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QuestionCalculate the average rate of change of f(x)=x2−2x+8f(x)=x^{2}-2x+8 from x1=1x_{1}=1 to x2=4x_{2}=4.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x−x+8f(x) = x^{} -x +8 . The initial x-value is x1=1x_{1} =1
3. The final x-value is x=4x_{} =4

STEP 2

The average rate of change of a function from x1x_{1} to x2x_{2} is given by the formulaAverage rate of change=f(x2)−f(x1)x2−x1Average\, rate\, of\, change = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}

STEP 3

First, we need to find the values of the function at x1x_{1} and x2x_{2}. Let's start with x1x_{1}.
f(x1)=f(1)=(1)2−2(1)+8f(x_{1}) = f(1) = (1)^{2} -2(1) +8

STEP 4

Calculate the value of f(x1)f(x_{1}).
f(x1)=(1)2−2(1)+8=1−2+8=7f(x_{1}) = (1)^{2} -2(1) +8 =1 -2 +8 =7

STEP 5

Now, let's find the value of the function at x2x_{2}.
f(x2)=f(4)=(4)2−2(4)+8f(x_{2}) = f(4) = (4)^{2} -2(4) +8

STEP 6

Calculate the value of f(x2)f(x_{2}).
f(x2)=(4)2−2(4)+8=16−8+8=16f(x_{2}) = (4)^{2} -2(4) +8 =16 -8 +8 =16

STEP 7

Now that we have the values of f(x1)f(x_{1}) and f(x2)f(x_{2}), we can substitute these into the formula for the average rate of change.
Average rate of change=f(x2)−f(x1)x2−x1=16−74−1Average\, rate\, of\, change = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} = \frac{16 -7}{4 -1}

STEP 8

Calculate the average rate of change.
Average rate of change=16−74−1=3=3Average\, rate\, of\, change = \frac{16 -7}{4 -1} = \frac{}{3} =3The average rate of change of the function from x1=1x_{1} =1 to x2=4x_{2} =4 is3.

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