TOUGH QUESTION---4th degree polynomial, I want to know where the slope of the equation is the greatest?

Question

I have a fourth degree polynomial, namely:

y = 0.0003x4 - 0.0272x3 + 0.6806x2 - 0.8345x + 0.5659

When plotting this line, from points x=560 to x=790, the way that that curve looks...I want to know at what point am i going to get the greatest increase in y, with the least increase in x...do you get waht i'm saying? At what point, like at say 700 to 701, is that where i am going to see the largest y increase with the least x increase. I want it to be proportaional,like for example:

Maybe from 700-701 is just an increase of 2 in the y direction, but if going from 703-705 i get an incrase of 5, thats more benificial for me, cause i more than doubled my input...get waht i'm saying??

Please help here, i'm not sure if i should be taking the derivitive or what.

Thanks

Answer 1

Find the derivative to get an equation for the slope and then analyze it. If you have a positive slope for the entire range that means your slope is greatest at the highest value. If you want to know whether or not the slope goes up and down in your range, set the derivative to zero and solve. If any of those zero values fall within your range (no), it means that the slope has changed directions and either gone up or down. Use that answer to evaluate your equation.

Answer 2

Derivation is a MUST.



The first thing you should do is get rid of the decimal numbers, in order to facilitate your calculations. Multiply all the polynomial's factors by 10000. Doing so does not mess with the result and makes life a little bit easier.



You should take the polynomal's derivative. e.g.

Let y=f(x) where y is your polynomial.

Calculate g=df/dx.

This is the derivative. It also called "rate of change". Its value is a measure of how steep your polynomial graph is.

If df/dx is positive, you the value of f is increasing, if df/dx is negative it is decreasing. The slope can be the steepest in either case.

The slope will be at the positive maximums of g or at the negative minimum, whichever has the largest absolute value.

In that way, you have to find the maximum and minimum values of g. You find mins and maxs by deriving the function and using Fermat's theorem: The function's critical points are wherever it's derivative's value is zero.

You do that by taking yet another derivative: h = dg/dx (=d^2f/dx^2)

Calculate the roots of h. That should be quite easy, since h has to be a polynomial of the second degree, since f is a polynomial of the fourth degree.

let x1 and x2 be the roots of h.

If g(x1) and g(x2) values are equal, there are two points P(x1, f(x1)) and Q(x2, f(x2)) at which the slope is steepest.

If not choose the one with the largest absolute value (g|x1|, or g|x2|)

Hope I have helped.

Answer 3

I get the derivative to be



0.0012X³ - 0.0816X² +1.3612X - 0.8345



In my TI-83, its graph crosses the X-axis around X = 40 going positive. In the 560 - 790 range you've given, the slope will be greatest at X = 790 at something like 54,1794.8!

Answer 4

It looks like the increase is greatest at the right end point (790).



I used Excel to compute the value for 560-790 at the integer points. I then took the difference between f(x) and f(x+1)

show the change between integer values.



It took on it's maximum value at 790.



Looking at the graph, it appears to be concave up so the max would be at the right end of the interval.



I hope this helps.

Answer 5

4th Degree Polynomial

Answer 6

The derivitive of a function tells you about the slope of the function, so it sounds like that is what you want. Find out more about differentiation here http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivs/deriv6.html



and try the online calculator and grapher at



http://www.compute.uwlax.edu/calc/



Also, keep in mind that for a fourth degree polynomial, just based on the shape the magnitude of the slope should be continuously increasing as x increases (for all x>0).

Answer 7

The easiest way to find the greatest rate of change is to find the maximum of the derivative of the original equation. In this case, it should turn out to be infinity since the largest exponent is positive.

Answer 8

Differenciate y: it gives y'.

Differentiate y': it gives y".

The solution of y"= 0 is the answer of your problem.

At that x the graph of y=f(x) shows a bending point.