Laboratory 4

Use the following module to explore the shape of the graph of the derivative.

DerivativeAsFunction:=Manipulate[
Module[{func},
Clear[x];
Clear[e];
e=Exp[1];
func[x_,formula_]=Function[x,formula];
Module[{f=func[x,expr],m},
m=D[f[x],x]/.{x->c};
Column[{TraditionalForm@Text@Style[Row[{"Tangent line slope = ",m, ", c = ",c}],14],Show[Plot[{f[x],m*(x-c)+f[c]},{x,a,b},PlotRange-> h,
ImageSize->{600,325},PlotStyle->{Black,Red}],ListPlot[{{{c,f[c]}},{{c,m}}},PlotStyle->{Blue,Red},PlotLegends->{"f(c)","f '(c)"}]]},Center]]],
{{expr,x^3-x,"f(x)="},ControlType -> InputField},
{{a,-5,"x min"},ControlType -> InputField },
{{b,5,"x max"},ControlType -> InputField },
{{h,5,"window height"},ControlType -> InputField},
{{c,-1.5,"c"},a+0.1,b-0.1,Appearance->"Labeled" }]

DerivativeAsFunction

DerivativeAsFunction

Experiment 4.1

Use the DerivativeAsFunction module on the following functions, using the given range on the x-axis. Sketch a graph of f ‘(x). Then state the number of “peaks” + number of “valleys” for both f(x) and f ‘(x) and at which value of c (approximately) these peaks and valleys occur.

1(a). f(x) = (x^2)-x-2, xmin = -3, xmax = 3, window height = 10
1(b). f(x) = (x^3)-(x^2)-6x, xmin = -4, xmax = 4, window height = 20
1(c). f(x) = (x^4)-(x^3)-5(x^2)-x+1, xmin = -4, xmax = 4, window height = 20
1(d). f(x) = (x^5)-12(x^2), xmin = -4, xmax = 4, window height = 30

Experiment 4.2

Use the DerivativeAsFunction module on the following functions, using the given range on the x-axis. Sketch a graph of f ‘(x) and answer the question for that number. (You may want to zoom in when thinking about your answer.)

2. f(x) = x * (e^x), xmin = -5, xmax = 2, window height = 2. Question: Is the derivative f ‘(c) less than f(c) for any value of c between x min and x max? If so, for which values of c?

3.  f(x) = (1/(1+(x^2))), xmin = -5, xmax = 2, window height = 2. Question: Is the derivative f ‘(c) ever equal to f(c) for any value of c between x min and x max? If so, for which values of c?

TraditionalForm[Sin[x]]