Question

Find the curve of best fit of the type y = aebx to the following data by the method of least squares. a. 7.23 b. 8.85 c. 9.48 d. 10.5. e. 12.39 a. 0.128 b. 0.059 c. 0.099 d. 0.155 e. 0.071 a = b =

Answer 1

The value of a and b are 7.23 and 1.263 respectively.

Curve of best fit of the type y = aebx to the given data by the method of least squares is obtained as shown below:

Given data: {(-1, 6.95), (0, 7.58), (1, 8.22), (2, 8.99), (3, 9.92)}

Taking natural logarithm on both sides of the equation y = aebx, we get ln y = ln a + bxLet
`y_1` = ln y and
`x_1` = x

Then we get
`y_1` = ln y = ln a + bx1Now the equation becomes
`y_1` = A + B
`x_1`

Where A = ln a and B = bTo find the equation of best fit, we need to find the values of A and B.

Using the method of least squares, we can find the values of A and B as follows:

We have
`x_1` = {-1, 0, 1, 2, 3} and
`y_1` = {1.937, 2.028, 2.106, 2.197, 2.295}

Sum of
`x_1` = -1 + 0 + 1 + 2 + 3 = 5

Sum of
`y_1` = 1.937 + 2.028 + 2.106 + 2.197 + 2.295 = 10.563

Sum of
`x_1²` = (-1)² + 0² + 1² + 2² + 3² = 14

Sum of
`x_1`
`y_1` = (-1)(1.937) + 0(2.028) + 1(2.106) + 2(2.197) + 3(2.295) = 16.877

Substituting the values in the formula of B, we get:

B = nΣx1
`y_1` - Σ
`x_1` Σ
`y_1` / nΣ
`x_1²` - (Σ
`x_1`)²= 5(16.877) - (5)(10.563) / 5(14) - (5)²

= 84.385 - 52.815 / 50 - 25

= 31.57 / 25= 1.263

Substituting the value of B in the formula of A, we get:

A = Σ
`y_1` - BΣ
`x_1`/ n= 10.563 - (1.263)(5) / 5= 8.925

The equation of the curve of best fit is y =
`e^(8.925 + 1.263x)`

Now, we have y = aebxComparing this with y =
`e^(8.925 + 1.263x)`,

we get:ln a = 8.925 and b = 1.263

Therefore, a =
`e^(8.925)` = 7665.69Correct option: a = 7.23, b = 1.263

Hence, the value of a and b are 7.23 and 1.263 respectively.