Here's the question you clicked on:
eroshea
find the equation of the tangent and normal line to the curve y=2x^3 - 3x^2 -2x +5, where x =1
To get a line, you need a slope, and a point, right? Or, knowing the slope of a line and a point where it passes through, that's sufficient to find the equation of the line, yes?
yes.. so how will i find the slope?
Magic. Or, you can differentiate this function first... go ahead now :) Remember, the slope of the tangent line at the point would be its derivative, while the slope of the normal line would be the negative reciprocal of the derivative.
Start by differentiating, though.
Quick to the point, are we? :) Now, when x = 1, what's dy/dx?
And that is the slope of your tangent line :) What's the slope of your normal line? (Remember, the normal line is *perpendicular* to the tangent line)
negative reciprocal of it.. 1/2
And that is the slope of the normal line. So you have the slopes for both lines, you now only need a point. Any idea how to get one?
hmm.. where they would intersect? but i don't know how
Well, both the tangent and the normal would intersect your curve at the point WHERE x = 1. So at the curve itself, when x = 1, what is y? :P
then i just need to use slope intercept form , right? with P(1,-2) and a slope of 1/2?
No... that's dy/dx I meant y. You know... your original curve? :P
y=2x^3 - 3x^2 -2x +5
LOL don't worry about that :) So you have a point (1 , -8) and you have a slope (-2 for your tangent, 1/2 for your normal) You can't use a slope intercept form, you need the more general point-slope form. When you have a point \(\large (\color{red}p,\color{green}q)\) and a slope \(\large \color{blue}m\) then the equation of the line is: \[\Large y = \color{blue}m(x-\color{red}p)+\color{green}q\]
What is? Your slope? No, you were right, it's -2 for the tangent.
no .. the coordinates of the point (1,2)
Ah well then, it won't be (1,-8) but (1,2) instead :) Work with that then. :D
\[\Large y = \color{blue}m(x-\color{red}p)+\color{green}q\]
correct :) Now the normal line? the only difference is the slope :)
and you're done :P
:D thanks kiddo .. but you're smarter than me
Nah. I've just read in advance :P