To answer these questions, we will need to follow a step-by-step approach. Let's go through each question one by one.
a. To find the average rate of change over the interval [-1,3], we will use the formula:
Average rate of change = (f(b) - f(a)) / (b - a)
Where a and b are the endpoints of the interval and f(x) is the given function.
In this case, a = -1, b = 3, and f(x) = x^2 - 6x + 12. Plugging in these values into the formula, we get:
Average rate of change = (f(3) - f(-1)) / (3 - (-1))
= [(3^2 - 6(3) + 12) - ((-1)^2 - 6(-1) + 12)] / 4
Simplifying further, we have:
Average rate of change = [(9 - 18 + 12) - (1 + 6 + 12)] / 4
= [3 - 19] / 4
= -16 / 4
= -4
Therefore, the average rate of change over the interval [-1,3] is -4.
b. To find f prime(x) using the definition of the derivative, we need to find the derivative of the given function f(x).
f(x) = x^2 - 6x + 12
To differentiate this function, we need to use the power rule for differentiation. According to the power rule, the derivative of x^n with respect to x is n*x^(n-1).
Differentiating each term in f(x) separately, we get:
f prime(x) = d/dx (x^2 - 6x + 12)
= d/dx (x^2) - d/dx (6x) + d/dx (12)
= 2x - 6
So, f prime(x) = 2x - 6.
c. To find the instantaneous rate of change at the point where x = 3, we can simply substitute x = 3 into the derivative function f prime(x).
f prime(3) = 2(3) - 6
= 6 - 6
= 0
Therefore, the instantaneous rate of change at the point where x = 3 is 0.
d. To find the value of x on the interval [-1,3] such that f prime(x) = -4, we need to set the derivative function f prime(x) equal to -4 and solve for x.
f prime(x) = -4
2x - 6 = -4
Adding 6 to both sides:
2x = 2
Dividing both sides by 2:
x = 1
Therefore, the value of x on the interval [-1,3] such that f prime(x) = -4 is x = 1.