SOLUTION: To find the slope of g at the point where x = 6, we must look at the
point on f where y = 6, namely, (2,6). Since
Example 35 __
Let y = f(x) = x^3 + x − 2, and let g be the inverse function. Evaluate g′(0).
SOLUTION: Since f ′(x) = 3x^2 + 1, g′(y) = . To find x when y = 0,
we must solve the equation x^3 + x − 2 = 0. Note by inspection that x = 1, so
I. THE MEAN VALUE THEOREM
If the function f(x) is continuous at each point on the closed interval a ≤ x ≤ b
and has a derivative at each point on the open interval a < x < b, then there is at
least one number c, a < c < b, such that . This important theorem,
which relates average rate of change and instantaneous rate of change, is
illustrated in Figure N3–9. For the function sketched in the figure there are two
numbers, c 1 and c 2 , between a and b where the slope of the curve equals the
slope of the chord PQ (i.e., where the tangent to the curve is parallel to the
secant line).