https://file.aiquickdraw.com/mathbot/file/1726933357927cwpycml3.jpg

To find the slope of the secant line between \( x = -1 \) and \( x = 2 \) on the graph of the function \( f(x) = -3x^3 + 4x^2 + 2 \), we use the formula for the slope of the secant line:

\[
\text{slope of secant line} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Here, \( x_1 = -1 \) and \( x_2 = 2 \).

### Step 1: Calculate \( f(-1) \)
Substitute \( x = -1 \) into the function:
\[
f(-1) = -3(-1)^3 + 4(-1)^2 + 2 = -3(-1) + 4(1) + 2 = 3 + 4 + 2 = 9
\]

### Step 2: Calculate \( f(2) \)
Substitute \( x = 2 \) into the function:
\[
f(2) = -3(2)^3 + 4(2)^2 + 2 = -3(8) + 4(4) + 2 = -24 + 16 + 2 = -6
\]

### Step 3: Calculate the slope
Now that we have \( f(-1) = 9 \) and \( f(2) = -6 \), we can find the slope:
\[
\text{slope} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-6 - 9}{2 + 1} = \frac{-15}{3} = -5
\]

So, the slope of the secant line is \( \boxed{-5} \).

Would you like further clarification or more details on this?

Here are 5 related questions:
1. How do you find the slope of a tangent line at a point on a curve?
2. What is the difference between a secant line and a tangent line?
3. Can the slope of a secant line ever equal the slope of a tangent line?
4. How would you interpret a negative slope in the context of a real-world problem?
5. What does the slope of a secant line tell you about the behavior of the function between two points?

**Tip:** The slope of the secant line gives you the average rate of change of the function between two points, while the tangent line slope gives the instantaneous rate of change at a specific point.

Find the slope of the secant line between x = -1 and x = 2 on the graph of the function f(x) = -3x^3 + 4x^2 + 2.

Learn how to calculate the slope of a secant line between x = -1 and x = 2 for the function f(x) = -3x^3 + 4x^2 + 2. This solution provides a step-by-step approach, applying the formula for the slope of a secant line.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation