use linear approximation, i.e. the tangent line, to approximate $\sqrt3{1.2}$ as follows:
let $f(x) = \sqrt3{x}$. the equation of the tangent line to $f(x)$ at $x = 1$ can be written in the form $y = mx + b$
where $m$ is:
and where $b$ is:
using this, we find our approximation for $\sqrt3{1.2}$ is

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Linear Approximation" ], "new_concepts": [], "current_concepts": [ "Linear Approximation", "Tangent Line Slope" ] } ## Find the derivative of the function Using the **Linear Approximation** knowledge point $$ \begin{aligned} f(x) &= x^{1/3}\ f'(x) &= \frac{1}{3}x^{-2/3} \end{aligned} $$ ## Calculate the slope of the tangent line Using the **Linear Approximation** knowledge point $$ \begin{aligned} m &= f'(1) = \frac{1}{3}(1)^{-2/3} = \frac{1}{3} \end{aligned} $$ ## Calculate the y-intercept of the tangent line Using the **Linear Approximation** knowledge point $$ \begin{aligned} f(1) &= 1^{1/3} = 1\ y &= mx + b \implies 1 = \frac{1}{3}(1) + b\ b &= 1 - \frac{1}{3} = \frac{2}{3} \end{aligned} $$ ## Approximate the value of the cube root Using the **Linear Approximation** knowledge point $$ \begin{aligned} L(x) &= \frac{1}{3}x + \frac{2}{3}\ \sqrt[3]{1.2} &\approx L(1.2) = \frac{1}{3}(1.2) + \frac{2}{3} = 0.4 + \frac{2}{3} = \frac{2}{5} + \frac{2}{3} = \frac{16}{15} \end{aligned} $$ Use linear approximation, i.e. the tangent line, to approximate $\sqrt[3]{1.2}$ as follows: Let $f(x) = \sqrt[3]{x}$. The equation of the tangent line to $f(x)$ at $x = 1$ can be written in the form $y = mx + b$ where $m$ is: $\frac{1}{3}$ and where $b$ is: $\frac{2}{3}$ Using this, we find our approximation for $\sqrt[3]{1.2}$ is $\frac{16}{15}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Linear Approximation" ] }

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Question

Explanation:

Find the derivative of the function

Calculate the slope of the tangent line

Calculate the y-intercept of the tangent line

Approximate the value of the cube root

Answer: