Find two or three classmates and a few feet of chalkboard. As a group, try
your hand at the
following exercises. Be sure to discuss how to solve the exercises - how you get
the solution is
much more important than whether you get the solution. If as a group you agree
that you all
understand a certain type of exercise, move on to later problems. You are not
expected to solve all
the exercises: in particular, the last few exercises may be very hard.
Many of the exercises are from Single Variable Calculus : Early
Transcendentals for UC Berkeley
by James Stewart, these are marked with an
.
Others are my own, or are independently marked.
Derivatives and graphing
1.
To the right, the graph of the
derivative f' of a function f is
shown. On what intervals is f increasing?
Decreasing? Where is f
concave up? Concave down? At
what values does f have a local
maximum or minimum? Inflection
points?
2.
Sketch the graph of a function f such that f'(1) = f'(-1) = 0, f'(x) < 0 if |x|
< 1, f'(x) > 0
if 1 < |x| < 2, f'(x) = -1 if |x| < 2, f''(x) < 0 if -2 < x < 0, and such that
(0, 1) is an
inflection point of y = f(x).
3.
Suppose that f(x) = 2, f'(3) = 1/2 , and f'(x) > 0 and f''(x) < 0 for all x.
(a) Sketch a possible graph of f.
(b) How many solutions does the equation f (x) = 0 have?
(c) Is it possible that f'(2) = 13?
4.
For each of the following functions, find: intervals when f is increasing,
intervals when f is
decreasing, intervals when f is concave up, intervals when f is concave down,
local extreme
of f, inflection points. Then sketch a graph of the function
5. Sketch a careful graph of
.
Label any interesting features (intercepts,
asymptotes, extrema, points of inflection).
6. (a) Let
.
Use calculus to sketch a graph of f(x), and label the zeros, local
extrema, and inflection points. Also label the y- intercept and any horizontal
asymptotes.
(b) Let
.
What is the behavior of f(x) as
?
Use the Mean
Value Theorem to show that if f(x) has one or two zeros , then it must have two
local
extrema.
(c) More generally, let f(x) = p(x)ex, where p(x) is a polynomial of degree
n. Show that
if f(x) has exactly n ( real , distinct) zeros, then it also has exactly n local
extrema and
exactly n inflection points.
(d) (Harder) Let's return to the case when
.
Prove that the zeros of
f correspond to the zeros of
,
and the number of these is
classified by the determinant
.
For n a non- negative integer , define qn(x) to be f(n)(x)=ex, the polynomial
part of the
nth derivative of ex. Prove that qn(x) is a quadratic for any n.
How if the de terminant of q 1(x) related to the determinant of q0(x)? Is this the
same as
the relationship between the determinants of qn(x) and qn+1(x) for arbitrary n?
Why
or why not?
Prove that for n large enough, qn(x), and hence
f(n)(x), will have two roots .
7.
Suppose the derivative of a function f is 
.
On what intervals
is f increasing? What are the local maxima of f?
8.
Use calculus to sketch the family of curves 
,
where a is a positive
constant.
9.
Find the value of x such that 
increases most
rapidly.
10.
Find a cubic function 
that has a local
maximum value of 3 at
x = -2 and a local minimum value of 0 at x = 1.
11.
For what values of the numbers a and b does the function
have the maximum
value f(2) = 1?
12.
Show that the curve 
has three points of
inflection and that they all lie on one
straight line.
13.
Show that the curves 
and
touch the curve
sin x at its inflection
points.
14.
Show that tan x > x for 
. Hint: show that
is increasing on
(0, π/2).
15.
Show that a cubic function always has precisely one point of inflection. Show
that if the
graph has three x-intercepts x1, x2, and x3,
then the x- coordinate of the inflection point is
(x1 + x2 + x3)/2.
What is the similar statement about local extrema of a
quadratic function?
16.
(a)
Show that f(x) = x^4 is such that f''(0) = 0 but (0, 0) is not an inflection
point of the
graph of f.
(b)
Show that g(x) = x|x| has an inflection point at (0, 0) but g''(0) does not
exist.
(c)
Let f be any function. Use the First Derivative Test and Fermat's Theorem on the
function g = f' to show that if (c, f(c)) is an inflection point and f'' exists
in an open
interval that contains c, then f''(c) = 0.
