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burhan101
 2 years ago
Determine the coordinates of the points on f(x)=x³+2x² where the tangent lines are perpendicular to yx+2=0.
How do I approach this question ?
burhan101
 2 years ago
Determine the coordinates of the points on f(x)=x³+2x² where the tangent lines are perpendicular to yx+2=0. How do I approach this question ?

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burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0:O why'd you delete it

genius12
 2 years ago
Best ResponseYou've already chosen the best response.1So basically you have to find the coordinates of points on f(x) such that the derivative, or the slope of the tangent, is the negative reciprocal of the slope of the given line y = x  2. This implies that the tangents at those coordinates on f(x) will be perpendicular to the line y = x  2. So we know that the derivative must be the negative reciprocal of the slope of y = x  2 at those coordinates so f'(x) must be 1 at those coordinates. Let's find f'(x): \[\bf f'(x)=3x^2+4x\] We know that the slope of the tangent at those coordinates of f(x) or the derivative is 1. So we equate f'(x) with 1 and solve for x:\[\bf 3x^2+4x=1 \rightarrow 3x^2+4x+1=0 \rightarrow (3x+1)(x+1)=0 \implies x = \frac{ 1 }{ 3 },1\]Plug these xvalues back in to f(x) to get the yvalues and that will give you the coordinates. @burhan101

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0@genius12 thank you for the explanation ! very clear :)

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0How would I determine the equations for the tangent for f(x) ?

genius12
 2 years ago
Best ResponseYou've already chosen the best response.1Why do you need the equation of the tangent lines at the coordinates? @burhan101

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0it's part b of the question

genius12
 2 years ago
Best ResponseYou've already chosen the best response.1You have the coordinates right?

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0Yes \[(\frac{ 1 }{ x },\frac{ 5 }{ 27 })\] and (1,1)

genius12
 2 years ago
Best ResponseYou've already chosen the best response.1whats the x there for.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0typo it was supposed to be a 3

genius12
 2 years ago
Best ResponseYou've already chosen the best response.1Anyway, to find the equation of the tangent line at the coordinates, we need a point and a slope. We already know that since the tangent lines at these coordinates are perpendicular to y = x  2, the slope for both tangents will be 1. We already have the points each tangent goes through. First one goes through (1/3, 5/27) and second one goes through (1, 1). The equation of tangent line will be int he form:\[\bf y= mx+b\]Where m is the slope, which is 1, and we need b. Since we know a point for each tangent, plug in the xvalue and yvalue of the coordinates and solve for b. This will then give you the slopeintercept equation of the tangents at the coordinates. @burhan101
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