7.Find the equation of the tangent to the graph of y = at (4, 7).8.Find the values of x where the tangent to the graph of y = 2x^3 − 8x has a slope equal to the slope of y
= x.9.Find the equation of the normal to the graph of y = at x = 3.10.Find the values of x where the normal to the graph of (x − 9)^2 is parallel to the y-axis.11.Find the coordinates where the tangent to the graph of y = 8 − 3x − x^2 is parallel to the x-axis.12.Find the values of a, b, and c where the curves y = x^2 + ax + b and y = cx + x^2 have a common
tangent line at (−1, 0).THE MEAN VALUE THEOREM FOR DERIVATIVES
If y = f(x) is continuous on the interval [a, b], and is differentiable everywhere on the interval (a, b),
then there is at least one number c between a and b such thatf′(c) = Remember that in order for The Mean Value Theorem for
Derivatives to work, the curve must be continuous on the
interval and at the endpoints.In other words, there’s some point in the interval where the slope of the tangent line equals the slope of
the secant line that connects the endpoints of the interval. (The function has to be continuous at the
endpoints of the interval, but it doesn’t have to be differentiable at the endpoints. Is this important? Maybe
to mathematicians, but probably not to you!) You can see this graphically in the following figure: