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I'm given a curve $$y=x^2+2x-4$$

How do I find the tangent line to this curve at $x = 2$?

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    For the sake of completeness, it would be nice to have an answer that showed the differentiating approach2015-09-27

5 Answers 5

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Here's an algebraic approach that avoids the explicit use of derivatives.

We are given a quadratic function $f(x) = x^2 + 2x -4$, and we want to find the equation of the tangent to the parabola $y = f(x)$ at the point $(2, 4)$. (Note that $f(2) = 2^2 + 2 \cdot 2 - 4 = 4$.) Assume that it is given by the equation $ y = m(x-2) + 4, \tag{$\ast$} $ where $m$ is its slope.

Let's consider the intersection of the parabola with the tangent; this is given by the system of equations $ \begin{cases} y &=& x^2 + 2x - 4, \\ y &=& m(x-2)+4. \end{cases} $ In other words, to find the intersection, we should solve the quadratic equation $ x^2 + 2x - 4 = m(x-2)+4$, or $ x^2 + (2-m)x+(2m-8) = 0. \tag{$\ast\ast$} $ using the quadratic formula like so $ \frac{-(2-m)\pm\sqrt{(2-m)^2-4.1.(2m-8)}}{2.1} $ Pictorially it is clear that the tangent meets the parabola meets in exactly one point, so we want $(\ast \ast)$ to have a unique solution. This implies that the discriminant of $(\ast \ast)$ vanishes: $ \begin{array}{crcl} &(2-m)^2 - 4 \cdot (2m-8) &=& 0 \\ \implies \qquad & m^2 - 12m + 36 &=& 0. \end{array} $ Conveniently (although this is not a numerical coincidence), this equation has a unique solution $m=6$: this is the solution we are after. Plugging $m=6$ in $(\ast)$, we get the equation of the tangent at $(2, 4)$ to be $y = 6x-8$.

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    @Max If this answered your question, don't forget to check the green checkmark by the votes to mark the question as closed.2015-10-04
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Another method that can be used which is traditionally taught before derivatives is the limit (they're very similar but this is more intuitive):

Let $f(x) = x^2 + 2x - 4$ and f'(x) be the tangent line at any point $a$ on the curve $f(x)$

Difference quotient:

f'(x) =\lim_{h \rightarrow 0}\frac{f(a + h) - f(a)}{h}

What this essentially means is that we we take the slope between two points $a$ and $a+h$ as $h$ gets very small or as $h\rightarrow0$ the point $a+h$ gets closer to $a$ until there mathematically the slope is tangent to the point $a$.

f'(x) =\lim_{h \rightarrow 0} \frac{[(a+h)^2+ 2(a+h) - 4] - [(a)^2 - 2(a) - 4]}{h} $=\lim_{h \rightarrow 0} \frac{[(h^2 + 2ah + a^2) + (2a + 2h) - 4] - [a^2 - 2a - 4]}{h}$ $=\lim_{h \rightarrow 0} \frac{h^2 + 2ah + 2h}{h}$ $=\lim_{h \rightarrow 0} h + 2a + 2$ $= 2a + 2$

There's your formula for the slope at any value $a$ along the curve $f(x)$

Sorry, I misread your post now I see that you want the equation!

We'll right the formula out in the form $y = mx + b$

Well we can begin with finding the slope of the line at $a = 2$

$m = 2a + 2$ $= 2(2) + 2$ $= 6$

The point at $x = 2$ is $(2,4)$ since if we plug $x = 2$ into $f(x)$ we end up with $y = 4$

So,

$y = mx+b$ $4 = 6(2) + b$ $4 - 12 = b$ $-8 = b$

Therefore your equation is:

$y = 6x - 8$