Graphing with the graphing calculator and calculus.

Example 1

Produce graphs of f that reveal all the important aspects of the curve.  In particular, you should use graphs of f' (x) and f'' (x)to estimate the intervals of increase/decrease, extreme values, concavity, and inflection points.

f(x) = 
        8  (x^5) + 45  (x^4) + 80  (x^3) + 90  (x^2) + 200  x

Notice, f(x) has at most 5 roots and one of these roots is at x = 0.

f(x) is in red - f ' (x) is in blue - f '' (x) is in aqua

Graph this on ZStandard: -10 ≤ x ≤ 
        10  and  - 10 ≤ y ≤ 10

Graph this on ZoomFit: -10 ≤ x ≤ 
        10  and  - 423000 ≤ y ≤ 1341000.  ZoomFit works in much the same way that the text says that a computer will find the range.

Changing our vertical scale to -1000 ≤ y ≤ 1000, we clearly have a zero at x = 0.

Plotting f(x) and f' (x)together on the same axes with the window -5 ≤ x ≤ 
        1 and  - 300 ≤ y ≤ 300, we have:

Solving for the roots of f' (x), we have x = -(5/2) and x = -2.

To find where the concavity changes, graph the second derivative.

The concavity changes at x = 
        (1/8) (-9 - (17^(1/3)) - 
                (17^(2/3))) ≅ -2.2728463261395224`

Example 2

Consider the family of functions j(x) =(1/((x^2) + 2  x + c)) for -3≤c≤5.

j(x) is in red - j' (x) is in blue - j'' (x) is in aqua

Example 3

Consider the family of functions h(x) =x(Sqrt[(c^2) - (x^2)]) for -3≤c≤3.

h(x) is in red - h' (x) is in blue - h'' (x) is in aqua

Scratch Work

Off[Plot::"plnr"];

h[x_] = x (Sqrt[ (c^2) - (x^2)]);
Table[Plot[{h[x], h'[x], h''[x]}, {x, -4, 4}, 
            PlotRange -> {{-3, 3}, {-5, 5}}, 
            PlotStyle -> {{Thickness[0.015], RGBColor[1, 0, 0]}, 
                RGBColor[0, 0, 1], RGBColor[0, 1, 1]}, 
            TextStyle -> {FontFamily -> "Times", 
                FontSize -> 12, FontSlant -> "Oblique"}, 
            AxesLabel -> {"x", "y"}];, {c, -3, 
          3,  .1}];

j[x_] = (1/((x^2) + 2  x + c));
Table[Plot[{j[x], j'[x], j''[x]}, {x, -6, 3}, 
            PlotRange -> {{-6, 3}, {-5, 5}}, 
            PlotStyle -> {{Thickness[0.015], RGBColor[1, 0, 0]}, 
                RGBColor[0, 0, 1], RGBColor[0, 1, 1]}, 
            TextStyle -> {FontFamily -> "Times", 
                FontSize -> 12, FontSlant -> "Oblique"}, 
            AxesLabel -> {"x", "y"}];, {c, 
          5, -3, - .1}];

f[x_] = 
        8  (x^5) + 45  (x^4) + 80  (x^3) + 90  (x^2) + 
          200  x;
Plot[{f[x], f'[x], f''[x]}, {x, -10, 10}, 
      PlotRange -> {{-5, 1}, {-300, 300}}, 
      PlotStyle -> {{Thickness[0.015], RGBColor[1, 0, 0]}, 
          RGBColor[0, 0, 1], RGBColor[0, 1, 1]}, 
      TextStyle -> {FontFamily -> "Times", FontSize -> 12, 
          FontSlant -> "Oblique"}, AxesLabel -> {"x", "y"}, 
      Ticks -> False]

Simplify[Solve[f''[x] == 0, x]] // N