No Tangent Line: Graph of y = |x| Explained

In summary, the conversation discusses the question of why the graph of y = lxl has no tangent line at the point (0,0). The definition of a tangent line is brought up and it is explained that the x-axis does not fit the definition, leading to the conclusion that there is no tangent line at x = 0. The concept of the tangent line being the limit of secant lines is also mentioned.
  • #1
Moneer81
159
2
Am I thinking right about this following question:

Give a geometric argument to show that the graph of y = lxl has no tangent line at the point (0,0).

Since that graph would be basically the bisectors of the 1st and second quadrants, why can't the x-axis be that tangent line?
(def. of a tangent line is a line that touches the graph at one point, which is the origin in this case)

thanks
 
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  • #2
The slightly more correct definition of a tangent line at a point is the limit of secant lines through that point and one near it, as the nearer point approaches infinitely close to the point you want to find the tangent line to. If you do this for the function f(x) = |x| you'll notice what happens when you take points on both sides of the origin to draw secant lines through.
 
  • #3
so the x-axis doesn't qualify as a tangent line?
 
  • #4
Moneer81 said:
so the x-axis doesn't qualify as a tangent line?

In a sense it is, but it doesn't fit the definition of a tangent line that they want you to use. I explained what they really want you to do in my previous post, and if you think about that you will see why there is no tangent line at x = 0.
 
  • #5
Moneer81 said:
(def. of a tangent line is a line that touches the graph at one point, which is the origin in this case)

If that were the definition of tangent line, then every straight line through the origin would be a tangent line there. Check your textbook for the specific definition of "the tangent line to a graph".
 

1. What does a graph with no tangent line mean?

A graph with no tangent line means that the graph is not differentiable at that point. This means that the slope of the graph is undefined at that point and it cannot be represented by a single straight line.

2. Why does the graph of y = |x| have no tangent line at the origin?

The graph of y = |x| has no tangent line at the origin because the function is not differentiable at that point. This is because the function changes direction abruptly at the origin, making it impossible to draw a single straight line that represents its slope at that point.

3. Is the graph of y = |x| continuous?

Yes, the graph of y = |x| is continuous. This means that the function is defined at all points and there are no breaks or holes in the graph. However, it is not differentiable at the origin, where there is a sharp point on the graph.

4. What does the absolute value function represent graphically?

The absolute value function represents the distance of a number from zero on a number line. Graphically, it is represented by a V-shaped graph, with the vertex at the origin. The absolute value of a number is always positive, regardless of its sign.

5. Can a function have multiple tangent lines?

No, a function can only have one tangent line at a given point. This is because a tangent line represents the slope of a function at a specific point, and a function can only have one slope at a given point. However, a function may have multiple tangent lines at different points if it is not continuous or differentiable.

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