Expression 10: "m" Subscript, "s" , Baseline equals StartFraction, "f" left parenthesis, "c" plus "h" , right parenthesis minus "f" left parenthesis, "c" , right parenthesis Over "h" , EndFractionms=fc+h−fch
equals=
33
10
Expression 11: "m" Subscript, "S" , Baseline equals StartFraction, "f" left parenthesis, "z" , right parenthesis minus "f" left parenthesis, "c" , right parenthesis Over "z" minus "c" , EndFractionmS=fz−fcz−c
equals=
33
11
We can express the secant line as a function in point slope form
12
Expression 13: "S" left parenthesis, "x" , right parenthesis equals "f" left parenthesis, "c" , right parenthesis plus "m" Subscript, "s" , Baseline left parenthesis, "x" minus "c" , right parenthesisSx=fc+msx−c
13
Desmos understands "prime" notation for derivatives, and can calculate derivatives, so we can calculate the slope of the tangent line at x=c
14
Expression 15: "m" Subscript, "t" , Baseline equals "f" prime left parenthesis, "c" , right parenthesismt=f′c
equals=
22
15
So we can graph the tangent line
16
Expression 17: "T" left parenthesis, "x" , right parenthesis equals "f" left parenthesis, "c" , right parenthesis plus "f" prime left parenthesis, "c" , right parenthesis left parenthesis, "x" minus "c" , right parenthesisTx=fc+f′cx−c
17
Hidden Label: left parenthesis, "c" plus "h" , "f" left parenthesis, "c" plus "h" , right parenthesis , right parenthesisc+h,fc+h
Label
18
Expression 19: "f" prime left parenthesis, "x" , right parenthesisf′x