for each case, find the ARC over the given interval. a) f(x) = x^4 - x^3 + x^2, [-1,1] b) f(x) = (2x - 1)/(2x + 1), [0,2]

Sure, let's calculate the Average Rate of Change (ARC) for each function over the given interval.

The formula for ARC is (f(b) - f(a)) / (b - a), where [a, b] is the interval.

a) For f(x) = x^4 - x^3 + x^2 over the interval [-1,1]:

First, calculate f(-1) and f(1): f(-1) = (-1)^4 - (-1)^3 + (-1)^2 = 1 - (-1) + 1 = 3 f(1) = 1^4 - 1^3 + 1^2 = 1 - 1 + 1 = 1

Then, substitute these values into the ARC formula: ARC = (f(1) - f(-1)) / (1 - (-1)) = (1 - 3) / (1 - (-1)) = -2 / 2 = -1

b) For f(x) = (2x - 1)/(2x + 1) over the interval [0,2]:

First, calculate f(0) and f(2): f(0) = (20 - 1)/(20 + 1) = -1/1 = -1 f(2) = (22 - 1)/(22 + 1) = 3/5

Then, substitute these values into the ARC formula: ARC = (f(2) - f(0)) / (2 - 0) = (3/5 - (-1)) / 2 = 8/5 / 2 = 0.8

So, the ARC for f(x) = x^4 - x^3 + x^2 over the interval [-1,1] is -1, and the ARC for f(x) = (2x - 1)/(2x + 1) over the interval [0,2] is 0.8.