What is the equation of the normal line of f(x)=tan⁡x at x=π6?

Explanation:The point which the tangent will intersect is (π6,33).The slope of the tangent line is f′(π6).f(x)=tan⁡xf′(x)=sec2xf′(π6)=sec2(π6)=1cos2(π6)=1(32)2=134=43Thus, the tangent line passes through the point (π6,33) and has a slope 43.Write in point-slope form:y−33=43(x−π6)In slope-intercept form:y=43x+33−2π9

Answered: "What is the equation of the normal line..."

High School
Calculus and Analysis
Calculus 1
Derivatives

Breanna Mcclure

Answered question

2022-04-08

What is the equation of the normal line of

f (x) = \tan x

x = \frac{π}{6}

Answer & Explanation

Dubliddissefyf8

Beginner2022-04-09Added 10 answers

Explanation:
The point which the tangent will intersect is

(\frac{π}{6}, \frac{\sqrt{3}}{3})

.
The slope of the tangent line is

f^{'} (\frac{π}{6})

f (x) = \tan x

f^{'} (x) = \sec^{2} x

f^{'} (\frac{π}{6}) = \sec^{2} (\frac{π}{6}) = \frac{1}{\cos^{2} (\frac{π}{6})} = \frac{1}{{(\frac{\sqrt{3}}{2})}^{2}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}

Thus, the tangent line passes through the point

(\frac{π}{6}, \frac{\sqrt{3}}{3})

and has a slope

\frac{4}{3}

.
Write in point-slope form:

y - \frac{\sqrt{3}}{3} = \frac{4}{3} (x - \frac{π}{6})

In slope-intercept form:

y = \frac{4}{3} x + \frac{3 \sqrt{3} - 2 π}{9}

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?