Question

Consider the function f(x)=7−7x² on the interval [−3,8]. Find the average or mean slope of the function on this interval, i.e.

f(8)−f(−3)/8−(−3) = ___ .
By the Mean Value Theorem, we know there exists a c in the open interval (−3,8) such that f
′ (c) is equal to this mean slope. For this problem, there is only one c that works. Find it.

Answer 1

The average slope of the function f(x)=7−7x² on the interval [−3,8] is −35. By the Mean Value Theorem, the value c that makes f′(c) equal to this mean slope is 2.5.

Step-by-step explanation:

To find the average slope of the function f(x) = 7 - 7x2 over the interval [−3,8], we use the slope formula which is the change in y divided by the change in x. We calculate it as follows:

f(8) - f(−3) = (7 - 7(8)2) - (7 - 7(−3)2)
= (7 - 448) - (7 - 63)
= -441 - (-56)
= -441 + 56
= -385.

Now, we find the change in x:

8 - (−3) = 8 + 3 = 11.

So, the average slope is:

−385 / 11 = −35.

According to the Mean Value Theorem, there exists at least one value c in the interval (−3,8) where the derivative of the function is equal to this mean slope. The derivative of f(x) is f′(x) = -14x. To find the value of c, we set the derivative equal to the mean slope:

−14c = −35
c = −35 / −14c = 2.5.

Therefore, at x = 2.5 within the interval, the instantaneous slope of the function equals the average slope of the function over the interval [−3,8].