Understanding Derivatives of U-Shaped Parabolas

  • Thread starter MathWarrior
  • Start date
  • Tags
    Derivatives
In summary, to determine the stability of a point on a graph without an equation, you can use the limit definition of the derivative to find the slope at that point and then determine the stability based on the sign of the slope.
  • #1
MathWarrior
268
5
The problem for this is a picture. Its basically a U shaped parabola such that it intersects the points (-3,0) (0,0) (3, 0)

you have a graph such as:

f(x)
|
|
|------- y
|
|

The problem I am having is this.. I know what it means when a point is unstable or stable we tended to draw arrows on either side of the phase lines. I am unsure of how you figure out which way the arrow goes basically, I think it has something to do with their derivative but how am I suppose to find the derivative of a visual graph without an equation?
 
Physics news on Phys.org
  • #2
The way to find the derivative of a graph without an equation is to use the limit definition of the derivative. This involves finding the slope of the line tangent to the graph at each point. For example, at the point (-3, 0), you can draw a line that is tangent to the graph at this point and calculate the slope of that line. Using this method, you can then determine whether the point is stable or unstable based on the sign of the slope. If the slope is positive, then the point is stable. If the slope is negative, then the point is unstable.
 

1. What is a U-shaped parabola?

A U-shaped parabola is a type of curve that has a symmetrical, U-like shape. It is formed by the graph of a quadratic equation in the form of y = ax^2 + bx + c, where the coefficient a is negative, causing the graph to open downwards.

2. How do you find the vertex of a U-shaped parabola?

The vertex of a U-shaped parabola is the highest or lowest point on the curve. To find the vertex, you can use the formula x = -b/2a, where b and a are the coefficients in the quadratic equation. The x-coordinate of the vertex is the solution to this formula, and the corresponding y-coordinate can be found by substituting the x-value into the equation.

3. How do derivatives relate to U-shaped parabolas?

Derivatives of U-shaped parabolas represent the slope of the curve at a specific point. The derivative of a quadratic equation is a linear equation, which means that it changes at a constant rate. This is why the slope of a U-shaped parabola increases or decreases uniformly as you move along the curve.

4. How do you use derivatives to find the maximum or minimum point of a U-shaped parabola?

The maximum and minimum points on a U-shaped parabola correspond to the vertex of the curve. To find these points using derivatives, you can set the derivative of the quadratic equation to zero and solve for the x-value. This will give you the x-coordinate of the vertex, and you can find the corresponding y-coordinate by substituting the x-value into the original equation.

5. What is the significance of understanding derivatives of U-shaped parabolas?

Understanding derivatives of U-shaped parabolas is important in many fields, including mathematics, physics, and engineering. It allows us to analyze the behavior of the curve, find the maximum or minimum points, and make predictions about how the curve will change over time. Additionally, derivatives of U-shaped parabolas are used in real-world applications such as optimization problems and projectile motion.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
348
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
983
  • Calculus and Beyond Homework Help
Replies
4
Views
937
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
Replies
13
Views
3K
Replies
1
Views
757
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • General Discussion
Replies
12
Views
1K
Back
Top