Skip to main content

calculus and analysis - What is the range of values $x$ for which $f(x)$ is higher than $k$ over a given domain?



What is the easiest way to answer the following question in Mathematica:


Given a function $f(x)=y$, what is the range of values $x$ for which $y$ is higher than some number $k$ over the domain of $x$ $[a;b]$?



For example consider the function:


enter image description here


What is the range of values of $x$, for which $f(x)$ is greater than $k=4.8$ over the domain [-2;2]?


As I am aiming to apply this method to a complicated function, I would like numerical approximations rather than an analytical solution.



Answer



Let


f[x_] := Sin[3 x^2] + x;


and criterion k = 1, all in the range [-2,2].


One can use Solve[], NSolve[] or FindRoot[] to get crossing points, but it helps to search for solutions using a number of starting points.


myCrossings = Sort@DeleteDuplicates[(x /. FindRoot[f[x] == 1, {{x, Range[-2, 2, .4]}}])]

Then, include the lower and upper limit of your x range:


PrependTo[myCrossings, -2];
AppendTo[myCrossings, 2]

Then, depending upon whether the value of f[x] at the lower limit of the range is greater or less than the criterion (here, k = 1), choose pairs of crossing points for the corresponding ranges of x:



If[f[-2] > 1, Partition[myCrossings, 2, 2], Partition[Drop[myCrossings, 1], 2, 2]]

(* {{0.443523, 1.02785}, {1.39912, 1.86883}, {1.94362, 2}} *)


If you want a more easily readable output:


#[[1]] < x < #[[2]] & /@ If[f[-2] > 1, Partition[myCrossings, 2, 2], Partition[Drop[myCrossings, 1], 2, 2]]

(* {0.443523 < x < 1.02785, 1.39912 < x < 1.86883, 1.94362 < x < 2} *)


It is true, in the extremely unlikely case that you have a local maximum or local minimum of f[x] at a root--and thus f[x] remains below the criterion or remains above the criterion on both sides of the root--you might have to check for solution ranges.


enter image description here


Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more