https://file.aiquickdraw.com/mathbot/file/1725934139485i12yawpx.jpg

To solve this problem, we need to use the equation of the tangent line to approximate the value of \( f(-2.85) \).

### Step 1: Find the slope of the tangent line
The slope of the tangent line at \( x = -3 \) is given by the derivative of \( f(x) \) evaluated at \( x = -3 \). The derivative is given as:
\[
f'(x) = -5x - 6
\]
So, at \( x = -3 \):
\[
f'(-3) = -5(-3) - 6 = 15 - 6 = 9
\]
The slope of the tangent line at \( x = -3 \) is 9.

### Step 2: Write the equation of the tangent line
The tangent line at \( x = -3 \) can be written as:
\[
y = f(-3) + f'(-3)(x + 3)
\]
Given \( f(-3) = 0 \) and \( f'(-3) = 9 \), the equation becomes:
\[
y = 0 + 9(x + 3) = 9(x + 3)
\]
Simplifying:
\[
y = 9x + 27
\]

### Step 3: Approximate \( f(-2.85) \)
Substitute \( x = -2.85 \) into the equation of the tangent line:
\[
f(-2.85) \approx 9(-2.85) + 27
\]
\[
f(-2.85) \approx -25.65 + 27 = 1.35
\]

### Final Answer:
The approximation for \( f(-2.85) \) using the tangent line at \( x = -3 \) is **1.35**.

Would you like further details or have any questions? Here are 5 related questions that build on this concept:

1. How would the approximation change if we used a different point close to \( x = -3 \)?
2. What does the tangent line tell us about the behavior of the function \( f(x) \) near \( x = -3 \)?
3. How could you use higher derivatives to get a better approximation of \( f(-2.85) \)?
4. What is the geometric significance of the slope of the tangent line?
5. How does the approximation method used here compare with using a Taylor series?

**Tip:** The closer the point of approximation is to the point where the tangent line is calculated, the more accurate the approximation will be.

Let f be a differentiable function such that f(-3) = 0 and f'(x) = -5x - 6. What is the approximation for f(-2.85) found by using the line tangent to the graph of f at x = -3?

This problem involves finding an approximation for f(-2.85) using the tangent line to a differentiable function at x = -3. The solution explores key concepts such as differentiation and linear approximation, making it suitable for high school calculus students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation