Find the difference quotient of​ f; that​ is, find StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
​, hnot equals​0, for the following function. Be sure to simplify.
​f(x)equalsStartFraction 7 Over x squared EndFraction

To find the **difference quotient** of the function \( f(x) = \frac{7}{x^2} \), we need to compute:

\[
\frac{f(x+h) - f(x)}{h}
\]

### Step 1: Compute \( f(x+h) \)

Since \( f(x) = \frac{7}{x^2} \), replacing \( x \) with \( x + h \) gives us:

\[
f(x+h) = \frac{7}{(x+h)^2}
\]

### Step 2: Set up the difference quotient

Now, substitute \( f(x+h) \) and \( f(x) \) into the difference quotient formula:

\[
\frac{f(x+h) - f(x)}{h} = \frac{\frac{7}{(x+h)^2} - \frac{7}{x^2}}{h}
\]

### Step 3: Simplify the expression

We need to combine the two fractions in the numerator. The least common denominator (LCD) of \( (x+h)^2 \) and \( x^2 \) is \( x^2(x+h)^2 \), so:

\[
\frac{f(x+h) - f(x)}{h} = \frac{7x^2 - 7(x+h)^2}{h \cdot x^2(x+h)^2}
\]

### Step 4: Expand the terms in the numerator

Expand \( (x+h)^2 = x^2 + 2xh + h^2 \), so the expression becomes:

\[
\frac{7x^2 - 7(x^2 + 2xh + h^2)}{h \cdot x^2(x+h)^2}
\]

Simplify the numerator:

\[
7x^2 - 7(x^2 + 2xh + h^2) = 7x^2 - 7x^2 - 14xh - 7h^2 = -14xh - 7h^2
\]

Now the difference quotient becomes:

\[
\frac{-14xh - 7h^2}{h \cdot x^2(x+h)^2}
\]

### Step 5: Factor and simplify

Factor out \( h \) from the numerator:

\[
\frac{h(-14x - 7h)}{h \cdot x^2(x+h)^2}
\]

Cancel \( h \) from the numerator and denominator:

\[
\frac{-14x - 7h}{x^2(x+h)^2}
\]

### Final simplified result:

\[
\frac{-14x - 7h}{x^2(x+h)^2}
\]

This is the simplified form of the difference quotient.

Would you like further explanation or more details on any step? 

Here are some related questions:
1. What is the importance of the difference quotient in calculus?
2. How does the difference quotient relate to the derivative?
3. What happens to the difference quotient as \( h \to 0 \)?
4. Can we compute the derivative of \( f(x) = \frac{7}{x^2} \) using the power rule?
5. How do the algebraic properties of fractions affect the simplification of difference quotients?

**Tip:** The difference quotient is the foundation for understanding the derivative of a function, which measures the rate of change.

Find the difference quotient of f; that is, find (f(x + h) - f(x)) / h, where h ≠ 0, for the function f(x) = 7 / x^2. Be sure to simplify.

Learn how to find the difference quotient for the rational function f(x) = 7 / x^2 with a detailed step-by-step solution. This example covers key concepts from calculus such as the difference quotient, expanding terms, and simplifying rational expressions. Suitable for students in Grades 11-12 or introductory calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation