the linear approximation at $x = 0$ to $\sin(3x)$ is $a + bx$ where $a$ is:

and where $b$ is:

Question

the linear approximation at $x = 0$ to $\sin(3x)$ is $a + bx$ where $a$ is:

and where $b$ is:

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Linear Approximation" ], "new_concepts": [], "current_concepts": [ "Linear Approximation", "Trigonometric Derivatives" ] } ## Identify the function and center point $$ f(x) = \sin(3x), \quad a = 0 $$ ## Evaluate the function and its derivative at the center $$ \begin{aligned} f(0) &= \sin(0) = 0 \ f'(x) &= 3\cos(3x) \ f'(0) &= 3\cos(0) = 3 \end{aligned} $$ ## Construct the linear approximation formula $$ \begin{aligned} L(x) &= f(a) + f'(a)(x - a) \ L(x) &= 0 + 3(x - 0) = 3x \ L(x) &= A + Bx \implies A = 0, \quad B = 3 \end{aligned} $$ The linear approximation at $x = 0$ to $\sin (3x)$ is $A + Bx$ where $A$ is: 0 and where $B$ is: 3 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Linear Approximation" ] }

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Question

Explanation:

Identify the function and center point

Evaluate the function and its derivative at the center

Construct the linear approximation formula

Answer: