Consider the function f(x)=1/2(x+1)^2-3. Use the preceding/following interval method to estimate the instantaneous rate of change at 𝑥 = 1.

To estimate the instantaneous rate of change at 𝑥 = 1, we can use the preceding/following interval method.

The instantaneous rate of change is given by the derivative of the function. Let's find the derivative of the function f(x):

f(x)=(1/2)(x+1)^2-3

Using the power rule for differentiation, we can find the derivative:

f'(x) = 2\cdot (1/2)(x+1)^{2-1} = (x+1)

Now, to estimate the instantaneous rate of change at 𝑥 = 1, we will find the average rate of change of the function on either side of 𝑥 = 1.

Let's find the average rate of change on the interval (0, 1):

Average \ rate \ of \ change = \frac{f(1) - f(0)}{1 - 0}

Substituting the values into the equation:

Average \ rate \ of \ change = \frac{(1/2)(1+1)^2-3 - [(1/2)(0+1)^2-3]}{1}

Simplifying the equation:

Average \ rate \ of \ change = \frac{(1/2)(2)^2-3 - (1/2)(1)^2-3}{1}

Average \ rate \ of \ change = \frac{(1/2)(4)-3 - (1/2)(1)-3}{1}

Average \ rate \ of \ change = \frac{2-3 - 1/2-3}{1}

Average \ rate \ of \ change = \frac{-1/2-7/2}{1}

Average \ rate \ of \ change = \frac{-8}{2}

Average \ rate \ of \ change = -4

Similarly, let's find the average rate of change on the interval (1, 2):

Average \ rate \ of \ change = \frac{f(2) - f(1)}{2 - 1}

Substituting the values into the equation:

Average \ rate \ of \ change = \frac{(1/2)(2+1)^2-3 - [(1/2)(1+1)^2-3]}{2-1}

Simplifying the equation:

Average \ rate \ of \ change = \frac{(1/2)(3)^2-3 - (1/2)(2)^2-3}{1}

Average \ rate \ of \ change = \frac{(1/2)(9)-3 - (1/2)(4)-3}{1}

Average \ rate \ of \ change = \frac{9/2-3 - 2-3}{1}

Average \ rate \ of \ change = \frac{9/2-6 - 5}{1}

Average \ rate \ of \ change = \frac{9/2-12/2 - 5}{1}

Average \ rate \ of \ change = \frac{-3/2-5}{1}

Average \ rate \ of \ change = \frac{-13/2}{1}

Average \ rate \ of \ change = -\frac{13}{2}

Therefore, the estimated instantaneous rate of change at 𝑥 = 1 is -4 on the interval (0, 1) and -13/2 on the interval (1, 2).

Answer: The estimated instantaneous rate of change at 𝑥 = 1 is -4 and -13/2 on the intervals (0, 1) and (1, 2), respectively.