Saturday, August 19, 2023

Casio fx-991CW and TI-30X Pro MathPrint: Arc Length of f(x)

Casio fx-991CW and TI-30X Pro MathPrint:  Arc Length of f(x)



Here is a tip for calculus class.  



Calculating The Arc Length of f(x)



The arc length of f(x) is defined as:


∫( √( 1 + (f'(x))^2 ) dx, x = a to x = b)



With the two function memory, f(x) and g(x), on the Casio fx-991CW and TI-30X Pro MathPrint, we can calculate arc length with the expression:  


∫( √( 1 + (g(x))^2 ) dx, x = a to x = b)


where g(x) = d/dx f(x)





Setup Instructions



Casio fx-991CW


To set up f(x) and g(x):


1.  Press [ f(x) ], select Define f(x) to define the function.  Press [ EXE ] to register it. 

2.  Press [ f(x) ], select Define g(x) to define the derivative.  Press [ EXE ] to register it.  

3.  Recall f(x) and g(x) by pressing [ f(x) ] and selecting the function.



Derivative: 


g(x)  =  ( (f(x + h) - f(x) )/h     [h is sufficiently small, like 0.001]


The integral function on the fx-991CW does not allow to have the derivative command in the integrand.  Therefore, the definition of the derivative must be used.  



TI-30X Pro MathPrint


To set up f(x) and g(x):


1.  Press [ table ].   Select Add/Edict Func.

2.  Edit f(x), then press [ enter ].

3.  Edit g(x), enter the derivative.  Press [ enter ].

4.  Press [ 2nd ] (quit) to exit Table mode. 

5.  Recall f(x) and g(x) by pressing [ table ] and selecting the function.


Derivative: 


g(x)  =  ( (f(x + h) - f(x) )/h     [h is sufficiently small, like 0.001]


 or


g(x) = d/dx( f(x) )|x=x


Note:  The TI-30X Pro MathPrint is a European version of the TI-36X Pro.   The TI-36X Pro has only one function memory slot.  


In any calculus problem, set the angle mode to Radians. 



Examples



1.  f(x) = x^3 + 1 from x = 0 to x = 3.  


For this first example, screens are from the Casio fx-991CW.






Arc length ≈ 27.65808204



2.  f(x) = sin x from x = 0 to x = π


Arc length ≈ 3.820197739



3.  f(x) = 4x^2 + 1 from x = 0 to x = 1


Arc length ≈ 1.478942858



Eddie


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