Find and simplify the difference quotient for the following function.
​f(x)equals=2xsquared+6xplus3

To find and simplify the difference quotient for the function \( f(x) = 2x^2 + 6x + 3 \), we will follow these steps:

### Step 1: Write the formula for the difference quotient
The difference quotient is given by:

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

### Step 2: Compute \( f(x+h) \)
Substitute \( x + h \) into the function \( f(x) \):

\[
f(x+h) = 2(x+h)^2 + 6(x+h) + 3
\]

Expanding this:

\[
f(x+h) = 2(x^2 + 2xh + h^2) + 6x + 6h + 3
\]

Simplify further:

\[
f(x+h) = 2x^2 + 4xh + 2h^2 + 6x + 6h + 3
\]

### Step 3: Compute \( f(x+h) - f(x) \)
Subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 + 6x + 6h + 3) - (2x^2 + 6x + 3)
\]

Simplify by canceling out the like terms:

\[
f(x+h) - f(x) = 4xh + 2h^2 + 6h
\]

### Step 4: Divide by \( h \)
Now, divide the result by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 + 6h}{h}
\]

Simplify by factoring out \( h \) from the numerator:

\[
\frac{f(x+h) - f(x)}{h} = 4x + 2h + 6
\]

### Final Answer
The simplified form of the difference quotient is:

\[
4x + 2h + 6
\]

This is the simplified expression for the difference quotient of the given function.

Would you like more details or have any questions?

Here are five related questions:
1. What does the difference quotient represent geometrically?
2. How would the difference quotient change if the function were cubic instead of quadratic?
3. What happens to the difference quotient as \( h \) approaches zero?
4. Can you apply the difference quotient to a trigonometric function like \( \sin(x) \)?
5. How does the difference quotient relate to the derivative of a function?

**Tip:** The difference quotient is a fundamental concept in calculus, as it provides the basis for defining the derivative, which measures the rate of change of a function.

Find and simplify the difference quotient for the following function: f(x) = 2x^2 + 6x + 3.

Learn how to find and simplify the difference quotient for the quadratic function f(x) = 2x^2 + 6x + 3. This problem covers key calculus concepts, including the difference quotient formula and expanding quadratic expressions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation