Question

y = y1 + (x - x1) / (x2 - x1) * (y2 - y1)
y = 0,517 + (75 - 100) / (50-100) * (0,579 – 0,517) =

Answer 1

Given ( , ), 0 0 x y ( , ), 1 1 x y and ( , ), 2 2 x y fit a quadratic interpolant through the data. Note that

y = f (x), ( ), 0 0 y = f x ( ), 1 1 y = f x and ( ), 2 2 y = f x assume the quadratic interpolant ( ) 2 f x

given by

( ) ( ) ( )( ) 2 0 1 0 2 0 1 f x = b + b x − x + b x − x x − x

At 0 x = x

( ) ( ) ( ) ( )( ) 0 2 0 0 1 0 0 2 0 0 0 1 f x = f x = b + b x − x + b x − x x − x

= b0

( ) 0 0 b = f x

At 1 x = x

( ) ( ) ( ) ( )( ) 1 2 1 0 1 1 0 2 1 0 1 1 f x = f x = b + b x − x + b x − x x − x

( ) ( ) ( ) 1 0 1 1 0 f x = f x + b x − x

then

1 0

1 0

1

( ) ( )

x x

f x f x b −

− =

At 2 x = x

( ) ( ) ( ) ( )( ) 2 2 2 0 1 2 0 2 2 0 2 1 f x = f x = b + b x − x + b x − x x − x

( ) ( )( ) ( ) ( ) ( ) ( ) 2 0 2 2 0 2 1

1 0

1 0

2 0 x x b x x x x

x x

f x f x f x f x − + − − −

− = +

then

2 0

1 0

1 0

2 1

2 1

2

( ) ( ) ( ) ( )

x x

x x

f x f x

x x

f x f x

b −

−

− − −

−

=

Hence the quadratic interpolant is given by

( ) ( ) ( )( ) 2 0 1 0 2 0 1 f x = b + b x − x + b x − x x − x

( )( )