Ethan B. answered • 11/22/22
a. The net change of a function is just the value of the function at the greater endpoint minus the value of the function at the lesser endpoint, so for g(x) = sqrt(x + 2) from [-2,2] we would get sqrt(2 + 2) - sqrt(-2 + 2) = sqrt(4) - sqrt(0) = 2 - 0 = 2.
b. The average rate of change would be the net change divided by the range of the endpoints, which is 2 - (-2) = 4, so 2/4 = 1/2 = 0.5.
c. In order to find values of c in the interval [-2,2] that satisfy the mean value theorem we have to find all values where the derivative is equal to the mean value (0.5). So what is the derivative of sqrt(x + 2), well we can do the chain rule, if f(x) = sqrt(x) and h(x) = x + 2, how do we find the derivative of f(h(x)), well we can do f’(h(x))*h’(x). And f’(x) = 1/(2sqrt(x)) because sqrt(x) = x^(1/2) and we can use the power rule of multiplying by the power then subtracting 1 from the power so (1/2)*x^(-1/2) which is the same as (1/2)*(1/(x^(1/2))) because something a negative power is the one over that thing to the positive power and finally we get 1/(2sqrt(x)). Therefore we can use h’(x) = 1 to get f’(x) = 1/(2sqrt(x + 2)) and we want to find values of c such that f’(c) = 1/2 so let’s do that 1/(2sqrt(c + 2)) = 1/2 ==> sqrt(c + 2) = 1 ==> c + 2 = 1 ==> c = -1. Which is our answer.
d. This one is not really possible to show without a picture, but I can describe it. The secant line is just the line from (-2,0) to (2,2), and the tangent line at (c,g(c)) is a parallel line at the point (-1, 1).
I hope this helped. If you need more help, please feel free to reach out!
