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# Homework Help

Posted 10 years ago
 Ok, so this is my homework and I'm stuck on Exercise 2 part d). I've spent the past 2 hours trying an array of different ways to solve this but I can't figure it out and my professor isn't answering her email. Please help! It's due at 11:59pm. Attachments:
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Posted 10 years ago
 The implicit differentiation gave you the function (-7 - x - 7 x^2 - x^3)/(-24 - 2 y + y^2) as the result for dy/dx. Notice that the numerator only depends on x and the denominator only depends on y. The horizontal tangents are where the expression is zero and this in turn is where the numerator is zero. So solve for the values of x where this is true: Solve[-7 - x - 7 x^2 - x^3 == 0, x] Once you have the x values from this (there is only one that is real in this case) you can find the corresponding y value by going back to the original implicit equation 84 x + 6 x^2 + 28 x^3 + 3 x^4 - 288 y - 12 y^2 + 4 y^3 == 5 substitute each x value and then solve for the corresponding y value for that x value. So now you have the {x,y} points where there are horizontal tangents. Similarly the vertical tangents are where the expression goes to + or - infinity and this is where the denominator becomes zero (so long as it also is not one of the points where the numerator is also zero). You can determine this from Solve[-24 - 2 y + y^2 == 0, y] And like in the horizontal case, use the implicit equation to now solve for the x values corresponding to each of these y-values. When you are done you can plot all of these values using ListPlot and then combine them with your original contour plot using Show. You can make the points in your ListPlot show nicely by using something like (I am using made up points here):ListPlot[{{1, 2}, {3, 5}}, PlotStyle -> Directive[Red, AbsolutePointSize[12]]]If this ListPlot is called plot4 then you would combine this with your contour plot (which was called plot3) usingShow[plot3,plot4]P.S. Give Professor Read our best regards ;-)
Posted 10 years ago
 Ok, this is what I got. Would you say that is right or do I need to switch the x and y values in the coordinates?Thank you so much for getting back to me! It was extremely helpful!!! Attachments:
Posted 10 years ago
 Wait! I think I did that wrong. Because when I plotted them the points weren't on the original contour plot. I think I had the x and y values switched. This is right now, yes? Attachments:
Posted 10 years ago
 In your latest case you will notice that the point at the top does correspond to a horizontal tangent ((just taking a look at it you see that the slope of the curve there is zero), but your other points are not vertical tangents since the slopes of the curve there is not vertical, it has a slope in both cases. The problem is that you substituted into the x values and solved for y rather than substituted for the y values and solved for x in each case,Here are those two cases, corrected: NSolve[(84 x + 6 x^2 + 28 x^3 + 3 x^4 - 288 y - 12 y^2 + 4 y^3 == 5) /. y -> -4, x] and NSolve[(84 x + 6 x^2 + 28 x^3 + 3 x^4 - 288 y - 12 y^2 + 4 y^3 == 5) /. y -> 6, x] These give the following points (confirm for yourself):{{-9.14477, -4}, {-2.97489, -4}}and{-9.86928, 6}so your ListPlot is ListPlot[{{-9.144774626644757, -4}, {-2.974889393649998, -4}, \ {-9.869278763054162, 6}}, PlotStyle -> {Black, PointSize[Large]}] `
Posted 10 years ago
 I can't find L(x)! I only have 30 min to figure this out. I don't know what's going on. Everything just comes out as giant complicated functions. Please help! Attachments:
Posted 10 years ago
 Use the equation that your professor supplied. Then, as you did, compute the derivative of f and substitute the value of x0 in that expression. Also put that same value into the expression for f itself and then use these in the linearization equation for L[x]. You do not need to use Solve. Think of the reason for (i.e., the idea behind) the expression for L[x]. And (fatherly advice) try to get your homework started well before the midnight deadline ;-) !
Posted 10 years ago
 Why does the tangent line cross at 2 different places? Attachments:
Posted 10 years ago