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# 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.

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## Answer to a math question 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.

Frederik
4.6
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.

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