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Find the equation of the line tangent to $f(x) = \tan x$ at $x = \frac{\pi}{4}$.

I'm trying to incorporate the slope point formula using $f'(x) = \sec^2 x$ but I'm nowhere near!

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4 Answers 4

up vote 2 down vote accepted

HINT:

We have $$f'\left(\frac\pi4\right)=\frac{y-f\left(\frac\pi4\right)}{x-\frac\pi4}$$

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Yes this is where I am stuck. $$\sec^2(\pi / 4 ) = \frac{ y-1 } { x- \pi / 4 }$$ But I have 2 unknowns and only one equation! –  user3200098 May 10 at 13:56
    
@user3200098, What are the unknown(s) here. We need to form the equation of a Straight line, right? –  lab bhattacharjee May 10 at 13:57
    
Right, I was just about to edit my answer. Sorry! Thanks...Sometimes I get confused. –  user3200098 May 10 at 13:58
    
Could you give me a hand here? math.stackexchange.com/questions/789047/… –  user3200098 May 10 at 13:58
    
@user3200098. It is the same problem. –  Claude Leibovici May 10 at 13:59

At $x = \pi/4,\;$ $f(x) =y = \tan(\pi/4) = 1$.

So $(\pi/4, 1)$ is the point of tangency of $f(x)$ and the desired line, which means it is certainly a point on the desired line.

Now, $f'(x) = \sec^2(x)$. At $x = \pi/4$, $$f'(x) = \sec^2(\pi/4) = \frac{1}{\cos^2(\pi/4)} = \left(\sqrt 2\right)^2 = 2$$ So the slope of the tangent line at the given point is $m = 2$.

Now use the point-slope form of an equation of a line, since you have needed point $(\pi/4, 1) = (a, b)$ and the slope $m = 2$:

$$(y - a) = m(x - b)$$

$$\begin{align} y - 1 = 2(x - \pi/4) & \iff y = 2x + (1 - \pi/2) \\ \\ &\iff y= 2x + \frac{2-\pi}{2}\end{align}$$

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The tangent line is described by the equation: $$y=mx+b.$$ And we know that the slope of a tangent line to the graph of a function $f(x)$ at a point $(x_0, f(x_0))$ is the derivative of $f(x)$ evaluated at $x_0$, that is: $$m=f'(x_0).$$ So let's find it! $$m=\dfrac{\mathrm d}{\mathrm dx}\tan(x)\left|\right._{x={\pi}/{4}}=\sec^2\left(\tfrac{\pi}{4}\right)=2.$$ This tangent line passes through the point $\big(\pi/4, \tan(\pi/4)\big)=(\pi/4,1)$. By the slope-point formula: $$y-y_0=m(x-x_0)\Rightarrow y-1=2(x-\pi/4).$$ Rearranging we get: $$y=2x+\dfrac{2-\pi}2$$ which is our desired equation.

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Equation of tangent at $x=x_0$ is given by $y-f(x_0)=f'(x_0)(x-x_0)$

Now put $x=\frac{\pi}{4}$

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