Prep Session: The Big Theorems
Many theorems in mathematics are expressed as conditional statements
in the form “If
hypothesis
, then
conclusion
.”
The
converse
of a conditional statement is found by exchanging the
hypothesis and the conclusion of the con
ditional.
The
inverse
of a conditional statement is found by negating the
hypothesis and conclusion of the statement.
The
contrapositive
of a conditional statement is found by exchanging
AND negating the hypothesis and conclusion.
A
counterexample
is an
example that proves a statement false.
Definitions in mathematics are usually expressed as
biconditional
statements. Biconditional statements are statements of the form
“
Hypothesis
is true if and only if
conclusion
is true.” Because of the
nature of th
e biconditional, the following is also true “
Conclusion
is true if
and only if
hypothesis
is true.”
Consider the following important theorem:
Differentiability implies Continuity
If a function is differentiable at a point, then it is continuous at the
point.
What is the converse of the statement? Is the converse true or false?
What is the inverse of
the statement? Is the inverse true or false?
What is the contrapositive of the statement? Is the contrapositive true or
false?
The Big Theor
ems
Intermediate Value Theorem
If f is continuous on a closed interval [a, b] and
,
then for every value of M
between
and
, there exist at least one value of c in the open interval (a, b)
such that
Extreme Value Theorem
If f is continuous on a closed interval [a, b], then
f
takes on a maximum and a minimum
value on that interval.
Mean Value Theorem
If
f
is continuous on the closed interval [a, b], and differentiabl
e on the open interval (a,
b), then there exists a number c in the open interval (a, b) such
t
hat
Rolle’s Theorem
If f is continuous on the closed interval [a, b], and differentia
ble on the open interval (a,
b)
, then there is at le
ast one value of c in the open interval (a, b) such
that
and
Increasing/Decreasing Theorem
If f is continuous on the closed interval [a, b], and differentiable on the open interval (a,
b), and if for all c in t
he open interval (a, b),
, then f is increasing
(decreasing
)
on the closed interval [a, b].
First Derivative Test
If f is differentiable and c is a critical point of f, then if
changes from positive to
negativ
e at x = c, then
is a local maximum of f.
If f is differentiable and c is a
critical point of f, then if
changes from negative to positive at x = c, then
is a local minimum of f.
Se
cond Derivative Test
Let f be a function such that
and the second derivative of f exists on an open
interval containing c, then if
, f has
a local minimum value at x = c. If
, f has a
local m
ax
imum value at x = c.
If
, then the second
derivative test cannot be used.
Fundamental Theorem of Calculus
If
then
.
L’Hopital’s Rule
For the indeterminate forms 0/0 or ∞/∞,
.
2007 BC 6a Form B
The function
g
is continuous for all real numbers
x
and is defined
by
for
.
(a) Use L’Hospital’s Rule to find the value of
g
(0). Show the work that
leads to your an
swer.
2008 AB 6d
Let f be the function given by
for all x > 0….
(d) Find
.
2003 BC 6a
The function
f
is defined by the power series
for all real numbers
x
.
(a) Find
and
. Determine whether
f
has a local minimum, a
local maximum or neither at
x
= 0. Give a reason for your answer.
2005 BC 3a Form B
The Taylor Series about
x
= 0 for a certain function
f
converges to
f
(
x
) for
all
x
in the interval of convergence. The nth derivative of
f
at
x
= 0 is
given by
for
. The graph of
f
has a horizontal
tangent at
x
= 0, and
.
(a) Determine whether
f
has a rel
ative maximum, a relative minimum, or
neither at
x
= 0. Justify your answer.
2006 BC 6b
The function
f
is defined by the power series
for all real numbers
x
for which the
series converges. The function
g
is defined by
the power series
for all real numbers
x
for which the
series converges.
(b) The graph of
passes through the point (0,

1).
Find
and
. Determine whether
y
has a relative minimum, a
relative maximum, or neither at
x
= 0. Give a reason for your answer.
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