The linear approximation at $$ x=0 $$ to $$ \sin (6 x) $$ is $$ A+B x $$ where $$ A $$ is: and where $$ B $$ is:

Question

UpStudy AI · Accepted Answer

**Step 1:**  
The linear approximation of a function $$ f(x) $$ at $$ x=0 $$ is given by  
$$
f(x) \approx f(0) + f'(0)x
$$

**Step 2:**  
Let $$ f(x) = \sin(6x) $$. Compute $$ f(0) $$:
$$
f(0) = \sin(6 \cdot 0) = \sin(0) = 0
$$
Thus, $$ A = 0 $$.

**Step 3:**  
Compute the derivative $$ f'(x) $$. Using the chain rule:
$$
f'(x) = \cos(6x) \cdot 6 = 6\cos(6x)
$$

**Step 4:**  
Evaluate $$ f'(0) $$:
$$
f'(0) = 6\cos(6 \cdot 0) = 6\cos(0) = 6 \cdot 1 = 6
$$
Thus, $$ B = 6 $$.

**Final Answer:**  
$$ A = 0 $$ and $$ B = 6 $$.
$$ A = 0 $$ and $$ B = 6 $$.

The linear approximation at \( x=0 \) to \( \sin (6 x) \) is \( A+B x \) where \( A \) is: and where \( B \) is: