ap exam review (chapter 2) differentiability. ap exam review (chapter 2) product rule

AP Exam Review (Chapter 2)

has a vertical tangent at 2,0 and horizontal tangents at

1,-1 and 3,1 . For what values of x, in the open interval

-2,4 , is not differentiable?

f

f

(A) 0 only

(B) 0 and 2 only

(C) 1 and 3 only

(D) 0, 1, and 3 only

(E) 0, 1, 2, and 3

Differentiability


Let and be differentiable functions with the following

properties:

f g

(i) ( ) 0 for all

(ii) (0) 1

g x x

f

If ( ) ( ) ( ) and '( ) ( ) '( ), then ( )h x f x g x h x f x g x f x

(A) '( ) (B) ( ) (C) (D) 0 (E) 1 xf x g x e

Product Rule

is a constant functionf

' ' 'h f g fg ' 0f g


2

What is the instantaneous rate of change at 2 of the

2function given by ( ) ?

1

x

xf f x

x

Quotient Rule

1 1(A) 2 (B) (C) (D) 2 (E) 6

6 2

2 2

2

1 2'

1

4 1 2' 2 2

1

x x xf x

x

f


(A) 2 1 (B) 1 (C) (D) 1 (E) 0y x y x y x y x y

An equation of the line tangent to the graph of cos

at the point 0,1 is

y x x

Derivative of Cosine

Point-Slope Form

' 1 sin

' 0 1

1

y x

f

y x


( ) tan(2 ) '6

f x x f

(A) 3 (B) 2 3 (C) 4 (D) 4 3 (E) 8

Chain Rule

Derivative of Tangent

22

2' 2sec 2

cos 22

' 86 1/ 4

f x xx

f


Let be the function given by ( ) . Which of the

following statements about are true?

f f x x

f

I. is continuous at 0.f x II. is differentiable at 0.f x

III. has an absolute minimum at 0.f x

(A) I only (B) II only (C) III only (D) I and III only (E) II and III only

ContinuityDifferentiability

Minimum Extrema

T

FT


4 2

Which of the following is an equation of the line tangent to

the graph of ( ) 2 at the point where '( ) 1?f x x x f x

(A) 8 5

(B) 7

(C) 0.763

(D) 0.122

(E) 2.146

y x

y x

y x

y x

y x

Point-Slope Form

3' 4 4 1

0.2367

0.2367 0.1152

' 0.2367 0.9998

0.1152 0.9998 0.2367

f x x x

x

f

f

y x

(A) is always increasing.

(B) is always decreasing.

(C) is decreasing only when .

(D) is decreasing only when .

(E) remains constant.

A

A

A b h

A b h

A


If the base of a triangle is increasing at a rate of

3 inches per minute while its height is decreasing

at a rate of 3 inches per minute, which of the following

must be true about the area of the

b

h

A triangle?

3; 3

1

21

2

13 3

2

3

2

db dh

dt dt

A bh

dA db dhh b

dt dt dt

dA dAh b

dth b

dt


2

A particle moves along the -axis so that its position at time

is given by ( ) 6 5. For what value of is the

velocity of the particle zero?

x

t x t t t t

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Power Rule

2( ) 6 5

' 2 6 0

2 3 0

x t t t

x t t

t


2If 10, then when 2, dy

x xy xdx

7 2 3 7

(A) (B) 2 (C) (D) (E) 2 7 2 2

Quotient Rule

2 ' 0

' 2

2 2 4 3'

22 4 2 10 3

x y xy

xy x y

x y x yy

x xx y y

Which of the following could be the derivative of the graph

shown above?

Relative Extrema

Location where

' 0 & where

' goes from to

f

f

2

The function is differentiable for all real numbers. The

1point 3, is on the graph of ( ), and the slope of

4

each point , on the graph is given by 6 2 .

f

y f x

dyx y y x

dx

2

2

2

1(a) Find and evaluate it at the point 3, .

4

(b) Find ( ) by solving the differential equation

1 6 2 with the initial condition (3) .

4

d y

dx

y f x

dyy x f

dx

2

23 22

12 6 2 2

8

d yy x y

dx

2

1

6 13y

x x

Product Rule

Differential Equations

Separation of Variables

2

22

2

22 2

23 2

6 2

2 6 2 2

2 6 2 2

2 6 2 2

dyy x

dx

d y dyy x y

dx dx

y y x y

y x y

2

2

2

2

6 2

6 2

6 2

16

dyy x

dxdy

x dxy

dyx dx

y

x x Cy

4 18 9 13C C

2

2

2

2

2

6 2

6 2

6 2

1

1

6 13

6

4 9 13

dyy x

dx

y dy x dx

y dy x dx

x x Cy

C C

yx x

Problem

2

Water in this container is evaporating so that the depth

-3 1is changing at the constant rate of / .

10 3

(a) Find the volume of water in the container when

5 cm. Indicate units

h

cm hr V r h

V

h

of measure.

(b) Find the rate of change of the volume of water in the

container, with respect to time, when 5 cm. Indicate

units of measure.

(c) Show that the rate of change of the volume of w

h

ater

in the container due to evaporation is directly

proportional to the exposed surface area of the water.

What is the constant of proportionality?

315 cm /

8

dVhr

dt

circledAdVk

dt dt

Problem_Worked_Out_Completely

3

10k

AP Exam Review (Chapter 2)2 3

2 2

3

Consider the curve given by 6.

3(a) Show that .

2

(b) Find all points on the curve whose -coordinate is 1,

and write an equation for the tangent line at each of these

poi

xy x y

dy x y y

dx xy x

x

nts.

(c) Find the -coordinate of each point on the curve where

the tangent line is vertical.

x2,0m 2 & 3y x y

0 &???x

Implicit Differentiation

Point-Slope Form

Vertical Tangent

2 2 3

2 2

3

2 3 0

3

2

dy dyy x y x y x

dx dx

dy x y y

dx xy x

2 6 3 2 0 1,3 & 1, 2y y y y

3 232 0

2 2

x xxy x y

x

22 3

22 23

62

62 2

xy xy x y

x xx x

5 55 5

5 5

6 2 244 2

24 25

x xx x

x x


3 2

What is the -coordinate of the point of inflection on the

1graph of 5 24?

3

x

y x x

10(A) 5 (B) 0 (C) (D) 5 (E) 10

3

Inflection Point


The graph of a twice-differentiable function is shown in

the figure above. Which of the following is true?

f

(A) (1) '(1) ''(1)

(B) (1) ''(1) '(1)

(C) '(1) (1) ''(1)

f f f

f f f

f f f

(D) ''(1) (1) '(1)

(E) ''(1) '(1) (1)

f f f

f f f

Graphical Behavior


2If ''( ) ( 1)( 2) , then the graph of has inflection

points when

f x x x x f

x

(A) -1 only (B) 2 only (C) 1 and 0 only

(D) 1 and 2 only (E) 1, 0, and 2 only

Inflection Point


4 2The function is given by ( ) 2. On which of

the following intervals is increasing?

f f x x x

f

1(A) ,

2

1 1(B) ,

2 2

(C) 0,

(D) ,0

1(E) ,

2

Increasing and Decreasing Functions


3 2

The maximum acceleration attained on the interval 0 3

by the particle whose velocity is given by

( ) 3 12 4 is

t

v t t t t

(A) 9 (B) 12 (C) 14 (D) 21 (E) 40

Maximum Extrema

Position, Velocity, and Acceleration FCTNS


The function is continuous on the closed interval 0,2 and

1has values that are given in the table above. If ( ) , then

2( ) ( ) must have at least two solutions in the interval 0,2

if

f

g x

f x g x

k

1(A) 0 (B) (C) 1 (D) 2 (E) 3

2Intermediate Value THM

0,1

1,0

2,2

1/ 2y

f x

g x


The graph of the function is shown above. Which of the

following statements about is false?

f

f

(A) is continuous .

(B) has a relative maxima at .

(C) is in the domain of .

f at x a

f x a

x a f

(D) lim ( ) lim ( )f x f xx a x a

(E) lim ( ) existsf xx a


2

The first derivative of the function is given by

cos 1'( ) . How many critical values does have

5on the open interval 0,10 ?

f

xf x f

x

(A) One

(B) Three

(C) Four

(D) Five

(E) SevenCritical Number


The graphs of the derivatives of the function , , and are

shown above. Which of the functions , , or have a

relative maximum on the open interval ?

f g h

f g h

a x b (A) only (B) only (C) only

(D) and only (E) , , and

f g h

f g f g h Relative Extrema


2

If is a differentiable function such that ( ) 0 for all real

numbers and if '( ) 4 ( ), which of the

following are true?

g g x

x f x x g x

(A) has a relative maximum at 2 and a relative minimum at 2.

(B) has a relative minimum at 2 and a relative maximum at 2.

(C) has a relative minima at 2 and at 2.

(D) has a relat

f x x

f x x

f x x

f

ive maxima at 2 and at 2.

(E) It cannot be determined if has any relative extrema.

x x

f

Relative Extrema

'f x


Let be a function that is differentiable on the open interval

1,10 . If (2) -5, (5) 5, and (9) -5, which of the

following must be true?

f

f f f

I. has at least 2 zeros.fII. The graph of has at least one horizontal tangent.fIII. For some , 2 5, ( ) 3.c c f c

(A) None (B) I only (C) I and II only (D) I and III only (E) I, II and III

Differentiability

2, 5

5,5

9, 5


2

Let be a function defined for all 0 such that (4) 3

2and the derivative of is given by '( ) for all 0.

h x h

xh h x x

x

(a) Find the values of for which the graph of has ax h

horizontal tangent, and determine whether has a local

maximum, a local minimum, or neither at each of these

values. Justify your answers.

h

(b) On what intervals, if any, is the graph of concave up?

Justify your answer.

(c) Write an equation for the line tangent to the graph of

at 4.

(d) Does the tangent line to the graph of

h

h

x

h

at 4 lie above or

below the graph of for 4? Why?

x

h x

2, both are rel. minsx

is concave up on its entire domain.h

717

2y x

Relative ExtremaConcavity

Below Point-slope Form

Tangent Lines

Two runners, and , run on a straight racetrack for 0 10

seconds. The graph above, which consists of two line segments,

shows the velocity, in meters per second, of Runner . The

velocity, in meters

A B t

A

per second, of runner is given by the

24function defined by ( ) .

2 3

B

tv v t

t

(3,10) (10,10)

Time in seconds

Velocity of Runner A(meters per second)

Question

(a) Find the velocity of runner and the velocity of Runner at

time 2 seconds. Indicate units of measure.

A B

t

(b) Find the acceleration of Runner and the acceleration

of Runner at time 2 seconds. Indicate units of

measure.

(c) Find the total distance run by Runner and the total

distanc

A

B t

A

e run by Runner over the time interval

0 10 seconds. Indicate units of measure.

B

t

Graph & Function

(2) 6 meters per second

(2) 48 / 7 meters per secondA

B

V

V

(2) 10 / 3 meters per second per second

(2) 72 / 49 meters per second per secondA

B

A

A

A

B

Distance 85 meters

Distance 83 meters

The graph above shows ', the derivative of , for 7 7.

The graph of 'has horizontal tangent lines at 3, 2,

and 5, and a vertical tangent line at 3.

f f x

f x x

x x

(a) Find all values of , for 7 7, at which attains a

relative minimum. Justify your answer.

(b) Find all values of , for 7 7, at which attains a

relative maximum. Justify your answe

x x f

x x f

r.

(c) Find all values of , for 7 7, at which ''( ) 0.

(d) At what value of , for 7 7, does attain its

absolute maximum? Justify your answer.

x x f x

x x f

1x

5x

7, 3 & 2,5

7, on -1,7 , ' 0x f

Definition

Relative ExtremaConcavity

Maximum Extrema

AP Exam Review (Chapter 4)2

1

2e xdx

x

2 22 21 1 5

(A) (B) (C) (D) 2 (E) 2 2 2 2

e ee e e e e

e

If is a linear function and 0 , then ''( )b

af a b f x

2 2

(A) 0 (B) 1 (C) (D) (E) 2 2

ab b ab a

Rewrite the integrand


2

3What are the values of for which 0

kk x dx

(A) 3 (B) 0 (C) 3 (D) 3 and 3 (E) 3,0 and 3

3

0If ( ) 1 , then '(2)

xF x t dt F

(A) -3 (B) 2 (C) 2 (D) 3 (E) 18

Special Definite Integrals

Second Fundamental THM of Calculus


0sin

xtdt

(A) sin (B) cos (C) cos (D) cos 1 (E) 1 cosx x x x x

Fundamental THM of Calculus


2

A particle moves along the -axis with velocity given by

( ) sin for 0.

y

x t t t t

(a) In which direction (up or down) is the particle moving

at time 1.5? Why?

(b) Find the acceleration of the particle at time 1.5. Is the

velocity of the particle increasing at 1.5?

t

t

t

Why or why

not?

(c) Given that ( ) is the position of the particle at time and

that (0) 3, find (2).

(d) Find the total distance traveled by the particle from

0 2.

y t t

y y

t to t

Up because ( ) 0x t

( ) 2.05, ( ) is decreasing

because acceleration is neg.

a t v t

(2) 3.83y

0.83 units

Direction of a particle

v(x) increasing or decreasing?

s(t), v(t), and a(t)

Area under v(t)


3 2

A cubic polynomial function is defined by

( ) 4

where , , and are constants. The function has a local

minimum at 1, and the graph of has a point of inflection

at 2

f

f x x ax bx k

a b k f

x f

x

1

0

.

(a) Find the values of and .

(b) If ( ) 32, what is the value of ?

a b

f x dx k24, 36a b

5k Relative Extrema

Inflection PointFundamental THM of Calculus

2' 12 2

' 1 12 2 0

'' 24 2

f x x ax b

f a b

f x x a

'' 2 48 2 0

24, 36

f

b

a

a

1

13 2 4 3 2

00

4 24 36 24 / 3 18

1 24 / 3 18 32

x x x k dx x x x kx

k


1

The graph of the function , consisting of three line segments,

is given below. Let ( ) ( ) .x

f

g x f t dt

(a) Compare (4) and (-2).

(b) Find the instantaneous rate of change of , with respect to

, at 1.

(c) Find the absolute minimum value of on the closed

interval -2,4 . Justify your answer

g g

g

x x

g

.

(d) The second derivative of is not defined at 1 and 2.

How many of these values are -coordinates of points of

inflection of the graph of ? Justify your answer.

g x x

x

g

(4) 5 / 2 ( 2) 6g g

4

2

Both

Special Definite Integral Rules

Inflection Point

An object moves along the -axis with initial position (0) 2.

The velocity of the object at time 0 is given by ( ) sin .3

(a) What is the acceleration of the object at time 4?

(b) Consider t

x x

t v t t

t

he following two statements.

Statement I: For 3 4.5, the velocity is decreasing.

Statement II: For 3 4.5, the speed is increasing.

Are either or both of these statements correct

t

t

? Justify

(c) What is the total distance traveled by the object over the time

interval 0 4?

(d) What is the position of the object at time 4?

t

t


( ) cos3 3

a t t

, Speed ( )Both v t

1.4 units3.4 units

s(t), v(t), and a(t)

Velocity and Speed

Area under v(t)

AP Exam Review (Chapter 4)If is continuous for and differentiable for

, which of the following could be false?

f a x b

a x b

( ) ( )(A) '( ) for some such that .

(B) '( ) 0 for some such that

(C) has a minimum value on .

(D) has a maximum value on .

(E) ( ) exists.b

a

f b f af c c a c b

b af c c a c b

f a x b

f a x b

f x dx

Instantaneous Velocity & Slope

FalseTrue

True

True

True


3

1

If is a continuous function and if '( ) ( ) for all

real numbers , then (2 )

f F x f x

x f x dx

(A) 2 (3) 2 (1)

1 1(B) (3) (1)

2 2(C) 2 (6) - 2 (2)

(D) (6) - (2)

1 1(E) (6) (2)

2 2

F f

F F

F F

F F

F F

Antiderivative of a Composite FCTN

The flow of oil, in barrels per hour, through a pipeline on

July 9 is given by the graph. Of the following, which best

approximates the total number of barrels of oil that passed

through the pipeline that day?

(A) 500 (B) 600 (C) 2,400 (D) 3,000 (E) 4,800

Area under v(t)


x 2 5 7 8

f(x) 10 30 40 20

8

2

The function is continuous on the closed interval 2,8 and

has values that are given in the table. Using the subintervals

2,5 , 5,7 , and 7,8 , what is the trapezoidal

approximation of ( ) ?

f

f x dx

(A) 110 (B) 130 (C) 160 (D) 190 (E) 210

Trapezoidal Approximation


0

The graph of the function consists of two line segments.

Let be the function given by ( ) ( ) .x

f

g g x f t dt

(a) Find (-1), '(-1), and ''(-1).

(b) For what values of x in the open

interval 2, 2 is increasing? Explain

(c) For what values of x in the open

interval 2, 2 is the graph of concave

g g g

g

g

down? Explain

(d) On the axes provided, sketch the graph

of on the closed interval -2,2 .g

3, 0, 3

2

in 2, 1 & 0,1x

in 0,2 , ''( ) 0x g x

AP Exam Review (Chapter 4)The rate at which water flows out of a pipe,

in gallons per hour, is given by the differential

function of time . The table shows the rate

as measured every three hours for a 24-hour period.

R t

24

0

(a) Use a midpoint Riemann sum with 4

subdivisions of equal length to approximate

( ) . Using correct units, explain the

meaning of your answer in terms of water flow.

(b) Is there s

R t dt

2

ome time , 0 24, such that '( ) 0?

Justify your answer.

(c) The rate of water flow ( ) can be approximated

1 by ( ) 768 23 . Use ( ) to

79 approximate the average rate of water

t t R t

R t

Q t t t Q t

flow during

the 24-hour time period. Indicate units of measure.

255 gallons in a 24 hour period

No, concavity doesn’t change

Approximately 10.8 gallons per hour

Let be a differentiable function. The table gives the values

of and ' for selected points in 1.5,1.5 . ''( ) 0 in

1.5,1.5 .

f

f f x f x x

(d) Let ( )={g x22 7 for 0x x x 22 7 for 0.x x x

The graph of passes through each of the points , ( )

given in the table. Is it possible that and are the same

function? Give a reason for your answer.

g x f x

f g

3/2

0(a) Evaluate (3 '( ) 4) . Show the work that leads

to your answer.

(b) Write an equation of the line tangent to the graph of

at the point where 1. Use this point to approximate the

f x dx

f

x

value of (1.2). Is this approximation greater than or less

than the actual value of (1.2)? Why?

(c) Find a positive real number having the property that

there must exist a value w

f

f

r

c ith 0 0.5 and ''( ) .

Give a reason for your answer.

c f c r

24

5 9y x (1.2) 3f

, because the tangent is below the curve

No, g is not differentiable at x=0

6r

2

The rate at which people enter an amusement park on a

given day is modeled by the function defined by

15600 ( )

24 160

The rate at which people leave the same amusement park on

th

E

E tt t

2

e same day is modeled by the function defined by

9890 ( )

38 370

Both ( ) and ( ) are measured in people per hour and time

is measured in hours after midnight. These functions

L

L tt t

E t L t

t

are

valid for 9 23, the hours during which the park is open.

At time 9, there are no people in the park.

t

t

(a) How many people have entered the park by 5:00 PM

( 17)? Round answer to the nearest whole number.

(b) The price of admission to the park is $15 until 5:00 PM

( 17). After 5:00 PM, the pri

t

t

9

ce of admission is $11.

How many dollars are collected from admissions to the

park on the given day? Round your answer to the nearest

whole number.

(c) Let ( ) ( ) - ( ) for 9 23. Tt

H t E x L x dx t he value of

(17) to the nearest whole number is 3725. Find the

value of '(17) and explain the meaning of (17) and

'(17) in the context of the park.

(d) At what time , for 9 23, doe

H

H H

H

t t s the model predict

that the number of people in the park is a maximum?

2

A car is traveling on a straight road with velocity 55 ft/sec

at time 0. For 0 18 seconds, the car's acceleration

( ), in ft/sec , is the piecewise linear function defined by

the graph.

t t

a t

(2,15) (18,15)

(10, 15) (10, 15)

2

( )

(ft / sec )

a t

(seconds)t

(a) Is the velocity of the car increasing at 2 seconds? Why

or why not?

(b) At what time in the interval 0 18, other than 0, is

the velocity of the car 55 / sec? Why or why not?

(c) On th

t

t t

ft

e interval 0 18, what is the car's absolute

maximum velocity, in ft/sec, and at what time does it

occur? Justify your answer.

(d) At what times in the interval 0 18, if any, is the car's

t

t

velocity equal to zero? Justify your answer.

Increasing because (2) 0.a

6, a(6)=0 and ( ) goes from + to .t a x (18) (0) (6)v v v

( ) 0, the absolute min is 0v x

(6) 115 ft/sec, (10)=85 ft/sec, (12)=55 ft/secv v v

The temperature, in degrees Celsius, of the water in a pond

is a differentiable function of time . The table shows the

water temperatures as recorded every 3 days over a 15-day

period.

W t

(a) Use data from the table to find an approximation for

(12). Show the computations that lead to your answer.

Indicate units of measure.

(b) Approximate the average temperature, in degrees

'W

Celsius,

of the water over the time interval 0 15 days by

using a trapezoidal approximation with subintervals of

3 days.

(c) A student proposes the function , given by

( ) 2

t

t

P

P t

(- /3)0 10 , as a model for the temperature of the

water in the pond at time , where is measured in days

and ( ) is measured in degrees Celsius. Find '(12).

Using appropriate units

tte

t t

P t P

, explain the meaning of your

answer in terms of water temperature.

(d) Use the function defined in part (c) to find the average

value, in degrees Celsius, of ( ) over the time interv

P

P t al

0 15 days.t

(15) (9) 1'(12) degrees Celsius per day

15 9 2

w ww

Trapezoid Rule

15(20 62 56 48 44 21) 376.5

10

1376.5 25.1 degrees Celsius

15

.549 C/day

25.757 C

2

1

1e xdx

x

2 22 21 1 3

(A) - (B) 2 (C) (D) 2 (E) 2 2 2 2

e ee e e ee

If ( )f x

(A) ln 2 (B) ln8 (C) ln16 (D) 4 (E) DNE

then lim ( ) isf x2x

{ln for 0 2x x

2 ln 2 for 2 4,x x

If and is a nonzero constant, then could bedy

ky k ydt

2

(A) 2 (B) 2 (C) +3

1 1(D) 5 (E)

2 2

kty kt kte e e

kty ky

2

3

Let be the function given by ( ) 3 and let be the

function given by ( ) 6 . At what value of do the

graphs of and have parallel tangent lines?

xf f x e g

g x x x

f g

(A) 0.701 (B) 0.567 (C) 0.391 (D) 0.302 (E) 0.258

Population grows according to the equation ,

where is a constant and is measured in years. If the

population doubles every 10 years, then the value of is

dyy ky

dtk t

k

(A) 0.069 (B) 0.200 (C) 0.301 (D) 3.322 (E) 5.000

If ( ) sin( ), then '( )xf x e f x

(A) cos( )

(B) cos( )

(C) cos( )

(D) cos

(E) cos

x

x x

x x

x x

x x

e

e e

e e

e e

e e

The temperature outside a house during a 24-hour period is

given by

(a) Sketch the graph of . F

( ) 80 10cos , 0 24,12

tF t t

where ( ) is measured in degrees Fahrenheit and is

measured in hours.

F t t

(b) Find the average temperature, to the nearest degree,

between 6 and 14. t t (c) An air conditioner cooled the house whenever the

outside temperature was at or above 78 degrees

Fahrenheit. For what values of was the air conditioner

cooling the house?

t

0,70

24,70

87.1625

5.23<x<18.77

The temperature outside a house during a 24-hour period is

given by

(d) The cost of cooling the house accumulates at a rate of

$0.05 per hour for each degree the outside temperature

exceeds 78 degrees Fahrenheit. What was the total cost,

to the nearest cent, to cool the house for this 24-hour

period?

( ) 80 10cos , 0 24,12

tF t t

where ( ) is measured in degrees Fahrenheit and is

measured in hours.

F t t

18.769

5.231

0.05 80 10cos 7812

$5.10dC t

dtdt

-2

Suppose that the function has a continuous second

derivative for all , and that (0) 2, '(0) 3, and

''(0) 0. Let be a function whose derivative is given by

'( ) [3 ( ) 2 '( )] for all .x

f

x f f

f g

g x e f x f x x

(a) Write an equation for the tangent line to the graph of

at the point where 0.

(b) Is there sufficient information to determine whether or

not the graph of has a point of inflection

f

x

f

-2

where 0?

Explain your answer.

(c) Given that (0) 4, write an equation of the line tangent

to the graph of at the point where 0.

(d) Show that ''( ) [ 6 ( ) '( ) 2 ''( )]. Does

x

x

g

g x

g x e f x f x f x

have a local maximum at 0? Justify your answer.g x

2 3y x

No, I don't know if ''( ) changes sign at 0.f x x

Inflection Points

4y

Yes

2


1point 3, is on the graph of ( ), and the slope at

4


f

y f x

dyx y y x

dx

2

2

1(a) Find and evaluate it at the point 3, .

4

d y

dx

2 2 2

22

2

2 2 2

6 2 6 2

12 2 2

12 6 2 2 2 6 2

dyy x y xy

dx

d y dy dyy y x y

dx dx dx

y y x y xy y x

2

2

d y

dx 13,

4

1

8

2


1point 3, is on the graph of ( ), and the slope at

4


f

y f x

dyx y y x

dx

2

(b) Find ( ) by solving the differential equation

1 6 2 with the initial condition (3) .

4

y f x

dyy x f

dx

2

2

2

22

6 2

6 2

1 16 , 3, 4 18 9

4

16

6 1

1

313

dyy x

dx

y dy xdx

x x C C

x yxy x x

y

2Let be a function given by ( ) 2 .xf f x xe

(a) Find the lim ( ) and lim ( ).

(b) Find the absolute minimum of . Justify your answer.

f x f x

fx x ,0

2 2

2

'( ) 2( 2 )

2 1 2 0

1

2' 1 0

1' 0 0 is the location of an absolute min.

21 1

is the absolute m2

in.

x x

x

f x e x e

e x

x

f

f x

fe

2Let be a function given by ( ) 2 .xf f x xe

(c) What is the range of ?

(d) Consider the family of functions defined by ,

where is a nonzero constant. Show that the absolute

minimum value of is the same for all nonzero

bx

bx

f

y bxe

b

bxe

values of .b

2

2 in [ , )y

e

' ( )

(1 ) 0 1

1

bx bx

bx

b

y b e x be

be x x

bf

e

The mins occur at the same location,

but they are not the same value????

2

Let be a function with (1) 4 such that for all points

3 1, on the graph of the slope is given by .

2

f f

xx y f

y

(a) Find the slope of the graph of at the point where 1.f x dy

dx 1,4

1

2

(b) Write an equation for the tangent line to the graph of

at 1 and use it to approximate (1.2).

f

x f

1.2

14 1

24.. 4 1.5 2

y x

f

2



2

f f

xx y f

y

2

(c) Find ( ) by solving the seperable differential equation

3 1 with the initial condition (1) 4.

2

f x

dy xf

dx y

2

2

2 3

2 3

3 1

2

2 3 1

, 1,4 1

14

6 2 14

dy x

dx y

ydy x dx

y x x

y x x

C C C

2



2

f f

xx y f

y

(d) Use your solution from part (c) to find (1.2).f

2 3

3

14

1.2 1.2 1.2 4.11414

y x x

f

2

2

3Consider the differential equation .

y

dy x

dx e

(a) Find a solution ( ) to the differential equation

1 satisfying (0) .

2(b) Find the domain and range of the function found in part

(a).

y f x

f

f

2 2

3

2 3

3

3

3

1

2

2

2 ln 2

ln 2 1 1 ln, 0,

2 2 2 2

y

u

y

e dy x dx

e du x C

e x C

y x C

x C Cy C e

2

2

u y

du dy

3ln 2

2

x ey

3: ( , )2

:

eD

R

3

3

3

2 0

2

2

x e

ex

ex

2

Determine the area of the region bounded by the graphs of

4 and 4.y x x y x

(A) 9 / 2

(B) 23/ 6

(C) 9 / 2

(D) 8 / 3

(E) None of these

4 2

1

9

4

2

4A x x x dx

3

Find the volume of the sold formed by revolving the region

bounded by , 1, and 2 about the - .y x y x x axis

(A) 127 / 7

(B) 120 / 7

(C) 240 / 7

(D) 1013 /10

(E) None of these

2 6

1

27

1

120

7

1

7

x dx

xx

2

Identify the definite integral that represents the area of the

surface formed by revolving the graph of ( ) on the

interval 0, 2 about the - .

f x x

x axis

2 2 4

0

2 2 2

0

2 2

0

2

0

(A) 2 1

(B) 2 1 4

(C) 2 1 4

1(D) 2 1

(E) None of these

x x dx

x x dx

x x dx

y dyy

22 1 '

b

aA r f x dx

2 2 2

02 1 4x x dx

2

Find the volume of the solid formed by revolving the region

bounded by the graphs of +1 and 0 about the

-axis.

y x y

x

(A) 2 / 3

(B) 8 /15

(C) 16 /15

(D) 4 / 3

(E) None of these

1

22

0

161

12

5V x dx

Identify the definite integral that represents the arc length of

the curve over the interval 0,3 .y x

3

0

3

0

3

0

3

0

1(A) 1

4

1(B) 1

2

(C) 1

(D)

(E) None of these

dxx

dxx

xdx

xdx

21 '

b

aS f x dx

3

0

11

4dxx

3

Which of the following integrals represents the volume of

the solid formed by revolving the region bounded by ,

1, and 2, about the line 2?

y x

y x x

83

1

2 23 2

1

28 23

1

2 3

1

(A) 2 2 1

(B) 1 1

(C) 1

(D) 2 2 1

(E) None of these

y y dy

x dx

y dy

x x dx

2 3

12 2 1x x dx

3

Identify the definite integral that represents the area of the

surface formed by revolving the graph of ( ) on the

interval 0,1 about the -axis.

f x x

y

1 4

0

1 4

0

1 3 2

0

1 2

0

(A) 2 1 9

(B) 2 1 9

(C) 2 1 3

(D) 2 1 3

(E) None of these

x x dx

x dx

x x dx

x x dx

1 4

02 1 9x x dx

22 1 '

b

ar f x dx

Find the volume of the solid generated by revolving the

region bounded by the graphs of ( ) 6 , 6, and the

- axis about the - axis.

f x x x

x y

(A) 864 / 5

(B) None of these

(C) 432

(D) 108

6

0

2 6V x xdx

6

6

6

u x

du dx

ux

363/ 2

018

864

5V u du

2

Find the volume of the solid formed by revolving the

region bounded by the graphs of 2 , 0, and 2

about the - axis.

y x x y

y

(A) / 4

(B) 2 /3

(C)

(D) 16 / 3

(E) None of these

1

2

0

2 2 2V x x dx

Use your calculator to approximate the volume of the solid

formed by revolving the region bounded by , 0,

0 and 1 about the -axis. Round your answer to three

decimal places.

xy e y

x x y

(A) 1.359

(B) 6.283

(C) 8.540

(D) 9.870

(E) None of these

1

0

2 xV xe dx

Find the area of the region bounded by the graphs of

( ) sin and ( ) cos , for / 4 5 / 4.f x x g x x x

2

1

1

2

3

2

5 / 4

5 / 4

/ 4/ 4

sin cos cos sin

2 2 2 22 2

2

2

2

2

2 2

x x dx x x

Consider the surface formed by revolving the graph of

( ) sin on the closed interval 0, about the -axis.f x x x

(a) Write the integral that computes the surface area of the

solid described.

(b) Use your calculator to approximate the surface area.

Round your answer to three decimal places.

2

02 sin 1 cosx xdx

14.424

22 1 '

b

a

S r f x dx

2


bounded by the graphs of and 4 about the -axis.y x y x

256

5

24

0

2 16V x dx


bounded by the graphs of 2, 0, and 6 about

the -axis.

y x y x

y

6

2

2 2V x x dx 2

2

u x

du dx

x u

41/ 2

0

43/ 2 1/ 2

0

45/ 2 3/ 2

0

2 2

2 2

2 42

704

1

5 3

5

V u u du

u u du

u u

2 2x y

2

22

0

706

13 2

4

5V y dy

Use the shell method to set up the integral that represents

the volume of the solid formed by revolving the region

1bounded by the graphs of and 2 2 5 about the

1line . (Do not evaluate the i

2

y x yx

y

ntegral.)

2

1/ 2

1 5 2 12

2 2

yy dy

y

2

2

1 5 22 5 2

2

2 5 2 0

2 1 2 0

1 1,2 2,

2 2

xx x

x

x x

x x

x y

1 1

5 22 2 5

2

y xx y

yx y x

is differentiable on a,b

is a continuous,

smooth curve on , and

does not have a vertical

tangent.

f

f

a b

f

Question 1

Question 2Click the box each time

you read this definition.

Click "Question X" to return. Question 3

2.3 The Product and Quotient Rules & Higher-Order Derivatives

( ) ( ) '( ) ( ) ( ) '( )d

f x g x f x g x f x g xdx

The Product Rule:

Question 1

Question 2

Click the box each time


Click "Question X" to return.

2.3 The Product and Quotient Rules & Higher-Order Derivatives

2( ) '( ) ( ) ( ) '( )

( ) ( )

d f x f x g x f x g x

dx g x g x

The Quotient Rule:

Question 1

Question 2




Basic Differentiation Rules for Elementary Functions

Question 2




1 1

The equation for the line that has slope and goes through

the point , is:

m

x y

1 1y y m x x

Point-Slope:

Question 1




Question 3

Question 4

2.4 The Chain Rule

The Chain Rule:

( ( )) '( ( )) '( )d

f g x f g x g xdx

Question 1




THM 3.8 Points of Inflection

( , ( )) is a point of inflection of

''( ) 0 or ''( ) DNE

c f c f

f c f c

Question 1

Question 2




Question 3

Question 4

:

'' must change sign at .

Note

f x c

Question 5

Question 6

Question 7

Question 8

2.2 Basic Differentiation Rules& Rates of Change

1

The Power Rule

n ndx nx

dx

Question 1

Question 2




3.3 Increasing and Decreasing Functions and the First Derivative Test

Test for Increasing and Decreasing Functions

Let f be a continuous on [a,b] and differentiable on (a,b).

1. '( ) 0 in ( , ) is increasing on [ , ].f x x a b f a b

2. '( ) 0 in ( , ) is decreasing on [ , ].f x x a b f a b

3. '( ) 0 in ( , ) is constant on [ , ].f x x a b f a b

Question 1




Question 3

Question 4

3.1 Extrema on an Interval Guidelines for Finding Extrema on a Closed Interval

To find the extrema of a continuous funtion f

on a closed interval [a,b], use the following steps.

4. The least of these values is the minimum.5. The greatest is the maximum.

1. Find the critical numbers of in ( , ).f a b

2. Evaluate at each critical number in ( , ).f a b

3. Evaluate at each endpoint of [ , ].f a b

Question 2Question 1



Click "Question X" to return. Question 4

Question 3

Definition of a Critical NumberLet be defined at .f c '( ) 0f c or '( ) is undefinedf c is a critical number of .c f

Question 4

Question 3

Question 2

Question 1




a

'y f

is the location of a relative minimuma

b 'y f

is the location of a relative maximumb

Question 2

Question 3

Question 4

Question 1




Question 5

Question 6

Question 7

5.6 Differential Equations:Growth & Decay

2Solve '

xy

y

' 2yy x' 2yy dx xdx

2ydy xdx 2

21 22

yC x C

2 22y x C You can use implicit

differentiation to check.

dy=y’dx

Multiply by 2Question

Click through this example if you need help.

Definition of Continuity on a Closed Interval

A function is continuous on a closed interval , if it is

continuous on the opoen interval , and

f a b

a b

x a x blim ( ) ( ) and lim ( ) ( ).f x f a f x f b

The function is continuous from the right at and

continuous from the left at .

f a

b

a b

Question 1




Guidelines for Implicit Differentiation

1. Differentiate both sides of the equation with respect to .

2. Collect all terms involving / on the left and move

all other terms to the right.

3. Fact

x

dy dx

or / out of the left side of the equation.

4. Solve for / .

dy dx

dy dx

Question 1




Vertical Tangent Line

( ) ( )lim

( ) ( )lim

The vertical tangent is .

f c x f c

xor

f c x f c

xx c

Question 1




0x

0x

. . The derivative is undefinedi e

Question 1

Question 4

Question 3

Question 2' is decreasing is concave down

' is increasing is concave up

f f

f f




Question 4




Question 3

Question 2

Question 1

Intermediate Value THM:

is continuous on [ , ] and is any number between ( )

and ( ) there exists at least one number in [ , ] such

that ( ) .

f a b k f a

f b c a b

f c k

Question 4




Question 3

Question 2

Question 1

Question 4




Question 3

Question 2

Question 1

Position ( ), velocity ( ), and Acceleration ( ) Functions

( ) = "Position of an object"

'( ) ( ) "Velocity of an object"

''( ) '( ) ( ) "Acceleration of an object"

s t v t a t

s t

s t v t

s t v t a t

Question 4




Question 3

Question 2

Question 1

Special Definite Integral Rules:

is defined at ( ) 0a

a

f x a f x dx

a b

b a

is integrable on a,b ( ) ( )f f x dx f x dx is integrable on three closed intervals determined by

, , and ( ) ( ) ( )b c b

a a c

f

a b c f x dx f x dx f x dx

Question 4




Question 3

Question 2

Question 1

The Second Fundamental THM of Calculus

is continuous on an open interval containing f I a

( ) ( ).x

a

df t dt f x

dx

Question 4




Question 3

Question 2

Question 1

THM 4.9 The Fundamental THM of Calculus

is continuous on , and and is an antiderivative of

on , ( ) ( ) ( ).b

a

f a b F f

a b f x dx F b F a

Question 4




Question 3

Question 2

Question 1

Direction of a Particle:

'( ) 0 The particle is moving up

'( ) 0 The particle is moving down

f x

f x

Question 4




Question 3

Question 2

Question 1

Is velocity increasing or decreasing?

( ) 0 ( ) is increasing

( ) 0 ( ) is decreasing

a x v x

a x v x

Question 4




Question 3

Question 2

Question 1

THE AREA UNDER ( ) EQUALS THE

DISTANCE A PARTICLE HAS TRAVELED

v t

Question 4




Question 3

Question 2

Question 1

( )Speed v t

Question 4




Question 3

Question 2

Question 1

a bc

Instantaneous Velocity and Slope( ) ( )f b f a

mb a

'( )f c m

Question 4




Question 3

Question 2

Question 1

( ( )) '( ) ( ( ))f g x g x dx F g x C If ( ), then '( ) and u g x du g x dx

( ) ( ) .f u du F u C

Integration by Substitution

Question 4




Question 3

Question 2

Question 1

Trapezoid Rule

THM 4.16 The Trapezoidal Rule

Let f be continuous on a,b .

b

0 1 1a( ) ( ) 2 ( ) ... 2 ( ) ( )

2 n n

b af x dx f x f x f x f x

n

Question 4




Question 3

Question 2

Question 1

Trapezoidal Approximation

1

10

?Area

h

r10 cm

10 cm2



10 3

h

cm hr V r h

3

10

dh

dt


5 cm. Indicate units of measure.

V

h

21

3V r h

Use similar right 's

to write in terms of .V h

5

10 2

r

h

hr

2 31

3 2 12

hh

hV

125

5

12V



units of measure.

h3

12

hV

2' 312

dhV t h

dt

33' 5 3 25

12 10

15 cm / hr

8V

3cm

h

r10 cm

10 cm2



10 3

h

cm hr V r h

3

10

dh

dt


5 cm. Indicate units of measure.

V

h

(c) Show that the rate of change of the volume of water

in the container due to evaporation is directly

proportional to the exposed surface area of the water.

What is the constant of proportionality?



units of measure.

h

2Let 5 :

12

1

5 3.5

8 0

h

k k

33 cm

12

hV

315' 5 cm / hr

8V

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