Curve Sketching using Data Points - Urgent Help Needed

In summary, the task is to find an equation that passes through five given points on a graph, with specific conditions such as having three inflection points, at least one local maximum and minimum, at least one critical point different from the given points, and being continuous and differentiable throughout. The equation should also have a piecewise definition, meaning it is not a single polynomial. To solve this, one can think of it as a connect-the-dots puzzle and use polynomials for each piece. The pieces must connect smoothly and have the same slope at the points where they meet. While there may be some difficulty in achieving this, it is important to make sure the graph is continuous and differentiable at the meeting points.
  • #1
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Homework Statement


The following five points lie on a function: (1, 20), (2, 4), (5, 3), (6, 2), and (10, 1).

Find an equation which passes through these points, and which also has the following:
. . .i) three inflection points
. . .ii) at least one local maximum
. . .iii) at least one local minimum
. . .iv) at least one critical point which differs from the listed points (above)
. . .v) the property that it is continuous and differentiable throughout
. . .vi) a piecewise definition (that is, it is not a single polynomial)


Homework Equations


none


The Attempt at a Solution


Actually, can you guys help me understand the question better. Also, the fact that the graph has to be piece-wise makes it harder. Can anyone offer me a reasonable solution?

I need this urgently since it's due tomorrow. Any help would be great. Thanks.
 
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  • #2
So this is like a connect-the-dots puzzle.

You'll probably want to make each 'piece' a polynomial because it is easy to differentiate polynomials, find their max/min, etc. You know how to write a polynomial that passes through certain points. Lines are the easiest, but lines don't have local max/min. Parabolas are also easy, and they do have max/min, but no inflection points. And so on.

Where the pieces meet, you'll need to make sure that there is continuity and differentiability there. So the pieces have to 'connect', and they have to have the same slope where they meet. It is probably easiest to have slope 0 at these points, which would then have to be the max/min/inflection points.
 
  • #3
Thanks for your help mutton. I think I better understand the question as you have described it well. I tried doing this question into two pieces.

Please see the attached image Testnew.PNG to see if we looks ok.

Actually, when I zoom into the point (3.5,6) there is a cut in the graph which I am not sure would be ok. But I am guessing that my teacher will probably not notice it. It's after a lot of zooming that I notice the "cut" in the graph. (Please see attached zoom.png)

I really would need help in this. It's due tomorrow please. Thanks.
 

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1. What is curve sketching using data points?

Curve sketching using data points is a method used to graphically represent a set of data points by drawing a smooth curve that connects them. It is a useful way to visualize trends and patterns in data.

2. Why is curve sketching using data points important?

Curve sketching using data points is important because it allows us to understand complex data sets in a more simplified and visual manner. It helps us identify relationships between variables and make predictions based on the trends observed.

3. How do you perform curve sketching using data points?

The first step is to plot the data points on a graph. Then, draw a smooth curve that connects the points, making sure to capture the overall trend of the data. You can use a ruler or a computer program to help you create a more accurate curve. Finally, label the axes and add a title to the graph.

4. What are some common mistakes to avoid when performing curve sketching using data points?

Some common mistakes to avoid include not including enough data points, not choosing an appropriate scale for the axes, and drawing a curve that does not accurately represent the data. It is also important to label the axes and provide a clear title for the graph.

5. How can I use curve sketching using data points to make predictions?

Curve sketching using data points can help you make predictions by identifying trends and patterns in the data. For example, if the curve is increasing at a steady rate, you can predict that the trend will continue in the future. However, it is important to note that predictions based on curve sketching are not always accurate and should be used with caution.

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