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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.

Given the function f(x) = 1 + 5x^2, calculate the following values:
1. f(a)
2. f(a+h)
3. (f(a+h) - f(a)) / h

Learn how to calculate the difference quotient for the function f(x) = 1 + 5x^2. This involves finding f(a), f(a+h), and simplifying the difference quotient (f(a+h) - f(a)) / h. Suitable for precalculus and calculus students.

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