Let f*f* be a differentiable function such that f, of, minus, 4, equals, minus, 5*f*(−4)=−5 and f, prime, of, minus, 4, equals, 3, .*f*′(−4)=3. What is the approximation for f, of, minus, 4, point, 1*f*(−4.1) found by using the line tangent to the graph of f*f* at x, equals, minus, 4, question mark*x*=−4?

To approximate \( f(-4.1) \) using the tangent line to the graph of \( f \) at \( x = -4 \), we use the linear approximation formula based on the point-slope form of the tangent line. Since \( f \) is differentiable at \( x = -4 \), the tangent line at this point can approximate values of \( f \) near \( x = -4 \).

### Step 1: Write the equation of the tangent line
The equation of the tangent line at \( x = -4 \) is:
\[
y = f(-4) + f'(-4)(x + 4)
\]

From the problem statement:
- \( f(-4) = -5 \)
- \( f'(-4) = 3 \)

Thus, the equation of the tangent line becomes:
\[
y = -5 + 3(x + 4)
\]

### Step 2: Substitute \( x = -4.1 \) into the tangent line equation
Now, we substitute \( x = -4.1 \) to approximate \( f(-4.1) \):
\[
f(-4.1) \approx -5 + 3(-4.1 + 4)
\]

Calculating inside the parentheses:
\[
-4.1 + 4 = -0.1
\]

So:
\[
f(-4.1) \approx -5 + 3(-0.1)
\]

Then:
\[
f(-4.1) \approx -5 - 0.3 = -5.3
\]

### Conclusion
The approximation for \( f(-4.1) \) using the tangent line is:
\[
f(-4.1) \approx -5.3
\]

Would you like more details or have any questions?

---

Here are some follow-up questions you might find helpful:

1. How does linear approximation relate to differentiability?
2. Why does the tangent line provide a good approximation near \( x = -4 \)?
3. What would change if \( f'(-4) \) were negative instead of positive?
4. How do we improve this approximation using higher-order terms?
5. How can we use this approximation method to estimate values at \( x \) farther from -4?

**Tip:** The linear approximation is more accurate for values close to the point of tangency.

Let f be a differentiable function such that f(−4) = −5 and f′(−4) = 3. What is the approximation for f(−4.1) found by using the line tangent to the graph of f at x = −4?

Find the approximation of f(−4.1) by using the tangent line at x = −4 for a differentiable function f, where f(−4) = −5 and f′(−4) = 3. This problem covers linear approximation and tangent line concepts, suitable for 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