If a diver of mass m stands at the end of a diving board wit

If a diver of mass m stands at the end of a diving board with length L and linear density p, then the board takes on the shape of a curve y=fx), where Ely\" = mg(L-x) + 2pg(L-x)2 E and I are positive constants that depend on the material of the board and g (

Solution

2)

let y=f(x)

replace x with \"a\"

the linear approximation at x=a is given by

y = f1(x)

y = f(a)+f\'(a)*(x-a)

tan(x) = x

f(x)=tan(x)

at x=0
f(0)=tan(0)

f(0)= 0

f\'(x)=1/cos²(x)

f\'(0)=1/cos²(0)
f\'(0)= 1
y=0+1*x
y=x is the linear approximation

The error due to the approximation is:
R=|f(x)-f1(x)|

R=|tan(x)-x|
|tan(x)-x|<0.1

For solving inequality,

find root
tan(x) - x=0.1

interval (0,pi/2)
tan(x) - x - 0.1 = 0,
zero of F(x)=tan(x)- x - 0.1
F\'(x)=1/cos²x-1

x(n+1) = x(n) - F(x(n)) / F\'(x(n))
x(0) = 1
x(1) = 1-(tan1-1 - 0.1) / (1/cos²1-1)

=0.8114

x(2) = 0.8114 - (tan 0.8114 - 0.8114 - 0.1) / (1/cos²0.8114 - 1)
x(2)=0.6694

x(3) = 0.6694 - (tan 0.6694 - 0.6694 - 0.1) / (1/cos²0.6694 - 1)
x(3)=0.63446

x(4) = 0.63446 - (tan 0.63446 - 0.63446 - 0.1) / (1/cos²0.63446 - 1)
x(4) = 0.63168
x=0.632

tan(x) = ( -0.632 , 0.632 )