Question

Estimate the value of 3 to the power of 2.2 using linear interpolation.

Answer 1

3^2=9
3^3=27

linear extrapolation on an exponential function? Goodness.
3^2.2= 9+.2(27-9)/(1)= 9+.2*18=12.6
and the audience says the real value is ...11.2

Answer 2

The linear interpolation formula is given as:

y = y1 + ( x − x1 ) ∙ ( y2 − y1 ) / ( x2 − x1 )

2.2 is between 2 and 3

Get the starting values:

x1 = 2 , y1 = 3^2 = 9 , x2 = 3 , y2 = 3^3 = 27

y = y1 + ( x − x1 ) ∙ ( y2 − y1 ) / ( x2 − x1 )

y = 9 + ( x − 2 ) ∙ ( 27 − 9 ) / ( 3 − 2 )

y = 9 + ( x − 2 ) ∙ 18 / 1

y = 9 + ( x − 2 ) ∙ 18

y = 9 + 18 x − 36

y = - 27 + 18 x

y = 18 x - 27

for x = 2.2

y = 18 ∙ 2.2 - 27 = 39.6 - 27 = 12.6

3^2.2 ≈ 12.6


bobpursley

This is a homework.
No exact value is required.
That is used to practice interpolation.