Question

Given the function f(x)=3x^2+4x−4

find the following.

(a) the average rate of change of f on [−4,0].

(b) the average rate of change of f on [x,x+ℎ].

Answer 1

(a) -8

(b) 6x +4 +3h

Explanation:

Sometimes, it works well to find the general solution first, then use that to solve the specific problem. That's what we'll do here.

(b)

The average rate of change on the interval [x, x+h] is defined as ...

m = (f(x+h) -f(x))/h

For the given f(x), this is ...

`m=(((3(x+h)^2+4(x+h)-4)-(3x^2+4x-4))/(h)\\\\m=(3x^2 +6xh+3h^2+4x+4h-4-3x^2-4x+4)/(h)\\\\m=(6xh+3h^2+4h)/(h)\\\\\boxed{m=6x+4+3h}\qquad\text{average rate of change on $[x,x+h]$}`

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(a)

Using the above result for the interval [-4. 0], we have x=-4 and h=4. The average rate of change on that interval is ...

m = 6(-4) +4 +3(4) = -24 +4 +12 = -8

The average rate of change on [-4, 0] is -8.

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Additional comment

In the above, we worked the generic problem first. That way, we only needed to expand f(x+h)-f(x) once. If you're not terribly comfortable with doing algebra on literal expressions, sometimes it is more comfortable to work the numerical problem--as this question would have you do. This lets you get an idea of the kinds of operations that are involved and the way terms combine.

We note that the average rate of change is the instantaneous rate of change (derivative of the function) at the midpoint of the interval. That derivative is the value of m when h=0: 6x+4. The midpoint of the interval [-4,0] is x=-2, and the derivative there is 6(-2)+4 = -8, as above.