Let
f(x) =

the square root of x − 3
.
Find the following and simplify. (Simplify your answers completely.)
(a)
f(x + h) =

(b)
f(x + h) − f(x) =

(c)
f(x + h) − f(x)
h
=

(d)
f '(x) =

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!

Let f(x) = the square root of x − 3. Find the following and simplify. (Simplify your answers completely.) (a) f(x + h) = (b) f(x + h) − f(x) = (c) f(x + h) − f(x) / h = (d) f '(x) =

Explore the square root function f(x) = √(x - 3). Learn how to simplify expressions, find the difference quotient, and calculate the derivative using the chain rule. Suitable for high school students in Grades 11-12.

Math Problem Statement

(c) f(x + h) − f(x) h

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation