Question

suppose we want to approximate \\(4 \sin (0.05)\\).

first find the linear approximation to \\(f(x) = 4 \sin x\\) at \\(x = 0\\).
\\(l(x) =\\)

use \\(l(x)\\) to approximate \\(4 \sin (0.05)\\).
\\(4 \sin (0.05) \approx\\)

compute the actual value of \\(f(0.05)\\). what is the error between the function value and the approximation?
answer as a positive value only.
\\(|\text{error}| =\\) (approximate to at least 7 decimal places.)

Explanation:

Find the linear approximation function

Using the Linear Approximation knowledge point

$$
LATEXBLOCK0
$$

Approximate the function value

Using the Linear Approximation knowledge point

$$
LATEXBLOCK1
$$

Calculate the absolute error

Using the Approximation Error knowledge point

$$
LATEXBLOCK2
$$

Answer:

Question 1

Suppose we want to approximate \(4 \sin (0.05)\).

First find the linear approximation to \(f(x) = 4 \sin x\) at \(x = 0\).
\(L(x) =\) <blank>\(4x\)</blank>

Question 2

Use \(L(x)\) to approximate \(4 \sin (0.05)\).
\(4 \sin (0.05) \approx\) <blank>\(0.2\)</blank>

Question 3

Compute the actual value of \(f(0.05)\). What is the error between the function value and the approximation?
Answer as a positive value only.
\(|\text{error}| =\) <blank>\(0.0000833\)</blank> (Approximate to at least 7 decimal places.)