Derivatives Min/Max Values

Linear Approximation

Theorems Potpourri

$100 $100 $100 $100 $100

$200 $200 $200 $200 $200

$300 $300 $300 $300 $300

$400 $400 $400 $400 $400

DERIVATIVES

$100

!

f '(x) = 2x cos(x2) 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

If

!

f (x) = ln(2x), then its derivative is this function.

What is….

(a)

!

f '(x) =1

2x (b)

!

f '(x) = ln2

(c)

!

f '(x) =ln2

2x (d)

!

f '(x) =1

2xln2

DERIVATIVES $300

If

!

f (x) = xx , then its derivative is this function.

What is….

DERIVATIVES $400

If

!

f (x) = sec( 2x + e3" 5

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 '(c) =f (2) " f ("1/2)

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

If

!

y5

+ 3x2y2

+ 5x4

=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 change with time. Knowing that

!

V = 1

3"r2h , the relationship

Between the changes with respect to time in the volume, radius, and height is this. What is….

(a)

!

dV

dt=1

3" 2r

dr

dth + r

2 dh

dt

#

$ %

&

' ( (b)

!

dV

dt=1

3" 2r

dr

dth + r

2 dh

dt

#

$ %

&

' (

(b)

!

dV

dt=1

3" 2rh + r

2 dh

dt

#

$ %

&

' ( (d)

!

dV

dt=1

3" r

2(1) + 2r

dr

dhh

#

$ %

&

' (

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

!

e0.5is approximated by using the tangent line to the graph

of

!

f (x) = ex

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….