(c) f(x + h) − f(x) h

Let $f(x) = \sqrt{x - 3}$.

We will solve the following parts:

(a) $f(x + h)$

To find $f(x + h)$, substitute $x + h$ into the function $f(x)$.

$f(x + h) = \sqrt{(x + h) - 3} = \sqrt{x + h - 3}$

(b) $f(x + h) - f(x)$

Now subtract $f(x)$ from $f(x + h)$:

$f(x + h) - f(x) = \sqrt{x + h - 3} - \sqrt{x - 3}$

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

This expression simplifies to:

$\frac{f(x + h) - f(x)}{h} = \frac{\sqrt{x + h - 3} - \sqrt{x - 3}}{h}$

(d) $f'(x)$ (The derivative of $f(x)$)

To find the derivative $f'(x)$, use the chain rule. Let $g(x) = x - 3$, so $f(x) = \sqrt{g(x)}$. The derivative of $f(x) = \sqrt{x - 3}$ is:

$f'(x) = \frac{d}{dx} \left( (x - 3)^{1/2} \right) = \frac{1}{2}(x - 3)^{-1/2} \cdot 1 = \frac{1}{2\sqrt{x - 3}}$

Summary of Results:

• (a) $f(x + h) = \sqrt{x + h - 3}$
• (b) $f(x + h) - f(x) = \sqrt{x + h - 3} - \sqrt{x - 3}$
• (c) $\frac{f(x + h) - f(x)}{h} = \frac{\sqrt{x + h - 3} - \sqrt{x - 3}}{h}$
• (d) $f'(x) = \frac{1}{2\sqrt{x - 3}}$

Do you want further details or have any questions?

Here are 5 related questions:

1. What is the domain of $f(x)$?
2. How does the derivative $f'(x)$ change as $x$ increases?
3. What happens to the value of $f(x)$ as $x \to 3^+$?
4. How can you use the difference quotient formula to approximate the derivative of $f(x)$?
5. What are some real-world applications of square root functions?

Tip: When finding derivatives of square root functions, remember to simplify expressions as much as possible before applying the chain rule!