<s>[INST] Finding point of contact and value of b If a line $y = 2x - b$ connects with a curve $y = 3x^2 + 2$, how do I approach finding the value of b? To find the coordinates, if I'm not mistaken, I need to set the derivative of $y = 3x^2 + 2$ equal to the equation of the line. Is this correct? Many thanks in advance!
 [/INST] By differentiation, we have
$$\dfrac{dy}{dx} = 6x$$
Since we know that the slope of a line is $2$,
$$\dfrac{dy}{dx} = 2 = 6x$$
$$x = \frac{1}{3}$$
Therefore,
$$y = 3(\frac{1}{3})^2 + 2$$
$$y = \frac{7}{3}$$
Thus,
$$y - \frac{7}{3} = 2(x - \frac{1}{3})$$
$$y = 2x + \frac{5}{3}$$
So $b = \frac{5}{3}$.
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