward understanding the derivative of a function, which gives the rate of change at any point.