What is the equation of the normal line of f(x)=3x2+2 at x=2?

f′(x)=ddx((3x2+2x)12)=12(3x2+2x)−12(6x+2)The tangent line should pass through the point (2, f(2)), and have a gradient of−1f′(2)f(2)=4f′(2)=74Equation of tangent line:y−f(2)x−2=−1f′(2)y−4x−2=−47y=−47x+367

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High School
Calculus and Analysis
Calculus 1
Derivatives

Izabelle Walsh

Answered question

2022-04-07

What is the equation of the normal line of

f (x) = \sqrt{3 x^{2} + 2}

at x=2?

Answer & Explanation

unduncjineei5r3

Beginner2022-04-08Added 19 answers

f^{'} (x) = \frac{d}{d x} ({(3 x^{2} + 2 x)}^{\frac{1}{2}})

= \frac{1}{2} {(3 x^{2} + 2 x)}^{- \frac{1}{2}} (6 x + 2)

The tangent line should pass through the point (2, f(2)), and have a gradient of

- \frac{1}{f^{'} (2)}

f(2)=4

f^{'} (2) = \frac{7}{4}

Equation of tangent line:

\frac{y - f (2)}{x - 2} = - \frac{1}{f^{'} (2)}

\frac{y - 4}{x - 2} = - \frac{4}{7}

y = - \frac{4}{7} x + \frac{36}{7}

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