Math 112
Review for Exam I
This test is all about derivatives!
•f0(m) is the slope of the tangent line to f(x) at x=m
•The derivative of distance is the instantaneous speed. That is, instantaneous speed is the
slope of a tangent line to the graph of distance.
•The derivative of T R is MR. That is, you can think of MR as the slope of a tangent line
to the graph of T R.
•The derivative of T C is M C.
We have two methods for computing derivatives. You must be able to do both!
•the long way:
To compute f0(m), compute the slope of the secant line through f(x) at x=mand x=m+h.
slope of secant = f(m+h)−f(m)
h.
Simplify this expression and let hgo to 0 to get the slope of the tangent line, f0(m).
•using the derivative rules (This should be your default method — do this unless you’re told
otherwise.)
Given the graph of f(x), you should be able to determine the general shape of f0(x):
•If f(x) is increasing, then f0(x) is positive (the graph of f0(x) is above the x-axis).
•If f(x) is decreasing, then f0(x) is negative (the graph of f0(x) is below the x-axis).
•If f(x) has a horizontal tangent, then f0(x) = 0 (the graph of f0(x) is hitting the x-axis).
Given the graph of f0(x), you should be able to determine the general shape of f(x):
•If f0(x) is positive, then f(x) is increasing.
•If f0(x) is negative, then f(x) is decreasing.
•If f0(x) = 0, then f(x) has a horizontal tangent line at x.
