Let's work through the problem step by step using the given function $f(x) = 1 + 5x^2$.

Step 1: Calculate $f(a)$

We substitute $a$ into the function for $x$:

$f(a) = 1 + 5a^2$

Step 2: Calculate $f(a+h)$

Now we substitute $a+h$ into the function for $x$:

$f(a+h) = 1 + 5(a+h)^2$

Expanding the square:

$f(a+h) = 1 + 5(a^2 + 2ah + h^2) = 1 + 5a^2 + 10ah + 5h^2$

Step 3: Compute the difference quotient $\frac{f(a+h) - f(a)}{h}$

First, subtract $f(a)$ from $f(a+h)$:

$f(a+h) - f(a) = \left(1 + 5a^2 + 10ah + 5h^2\right) - \left(1 + 5a^2\right)$

Simplifying:

$f(a+h) - f(a) = 10ah + 5h^2$

Now, divide this result by $h$:

$\frac{f(a+h) - f(a)}{h} = \frac{10ah + 5h^2}{h}$

Simplifying:

$\frac{f(a+h) - f(a)}{h} = 10a + 5h$

Final Answers:

1. $f(a) = 1 + 5a^2$
2. $f(a+h) = 1 + 5a^2 + 10ah + 5h^2$
3. $\frac{f(a+h) - f(a)}{h} = 10a + 5h$

Would you like further details or any clarifications?

Related Questions:

1. What is the derivative of $f(x)$ with respect to $x$?
2. How does the difference quotient relate to the concept of the derivative?
3. What happens to $\frac{f(a+h) - f(a)}{h}$ as $h \to 0$?
4. Can you apply this method to find the slope of the tangent line at $x = a$?
5. How would the function change if the coefficient of $x^2$ were different?

Tip:

The difference quotient $\frac{f(a+h) - f(a)}{h}$ is a key step in finding the derivative of a function, as it represents the slope of the secant line, which approaches the slope of the tangent line as $h$ approaches 0.