Derivatives Min/Max
Linear
Theorems Potpourri
Values Approximation
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DERIVATIVES
$100
f '(x) = 2x cos(x 2 ) is the derivative of some function f(x).
What is….
(a) f(x)=sin(2x)
(b) f(x)=(2x)sin(2x)
(c) f(x)=sin(x2)
(d) f(x)=(2x)sin(x2)
DERIVATIVES
$200
x
f
(x)
=
ln(2
) , then its derivative is this function.
If
What is….
!
!
!
1
(a) f '(x) = 2 x
(b) f '(x) = ln2
ln2
(c) f '(x) = 2 x
1
(d) f '(x) = 2 x ln2
!
!
!
DERIVATIVES
$300
x
f
(x)
=
x
If
, then its derivative is this function.
What is….
!
DERIVATIVES
$400
3
5
f
(x)
=
sec(
2x
+
e
"
If
x ) , then its derivative is this
function.
What is…
THEOREMS
$100
The following most clearly illustrates the Mean Value
Theorem applied to the situation of a car driving from Ithaca
to Watkin’s Glen.
What is….
(a) At some point, the acceleration of the car is equal to the rate of
change of its speed
(b) At some point, the car’s instantaneous speed is equal to its average
speed for the trip
(c) At some point, the car is exactly half way between Ithaca and
Watkin’s Glen
(d) At some point, the car’s distance from Ithaca is an absolute
maximum
THEOREMS
$200
DAILY DOUBLE!!!!!!!
If a function f is continuous on a closed interval [a,b] then
you are guaranteed this.
What is……
THEOREMS
$300
Whether or not a function f such that f(1)=-2, f(3)=0,
and f’(x)>1 exists.
What is….
(a) Does not exist – state why…..
(b) Does exist – here’s an example…….
THEOREMS
$400
True or false: For f (x) =| x | on the interval [-1/2, 2], you can
find a point c in (-1/2, 2) such that
f (2) " f ("1/2)
f '(c) =
2 " ("1/2)
!
What is…..
!
MIN/MAX VALUES
$100
TRUE OR FALSE: If f(x) is continuous on a closed interval,
then it is enough to look at the points where f’(x)=0 in order
to find its absolute maxima and minima.
What is _________? Because ________________
MIN/MAX VALUES
$200
The second derivative test finds this.
What is…..
MIN/MAX VALUES
$300
(1) Example of a function where f’(c)=0 but f does not have
a local maximum or minimum at c.
What is….
(2) True or false: If f has an absolute maximum at x=c then
f’(c)=0.
What is….
MIN/MAX VALUES
$400
TO THE BOARD!!!!!!!!
A graph of a function that has a local maximum at x=2 but
is not continuous at x=2.
POTPOURRI
$100
Two cars start moving from the same point. One travels
south at 60 mph and the other travels west at 25 mph. If you
need to find out what rate the distance between the cars
increasing two hours later, then you can use this to relate the
quantities.
What is…
(a) similar triangles
(b) trig functions to relate the angle to the side length
(c) Pythagorean theorem
(d) law of cosines
POTPOURRI
$200
5
2 2
4
If y + 3x y + 5x = 12 , then dy/dx is this.
What is….
!
POTPOURRI
$300
An article in the Wall Street Journal’s “Heard on the Street
Column” reported that investors often look at the “change in the
rate of change” to help them “get into the market before any big
rallies”. Your stock broker alerts you that the rate of change of a
stock’s price is increasing. You do the following as a result.
What is…
(a) can conclude the stock’s price is decreasing
(b) can conclude the stock’s price is increasing
(c) cannot determine whether the price is increasing or
decreasing
POTPOURRI
$400
As gravel is being poured into a conical pile, its volume V
changes with time. As a result, the height h and radius r also
2
1
V
=
"
r
h , the relationship
change with time. Knowing that
3
Between the changes with respect to time in the volume,
radius, and height is this.
What is….
!
!
!
(a)
dV 1 # dr
2 dh &
= " % 2r h + r
(
dt 3 $ dt
dt '
dV 1 # dr
2 dh &
(b) dt = 3 " %$2r dt h + r dt ('
(b)
dV 1 #
2 dh &
= " % 2rh + r
(
dt 3 $
dt '
dV 1 # 2
dr &
= " % r (1) + 2r
h(
dt 3 $
dh '
!
!
(d)
LINEAR APPROXIMATION
$100
Suppose that f ''(x) <0 for all x near a point, a. Then for x
near L(x), the linearization of f is this.
What is….
!
(a) an overestimate for f(x)
(b) an underestimate for f(x)
(c) unknown without more information
LINEAR APPROXIMATION
$200
The line tangent to the graph of f(x)=sin(x) at (0,0) is y=x.
This implies what relationship.
What is…
(a) sin(0.0005) is approximately 0.0005
(b) The line y=x touches the graph f(x)=sin(x) at exactly
one point (0,0)
(c) y=x is the best straight line approximation to the graph
of f(x) for all x
!
!
LINEAR APPROXIMATION
$300
If e 0.5 is approximated by using the tangent line to the graph
of
x
f (x) = e at (0,1), the approximation is this.
What is….
(a) 0.5
(b) 1 + e0.5
(c) 1 + 0.5
LINEAR APPROXIMATION
$400
Let f(x) be continuous with an absolute maximum at x=2.2.
If f(2)=3 and f’(2)=0.4, then this approximates the maximum
value of f(2.2).
What is….