55
Math 111 – Calculus I.
Week Number Ten Notes
Fall 2003
I.
The Mean Value Theorem
We will now state a theorem that explicitly shows us how important derivatives are to
understanding the behavior of a function. It is called the Mean Value Theorem.
Theorem 10.1 (sec 4.3 of text, p. 281 – THE MEAN VALUE THEOREM): Assume f is a
differentiable function on [a,b]. Then, there exists a c on (a,b) such that
f' (c) 
f(b) - f(a)
or equivalent ly f' (c)(b - a)  f(b) - f(a)
b-a
Pictorial Interpretation of the Mean Value Theorem
(i) Slope of tangent line at local max
agrees with slope of secant line to
function f at interval endpoints
(ii) Slope of tangent line to f(x) = x2
at (1,1) agrees with slope of secant
line to f between (0,0) and (2,4)
(closed x-interval [0,2])
Here is one very practical application of the mean value theorem used by police officers
in the State of Pennsylvania on the PA Turnpike.
Example 10.2: Assume I get on the PA Turnpike at the Toll Plaza near Valley Forge at
10:00 PM. Assume that I exit 90 miles later at the Wilkes-Barre/Route 315 Toll Plaza at
11:10 PM. Assume that there is an officer at the toll plaza in Wilkes-Barre and the
officer gives me a speeding ticket. Will this “stand up in court”? (YES/NO and
JUSTIFY)
56
The Mean Value Theorem also allows us to formally prove a fact we have already been
utilizing. Here it is.
Theorem 10.3: Assume f is differentiable on an interval [a,b] and that the derivative of f
is positive (respectively negative) on (a,b). Then, f is increasing (respectively
decreasing) on (a,b).
Proof: Assume x < y are two values on (a,b). The Mean Value Theorem tells us that
there is a c between x and y such that
(*)
f(y) - f(x)  f' (c)(y - x).
By assumption we know that the derivative of f of c is positive and so the RHS of
equation (*) above is positive. Thus, f(y) – f(x) is positive and so f(y) > f(x) which
implies that f is increasing as required. A similar proof yields that if x > y then f(x) > f(y)
and so f is decreasing on (a,b).
We are now in a position that to state some results that will be invaluable in helping us
graph functions.
II.
The First and Second Derivative Tests
The First Derivative Test: Assume c is a critical number of a function f that is
continuous on [a,b] and differentiable on [a,b] except possibly at c.
(1) If the derivative of f changes from positive (negative) to negative (positive) at c,
then f has a local maximum (minimum) at c.
(2) If the derivative of f does not change sign at c, then c is NOT a local extreme value
of f.
57
Now some more terminology for concepts we have already studied.
Definition 10.4: A function f is concave upward (respectively downward) on an interval
(a,b) if the derivative of f is increasing (respectively decreasing) on that interval.
Equivalently, if f is twice differentiable on (a,b), f is concave upward (respectively
downward) if the second derivative of f is positive (respectively negative) on (a,b). A
value c for which the second derivative of f is zero (i.e. it is a critical point of the
derivative of f) and the function f changes concavity from upward to downward (or vice
versa) is called an inflection point of f.
Here is a technique virtually equivalent to the First Derivative Test called the Second
Derivative Test.
The Second Derivative Test: Assume f is twice differentiable and the second derivative
of f is continuous on an open interval about a real number c.
(1) If f' (c)  0 and f' ' (c)  0, then f has a local minimum at c.
(2) If f' (c)  0 and f' ' (c)  0, then f has a local maximum at c.
III. Techniques for Graphing Functions
Here is my checklist for analyzing/graphing a function f.
I.
Analysis of f
1.
2.
3.
4.
What is f’s domain?
On what intervals is f positive, f negative, f zero.
Symmetries of f (y-axis,origin)
Vertical/Horizontal Asymptotes of f
II.
Analysis of the derivative of f
1. Find all critical points of f.
2. Locate local extreme values of f using the first derivative test
3. On what intervals is f increasing, f decreasing.
III.
Analysis of the second derivative of f
1. Find all critical points of f.
2. Locate all inflection points of f.
3. On what intervals is f concave upward, concave downward
58
Example 10.5: Using the above checklist, analyze and sketch the following functions.
Also, use you graphing calculator to aid this analysis.
(a) f(x) 
x
x 9
2
(b) g(x)  ln(sin 2 (x))
59
Problems Associated with Week #10 Notes – Sections 4.3 –4.4
Section 4.3: 1,3,5,7-33 odd, 38,39,47,48
Section 4.4: 1-8,10,11,17,21,22
Read Section 4.5-4.7 of the textbook