Horizontal Tangent Lines: Intersection of Cylinder and Plane

In summary, the homework statement is that you need to find a vector function that describes the intersection of a plane and a cylinder. You find the points where the tangent to the function is horizontal, and then find an equation for the tangent line at each of these points.
  • #1
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Homework Statement



Consider the plane z = x + 2y and the cylinder x^2 + y^2 = 1

(a) Find a vector function r(t) describing their intersection.
(b) Find the points if any where the tangent to ~r is horizontal
(c) Find an equation for the tangent line to ~r at each of these points.

Homework Equations

The Attempt at a Solution



r(t) = cos(t)i+sin(t)j+(cos(t)+2sin(t))k
r'(t) = -sin(t)i + cos(t)j + (-sin(t)+2cos(t))k

These equations are correct, as I have plotted them in Maple 14 and the images appear correct. I'm having issues finding the horizontal tangent, and then finding the equations afterwards.
The horizontal tangent should be when r'(t) = 0 correct? I cannot find a value that makes this true. Maybe I'm making a mistake in my thinking?

Thank you for any help.
 
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  • #2
Hi Lothar! :smile:

(try using the X2 icon just above the Reply box :wink:)
Lothar said:
The horizontal tangent should be when r'(t) = 0 correct?

No.

r' is a vector, it will be horizontal when … ? :wink:

(if r' was zero somewhere, you'd have a dodgy parameter! :rolleyes:)
 
  • #3
tiny-tim said:
Hi Lothar! :smile:

(try using the X2 icon just above the Reply box :wink:)


No.

r' is a vector, it will be horizontal when … ? :wink:

(if r' was zero somewhere, you'd have a dodgy parameter! :rolleyes:)

Where r' is equal to pi or 2pi? Maybe? I'm kind of lost now.
 
  • #4
Are we talking about the same thing? :confused:

r' is a vector, of the form (a,b,c) or ai + bj + ck

it can't be a number like π ror 2π …

when will that vector be horizontal?​
 
  • #5
tiny-tim said:
Are we talking about the same thing? :confused:

r' is a vector, of the form (a,b,c) or ai + bj + ck

it can't be a number like π ror 2π …

when will that vector be horizontal?​

I'm not sure. That's why I asked.
When the j component is equal to zero?
 
  • #6
Lothar said:
When the j component is equal to zero?

Nearly :rolleyes: … the k component! :wink:

(isn't that obvious … "horizontal" means moving only in the x,y plane, so no z ?)
 
  • #7
Well I'm stupid. Still though, I don't see a point on the unit circle where 2cost - sint is equal to zero.
 
  • #8
tant = 2 :smile:
 

1. What is a horizontal tangent line?

A horizontal tangent line is a line that is tangent to a curve or surface at a point and is parallel to the horizontal axis. It means that the slope of the tangent line at that point is zero.

2. How does a horizontal tangent line relate to the intersection of a cylinder and a plane?

In the case of the intersection of a cylinder and a plane, a horizontal tangent line is formed when the plane is parallel to the base of the cylinder. This creates a cross section of the cylinder that is a circle, and the tangent line is then perpendicular to the plane and parallel to the base of the cylinder.

3. Can a horizontal tangent line exist at multiple points of intersection between a cylinder and a plane?

Yes, it is possible for a horizontal tangent line to exist at multiple points of intersection between a cylinder and a plane. This occurs when the plane is parallel to the base of the cylinder at multiple points, creating multiple cross sections that are circles.

4. How can the equation of a horizontal tangent line be determined for the intersection of a cylinder and a plane?

The equation of a horizontal tangent line can be determined by finding the slope of the tangent line at the given point of intersection and using the point-slope form of a line. The slope can be found by taking the derivative of the equation of the cylinder at the point of intersection, and the point can be found by using the coordinates of the point of intersection.

5. What is the significance of a horizontal tangent line in the study of geometry and calculus?

Horizontal tangent lines play an important role in the study of geometry and calculus as they represent a point of zero slope on a curve or surface. They also help in determining the rate of change of a function at a specific point, and are crucial in finding the maximum and minimum values of a function. In geometry, horizontal tangent lines help in identifying symmetrical shapes and curves.

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