https://file.aiquickdraw.com/mathbot/file/1729023194673tbms891g.jpg

Determine the slope of the secant line of f for each of the intervals indicated in the table

We are asked to find the slope of the secant line for the function \( f(x) = 2x^2 - 12x + 10 \) over the interval \([4, x_2]\) for various values of \(x_2\) as indicated in the table. 

The formula for the slope of the secant line between two points \(x_1\) and \(x_2\) on the function \(f(x)\) is given by:

\[
\text{Secant Slope} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

In this case, \(x_1 = 4\) and we will calculate the secant slope for each \(x_2\) given in the table: 4.001, 4.01, 4.1, and 5.

First, let’s calculate \(f(x)\) at the points \(x_1 = 4\) and \(x_2\) values.

1. **Calculate \(f(4)\):**

\[
f(4) = 2(4)^2 - 12(4) + 10 = 2(16) - 48 + 10 = 32 - 48 + 10 = -6
\]

Now, we will calculate \(f(x_2)\) for each \(x_2\) value and then use the secant slope formula:

### 1. For \(x_2 = 4.001\):

\[
f(4.001) = 2(4.001)^2 - 12(4.001) + 10 = 2(16.008001) - 48.012 + 10 = 32.016002 - 48.012 + 10 = -5.995998
\]

Secant slope:
\[
\frac{f(4.001) - f(4)}{4.001 - 4} = \frac{-5.995998 - (-6)}{0.001} = \frac{0.004002}{0.001} = 4.002
\]

### 2. For \(x_2 = 4.01\):

\[
f(4.01) = 2(4.01)^2 - 12(4.01) + 10 = 2(16.0801) - 48.12 + 10 = 32.1602 - 48.12 + 10 = -5.9598
\]

Secant slope:
\[
\frac{f(4.01) - f(4)}{4.01 - 4} = \frac{-5.9598 - (-6)}{0.01} = \frac{0.0402}{0.01} = 4.02
\]

### 3. For \(x_2 = 4.1\):

\[
f(4.1) = 2(4.1)^2 - 12(4.1) + 10 = 2(16.81) - 49.2 + 10 = 33.62 - 49.2 + 10 = -5.58
\]

Secant slope:
\[
\frac{f(4.1) - f(4)}{4.1 - 4} = \frac{-5.58 - (-6)}{0.1} = \frac{0.42}{0.1} = 4.2
\]

### 4. For \(x_2 = 5\):

\[
f(5) = 2(5)^2 - 12(5) + 10 = 2(25) - 60 + 10 = 50 - 60 + 10 = 0
\]

Secant slope:
\[
\frac{f(5) - f(4)}{5 - 4} = \frac{0 - (-6)}{1} = \frac{6}{1} = 6
\]

### Table of Secant Slopes:

| \(x_2\)   | Secant Slope |
|-----------|--------------|
| 4.001     | 4.002        |
| 4.01      | 4.02         |
| 4.1       | 4.2          |
| 5         | 6            |

Do you have any questions or need more details?

Here are some relative questions for practice:
1. How would you compute the secant slope if the function was \(f(x) = 3x^2 + 5x - 7\)?
2. What happens to the secant slope as \(x_2\) approaches \(x_1\)?
3. Can you calculate the secant slope if \(x_1 = 3\) and \(x_2 = 6\) for the function \(f(x) = x^2 - x\)?
4. What is the relationship between secant slope and tangent slope?
5. How does the secant line change as \(x_2\) becomes further from \(x_1\)?

**Tip**: The secant line slope approaches the tangent slope as the interval between the two points gets smaller.

Determine the slope of the secant line of f for each of the intervals indicated in the table for the function f(x) = 2x^2 - 12x + 10.

Learn how to calculate the slope of the secant line for the quadratic function f(x) = 2x^2 - 12x + 10 over various intervals. This problem involves key algebraic concepts such as quadratic functions and secant slopes, using the difference quotient formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation