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Through a change of variable (specify it!), the Gaussian probability distribu-
tion function can be written as a function of the normalized distribution
function,

 x − a 
Fx() = F X
X   σ X  

where

 ξ

Fx( ) = 1 ∫ x exp − 2   dξ
2π −∞  2 
a. Develop the function M-file for the normal distribution function.
b. Show that for negative values of x, we have:

F(–x) = 1 – F(x)

c. Plot the normalized distribution function for values of x in the
interval 0 ≤ x ≤ 5.

Pb. 4.35 The computation of the arc length of a curve can be reduced to a
one-dimensional integration. Specifically, if the curve is described parametri-
cally, then the arc length between the adjacent points (x(t), y(t), z(t)) and the
point (x(t + ∆t), y(t + ∆t), z(t + ∆t)) is given by:

 dx 2  dy 2  dz 2
∆s = + + ∆t
 dt   dt   dt 

giving immediately for the arc length from t to t , the expression:
1
0

s = ∫ t 1   dx 2 +   dy 2 +   dz 2 dt
t 0 dt  dt  dt 

a. Calculate the arc length of the curve described by: x = t and y =
2
t between the points: t = 0 and t = 3.
3
b. Assuming that a 2-D curve is given in polar coordinates by r = f(θ),
and then noting that:

x = f(θ) cos(θ) and y = f(θ) sin(θ)

use the above expression for the arc length (here the parameter is
θ) to derive the formula for the arc length in polar coordinates to be

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