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Average rate of change on [1,4]

Consider the function f(x)=x2−3x+2f(x) = x^2 - 3x + 2 on the interval [1,4][1, 4].

Find the average rate of change of f(x)=x2−3x+2f(x) = x^2 - 3x + 2 on the interval [1,4][1, 4].

Hint

The average rate of change on [a,b][a,b] is f(b)−f(a)b−a\dfrac{f(b) - f(a)}{b - a}. Compute f(1)f(1) and f(4)f(4) first, then form the difference quotient.

Step-by-step solution

Evaluate the function at the endpoints: f(1)=12−3(1)+2=0f(1) = 1^2 - 3(1) + 2 = 0 and f(4)=42−3(4)+2=6f(4) = 4^2 - 3(4) + 2 = 6.

The average rate of change is f(4)−f(1)4−1=6−03=2\dfrac{f(4)-f(1)}{4-1} = \dfrac{6 - 0}{3} = 2.

Equation of the secant line

Using the two points on the graph of f(x)=x2−3x+2f(x) = x^2 - 3x + 2 corresponding to x=1x=1 and x=4x=4, find the equation of the secant line through these points.

Hint

The slope of this secant line is the average rate of change from the previous part. Use point-slope form with one of the two points (1,f(1))(1, f(1)) or (4,f(4))(4, f(4)).

Step-by-step solution

From the previous part, f(1)=0f(1) = 0, f(4)=6f(4) = 6, and the slope of the secant line is m=2m = 2.

Using point-slope form with the point (1,0)(1,0): y−0=2(x−1)y - 0 = 2(x - 1), which simplifies to y=2x−2y = 2x - 2.

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