Answer:
Explanation:
The average rate of change of a function on an interval is the ratio of the change in function value to the length of the interval.
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f(x)
The average rate of change of f(x) on the interval [5, 10] is ...
(f(10) -f(5))/(10 -5) = (-0.1(10²) -(-0.1(5²)))/(10 -5) = (-0.1(10 -5)(10 +5))/(10 -5)
= -0.1(10 +5) = -1.5 . . . . average rate of change of f(x)
g(x)
The average rate of change of g(x) is calculated the same way, and the simplification of the calculation is the same. The average rate of change of g(x) is ...
-0.4(10 +5) = -6 . . . . average rate of change of g(x)
ratio
The ratio of the average rates of change is ...
g'(x)/f'(x) = -6/-1.5 = 4
The rate of change of g(x) is 4 times that of f(x).
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Additional comment
In the above, we made use of the factoring of the difference of squares in order to simplify the expression: a² -b² = (a +b)(a -b). This let us cancel the denominator factor to show us an interesting fact about the rate of change of a quadratic function.
If we generalize the result we found above, we see that the average rate of change of h(x) = kx² on the interval [a, b] is ....
h'(x) = k(b +a) . . . . . squared term coefficient times the sum of the ends of the interval
You will notice that (b+a)/2 is the midpoint of the interval, so the average rate of change can also be expressed as ...
h'(x) = 2k × (midpoint of interval [a, b])