Applications of the Derivative

by Richard Monke

Initialization Cells

<< Graphics`Arrow`

<< Graphics`Colors`

<< Graphics`Legend`

$TextStyle = {FontSize -> 12}

wc[v_, t_] := 
      91.4 - (0.474677 - 0.020425*v + 0.303107*Sqrt[v])*
          (91.4 - t)

WINDCHILL[c_] := 
      (Plot[wc[v, c], {v, 5, 40}, PlotStyle -> {Blue, Thickness[0.01]}, 
          AxesLabel -> {"mph", "°F"}, 
          PlotLabel -> "Wind Chill vs Wind Speed Graph", 
          ImageSize -> {360, 360/GoldenRatio}, PlotRange -> {-60, 50}, 
          Epilog -> {Blue, Text[N[c] "°F", {30, 35}]}]; )

\(\(
\(HI[t_, h_] := 
      16.923 + \((1.85212*10\^\(-1\)*t)\) + \((5.37941*h)\) - 
        \((1.00254*10\^\(-1\)*t*h)\) + \((9.41695*10\^\(-3\)*t\^\(2\ \))\) + 
        \((7.28898*10\^\(-3\)*h\^2)\) + \((3.45372*10\^\(-4\)*t\^2*h)\) - 
        \((8.14971*10\^\(-4\)*t*h\^2)\) + 
        \((1.02102*10\^\(-5\)*t\^2*h\^2)\) - \((3.8646*10\^\(-5\)*t\^3)\) + 
        \((2.91583*10\^\(-5\)*h\^3)\) + \((1.42721*10\^\(-6\)*t\^3*h)\) + 
        \((1.97483*10\^\(-7\)*t*h\^3)\) - 
        \((2.18429*10\^\(-8\)*t\^3*h\^2)\) + 
        \((8.43296*10\^\(-10\)*t\^2*h\^3)\) - 
        \((4.81975*10\^\(-11\)*t\^3*h\^3)\); \)\)\)

HEATINDEX[c_] := 
      (Plot[HI[c, h], {h, 0, 100}, PlotStyle -> {Thickness[0.01], Red}, 
          AxesLabel -> {"%", "°F"}, 
          PlotLabel -> "Heat Index vs Relative Humidity", 
          ImageSize -> {360, 360/GoldenRatio}, PlotRange -> {80, 175}, 
          Epilog -> {Red, Text[N[c] "°F", {60, 160}]}]; )

Increasing and Decreasing Functions

Definition

Definition of Increasing and Decreasing Functions
If (Subscript[x, 1]) < (Subscript[x, 2]) implies f((Subscript[x, 1])) < f((Subscript[x, 2])) for every (Subscript[x, 1]), (Subscript[x, 2]) ϵ (a, b), then f(x) is increasing on (a, b).  See Figure 1.
If (Subscript[x, 1]) < (Subscript[x, 2]) implies f((Subscript[x, 1])) > f((Subscript[x, 2])) for every (Subscript[x, 1]), (Subscript[x, 2]) ϵ (a, b), then f(x) is decreasing on (a, b).  See Figure 2.

\(\(increasing\  = \ 
      Plot[x\^2, {x, 0, 3}, PlotStyle -> {Thickness[0.01], Red}, 
        PlotLabel -> "Increasing function", 
        AxesLabel -> {"x", "y"}, 
	
        Epilog -> {GrayLevel[0.5], Dashing[{0.05}], Line[{{1, 0}, {1, 1}}], 
            Line[{{1, 1}, {0, 1}}], Line[{{2, 0}, {2, 4}}], 
            Line[{{2, 4}, {0, 4}}]}, 
        Ticks -> {{{1, \*""\!\(x\_1\)""}, {2, 
                \*""\!\(x\_2\)""}}, {{1, \*""f(\!\(x\_1\))""}, {
                4, \*""f(\!\(x\_2\))""}}}]; \)\)

\(\(decreasing\  = \ 
      Plot[9 - x\^2, {x, 0, 3}, PlotStyle -> {Thickness[0.01], Blue}, 
        PlotLabel -> "Decreasing function", 
        AxesLabel -> {"x", "y"}, 
        Ticks -> {{{1, \*""\!\(x\_1\)""}, {2, 
                \*""\!\(x\_2\)""}}, {{8, \*""f(\!\(x\_1\))""}, {
                5, \*""f(\!\(x\_2\))""}}}, 
        Epilog -> {GrayLevel[0.5], Dashing[{0.05}], Line[{{1, 0}, {1, 8}}], 
            Line[{{1, 8}, {0, 8}}], Line[{{2, 0}, {2, 5}}], 
            Line[{{2, 5}, {0, 5}}]}]; \)\)

Examples

Wind Chill Factor

wc[v_, t_] := 
      91.4 - (0.474677 - 0.020425*v + 0.303107*Sqrt[v])*
          (91.4 - t)Plot[wc[v, 32], {v, 4, 40}, PlotStyle -> {Thickness[0.01], Blue}, 
      AxesLabel -> {"mph", "°F"}, 
      PlotLabel -> "Wind Chill vs Wind Speed at 32°F"];

Heat Index

\(Clear[HI, t, h]; 

    HI[t_, h_] := 
      16.923 + \((1.85212*10\^\(-1\)*t)\) + \((5.37941*h)\) - 
        \((1.00254*10\^\(-1\)*t*h)\) + \((9.41695*10\^\(-3\)*t\^\(2\ \))\) + 
        \((7.28898*10\^\(-3\)*h\^2)\) + \((3.45372*10\^\(-4\)*t\^2*h)\) - 
        \((8.14971*10\^\(-4\)*t*h\^2)\) + 
        \((1.02102*10\^\(-5\)*t\^2*h\^2)\) - \((3.8646*10\^\(-5\)*t\^3)\) + 
        \((2.91583*10\^\(-5\)*h\^3)\) + \((1.42721*10\^\(-6\)*t\^3*h)\) + 
        \((1.97483*10\^\(-7\)*t*h\^3)\) - 
        \((2.18429*10\^\(-8\)*t\^3*h\^2)\) + 
        \((8.43296*10\^\(-10\)*t\^2*h\^3)\) - 
        \((4.81975*10\^\(-11\)*t\^3*h\^3)\); 

    Plot[HI[90, h], {h, 50, 100}, PlotStyle -> {Thickness[0.01], Red}, 
      AxesLabel -> {"%", "°F"}, 
      PlotLabel -> "Heat Index vs Relative Humidity at 90°F"]; 
    \)

Relationship with the derivative

\(f[x_] := x^3 - 3  x^2; 

    Plot[f[x], {x, \(-2\), 4}, PlotRange -> {\(-15\), 12}, 
      AxesLabel -> {"x", "y"}, ImageSize -> {360, 360/GoldenRatio}]; 
    
Do[Plot[{f[x], \(f\^′\)[a]*x - \(f\^′\)[a]*a + f[a]}, 

        		{x, \(-2\), 4}, 
        PlotStyle -> {Black, 
            If[\(f\^′\)[a] > 0, Red, 
              If[\(f\^′\)[a] < 0, Blue, Green]]}, 
        PlotRange -> {\(-15\), 12}, AxesLabel -> {"x", "y"}, 
        ImageSize -> {360, 360/GoldenRatio}], {a, \(-2\), 4, 0.25}]\)

Test for Increasing and Decreasing Functions
Let f(x) be a differentiable function on (a, b).
1.  If (f^′)(x) > 0 for every xϵ(a, b), then f(x) is increasing on (a, b).
2.  If (f^′)(x) < 0 for every xϵ(a, b), then f(x) is decreasing on (a, b).
3.  If (f^′)(x) = 0 for every xϵ(a, b), then f(x) is constant on (a, b).

\(a\  = \ 
      Plot[{1\/3*x\^3 - x, x\^2 - 1}, {x, \(-3\), \(-1\)}, 
        AxesLabel -> {"x", "y"}, 
        PlotStyle -> {{Thickness[0.01], Red}, {Thickness[0.01], Green}}, 
        PlotRange -> {\(-3\), 3}, DisplayFunction -> Identity]; 

    b\  = \ Plot[{1\/3*x\^3 - x, x\^2 - 1}, {x, \(-1\), 1}, 
        AxesLabel -> {"x", "y"}, 
        PlotStyle -> {{Thickness[0.01], Blue}, {Thickness[0.01], Yellow}}, 
        PlotRange -> {\(-3\), 3}, DisplayFunction -> Identity]; 

    c\  = \ Plot[{1\/3*x\^3 - x, x\^2 - 1}, {x, 1, 3}, 
        AxesLabel -> {"x", "y"}, 
        PlotStyle -> {{Thickness[0.01], Red}, {Thickness[0.01], Green}}, 
        PlotRange -> {\(-3\), 3}, DisplayFunction -> Identity]; 

    Show[{a, b, c}, DisplayFunction -> $DisplayFunction, 
      Ticks -> {None, None}, 
      Epilog -> {Arrow[{\(-3\), 1}, {\(-2\), 2}], 
          Text["derivative", {\(-2.5\), 0.5}], 
          Arrow[{3, \(-1.5\)}, {2, \(-0.5\)}], 
          Text["function", {2.5, \(-2\)}], GrayLevel[0.5], 
          Dashing[{0.02}], Line[{{\(-1\), \(-3\)}, {\(-1\), 3}}], 
          Line[{{1, \(-3\)}, {1, 3}}]}]; \)

Definition of Critical Values
If f(x) is defined at c, then c is a critical value of f if (f^′)(x) = 0 or (f^′)(x) is undefined at c.

Exercises

f(x) = -(((x + 1))^2)

f(x) = (((x + 2))^(2/3))