Finding Parallel Tangent Line Points on Curve: y=1/3x^3-4x^2+18x+22

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In summary, the problem asks to find the two points on the curve y=1/3 x^3 - 4x^2 + 18x + 22 where the tangent line is parallel to the line 6x - 2y = 1. The suggested approach is to find the slope of the given line and use it to solve for the points where the slopes of the tangent lines are equal to it. No additional equation needs to be written.
  • #1
maxpayne_lhp
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Hey... I am kinda got stuck with this problem... can you guys give me a hint or something...

Problem says:

The tangentline to the curve y=1/3 x^3 - 4x^2 + 18x + 22 is parallel to the line 6x - 2y = 1 at 2 points on the curve... find the 2 points.

Well I was thinking about writing the function for another line which has the same lope with the one given... so it should be something like y = 3x + m

Now what?

Thanks for your help.
 
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  • #2
You don't need to write an equation for another line, you just have to know what the slope of 6x - 2y = 1 is. Then you need to find an equation for the slopes of the lines tangent to your cubic curve. Then you have to solve (a quadratic) for when the slopes of the tangents equal the slope of 6x - 2y = 1.
 

FAQ: Finding Parallel Tangent Line Points on Curve: y=1/3x^3-4x^2+18x+22

What is the equation of the given curve?

The equation of the given curve is y = 1/3x^3 - 4x^2 + 18x + 22.

How do you find the slope of the tangent line at a specific point on the curve?

To find the slope of the tangent line at a specific point on the curve, you will need to take the derivative of the given curve. This will give you the slope of the tangent line at any point on the curve.

What is the formula for finding the parallel tangent line to a curve?

The formula for finding the parallel tangent line to a curve is y = mx + b, where m is the slope of the tangent line and b is the y-intercept. To find the slope, you will need to take the derivative of the given curve at the specific point. Then, plug in the coordinates of the point into the equation and solve for b.

Can there be multiple parallel tangent lines to the same curve?

Yes, there can be multiple parallel tangent lines to the same curve. This is because there can be multiple points on the curve with the same slope, which would result in multiple parallel tangent lines.

How do you know if a point on the curve has a parallel tangent line?

A point on the curve has a parallel tangent line if its slope is equal to the slope of the tangent line at that point. This can be determined by taking the derivative of the given curve and plugging in the coordinates of the point. If the resulting slope is the same, then the point has a parallel tangent line.

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