Let 
\[f\] be a differentiable function with 
\[f(2)=-3\] and 
\[f'(2)=-4\].
What is the value of the approximation of 
\[f(1.9)\] using the function's local linear approximation at 
\[x=2\]?
Choose 1 answer:
Choose 1 answer:
(Choice A)   
\[-2.9\]
A
\[-2.9\]
(Choice B)   
\[-2.8\]
B
\[-2.8\]
(Choice C)   
\[-2.7\]
C
\[-2.7\]
(Choice D)   
\[-2.6\]
D
\[-2.6\]

To approximate \( f(1.9) \) using the function's local linear approximation at \( x = 2 \), we use the formula for the linear approximation:

\[
L(x) = f(a) + f'(a)(x - a)
\]

where:
- \( a = 2 \)
- \( f(2) = -3 \)
- \( f'(2) = -4 \)

We want to find \( L(1.9) \). Plugging in the values:

\[
L(1.9) = f(2) + f'(2)(1.9 - 2)
\]

Now substitute the given values:

\[
L(1.9) = -3 + (-4)(1.9 - 2)
\]

Calculate the difference \(1.9 - 2\):

\[
1.9 - 2 = -0.1
\]

Then multiply by \(f'(2) = -4\):

\[
-4 \times (-0.1) = 0.4
\]

Now, add this result to \(f(2) = -3\):

\[
L(1.9) = -3 + 0.4 = -2.6
\]

Thus, the value of the approximation of \(f(1.9)\) is:

\[
\boxed{-2.6}
\]

The correct choice is:

(Choice D) \(-2.6\)

Would you like more details on this calculation or have any questions?

Here are 5 related questions that might help further your understanding:

1. What is the significance of using a linear approximation for estimating function values?
2. How would you use the same method to approximate \(f(2.1)\)?
3. What does the derivative \(f'(2) = -4\) tell us about the function \(f(x)\) at \(x = 2\)?
4. How would the linear approximation change if \(f'(2)\) were positive instead of negative?
5. What is the error involved in using a linear approximation versus using the actual function?

**Tip:** When using linear approximations, remember that they are more accurate when \(x\) is close to the point \(a\) where the function is being approximated.

Learn how to approximate f(1.9) using the local linear approximation at x = 2. Understand the differentiation concept and the formula for linear approximation. Suitable for advanced high school students and anyone interested in calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation