Mathematica Labs | Denis Shubleka

Subject: Calculus

Topic: Derivatives Graphically

Goal: Use the Manipulate command to visualize the derivative as a function

Task 1

The interactive window shows that as h approaches zero, the graph of the 
quotient approaches that of -Sin[x], verifying that the derivative of Cos[x]
is -Sin[x]. Feel free to alter the expression below using another function.

Manipulate[Plot[{((Cos[x+h]-Cos[x])/h),-Sin[x]},{x,-2π,2π}],{h,1,0.01}]

Now we plot y=2^x and its tangent line at the point (1, 2), and zoom in
closely at (1,2). What do you notice?

f[x_]:=(2^x);
Manipulate[Plot[{f[x],f'[1]*(x-1)+f[1]},{x,1-m,1+m},
Frame->True, Axes->False, Epilog ->{Red,Point[{1,2}]},
GridLines->{Range[0,3,0.05],Range[-1,10,0.2]},GridLinesStyle->Gray,
FrameTicks ->None,Filling->{1->{2}}],{{m,1,"zoom"},1,0.1}]

Related Exercises/Notes:

ap-calc.github.io