Function of a concave curve given pts and slopes

In summary, to derive a function for a concave downward sloping curve given two or more points and their respective slopes, you can use a third degree polynomial and solve a linear system of equations for the coefficients. This will allow you to determine a general function for the given curve.
  • #1
el_llavero
29
0
It's been a while since my last calc course and I'm not sure if I even encountered how to do this but is there a way to derive a function of a concave downward sloping curve given two or more points and their respective slopes? I'm obviously aware how the curve looks but I would like to know how to derive the function if at all posible. i.e. (2,10) slope -8 and (6,4) slope -1/2, this curve has a negative decreasing slope and is concave towards the (0,0) origin. The curve's appearance is obvious from the information given but is there a way to derive a general function for this curve?
 
Physics news on Phys.org
  • #2
Take a third degree polynomial f(x) = a0 + a1x + a2x^2 + a3x^3, plug your four conditions in, and solve the linear system of equations for the coefficients a0, a1, a2 and a3.
 

1. What is the function of a concave curve given points and slopes?

The function of a concave curve is a mathematical expression that describes the relationship between two variables, typically represented as x and y. It is concave when the graph of the function curves downward, resembling the interior of a bowl.

2. How do you find the function of a concave curve given points and slopes?

To find the function of a concave curve, you will need to use the slope-intercept form of a line, which is y = mx + b. The slope (m) can be calculated using the points given, and the y-intercept (b) can be determined by plugging in one of the points into the equation.

3. Can a concave curve have multiple functions?

No, a concave curve can only have one function that describes its relationship between points and slopes. However, there can be different ways of expressing the same function, such as using different forms or rearranging terms.

4. How do you know if a curve is concave?

A curve is concave if it curves downward and resembles the interior of a bowl. This means that the slope decreases as you move from left to right on the graph, and there is a point where the slope is equal to zero, known as the point of inflection.

5. Can a concave curve have both positive and negative slopes?

Yes, a concave curve can have both positive and negative slopes. The curve may start with a positive slope, then decrease until it reaches the point of inflection where the slope is zero, and then continue with a negative slope. This pattern can repeat throughout the curve.

Similar threads

I Gradient Vectors: Perpendicular to Level Curves
    • Calculus
Replies
4
Views
2K
B What does the derivative of a function at a point describe?
    • Calculus
2
Replies
36
Views
3K
Graphing Derivatives: Concavity and Inflection Points
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
B Calculating shaded area: I'm getting discrepancies between methods
    • Calculus
Replies
3
Views
329
I How to understand the notion of a limit of a function
    • Calculus
Replies
8
Views
1K
A Dirac delta function confusion
    • Calculus
Replies
2
Views
1K
MHB Need help plotting a curve, no calculus knowledge....
    • Calculus
Replies
1
Views
1K
How can I use slope calculations to solve for sin(x)?
    • Calculus
Replies
3
Views
2K
Derivative and integral (inverse operations?)
    • Calculus
Replies
3
Views
11K
I How to find the maximum arc length of this equation?
    • Calculus
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top