young-seonlee math1100-4,spring2004 - university of utah · young-seonlee math1100-4,spring2004...

Young-Seon Lee Math 1100-4, Spring 2004WeBWorK problems. WeBWorK assignment 1 due 1/26/04 at 11:59 PM.

1.(1 pt) An equation of the line through (-5, 5)

which is parallel to the line y � 2x � 2 has slope:and y intercept at:

2.(1 pt) Find the difference quotient as specified

below and simplify your answer.

(a) f � x � � 5x � x2. Then given that h �� 0,f � 5 � h � f � 5 �

hcan be written as a linear expression

Ah�B, where

A = , and

B = .

(b) g � t � � 1

t 2 . Then given that t �� 1, g � t � g � 1 �t 1 can

be written as 1

Ct � D , whereC = , and

D = .

3.(1 pt) Simplify the complex fraction:�x � 1

2 � x�x

One way to write the simplified expression is in the

form A � 1B, where A is and B is .

4.(1 pt) Match the expressions below with the let-

ters labeling their equivalent expressions.

You must get all of the answers correct to receive

credit.

1. x2 11x � 28x2 7x � 12

2. x2 6x 7x2 � 7x � 6

3. x2 � 2x 24x2 2x 8

A. x 7x 3

B. x 7x � 6

C. x � 6x � 2

5.(1 pt) Enter a T or an F in each answer space be-

low to indicate whether the corresponding equation

is true or false. An equation is true only if it is true

for all values of the variables. Disregard values that

make denominators 0.

You must get all of the answers correct to receive

credit.

1. xx � y � 1

1 � y2. 58

96 � x � 5896 � 58x3. 58 � a

58

� 1 � a58

4. x � 58y � 58 � xy

6.(1 pt) Suppose a mining company will supply

103000 tons of ore per month if the price is 70 dol-

lars per ton but will supply 70500 tons per month if

the price is 10 dollars per ton. Let x be the tons of ore

supplied and y be the price per ton since the price is

dependent on the supply amount. Assuming the sup-

ply function is of the form y � mx � b, find the slope,m and y-intercept, b

m :

b:

7.(1 pt) Let f � x � � 4x2 � 3x � 5 and let g � h � �f � 2 � h � f � 2 �

h.

Determine each of the following:

(a) g � 1 � �(b) g � 0 1 � �(c) g � 0 01 � �You will notice that the values that you entered are

getting closer and closer to a number L. This number

is called the limit of g(h) as h approaches 0 and is also

called the derivative of f(x) at the point when x = 2.

Enter the value of L:

8.(1 pt) Evaluate the limit

limx �� 3 x � 8

3x2 � 3x � 59.(1 pt) Evaluate the limit

limx �� 7 x2 � 14x � 49x

�7


limx � 1 x � 1x2�7x � 8

11.(1 pt) Let f � x � � x � 3 if x �� 5 andf � x � � 3 if x �� 5.Sketch the graph of this function for yourself and find

following limits if they exist (if not, enter N).1

1. limx �� 5 � f � x �

2. limx �� 5 � f � x �

3. limx �� 5 f � x �

12.(1 pt) Let f � x � � 2 if x � 6,f � x � � 4 if x � 6,f � x � � � x � 7 if � 7 � x � 6,f � x � � 14 if x �� 7.Sketch the graph of this function and find following

limits if they exist (if not, enter DNE).

1. limx � 6 � f � x �

2. limx � 6 � f � x �

3. limx � 6 f � x �

4. limx �� 7 � f � x �

5. limx �� 7 � f � x �

6. limx �� 7 f � x �

13.(1 pt) Let limx � a f � x � � 5 , limx � ag � x � � 0,

limx � ah � x � � � 5.Find following limits if they exist. If not, enter DNE

(’does not exist’) as your answer.

1. limx � a f � x � � g � x �

2. limx � a f � x �� g � x �

3. limx � a f � x �� h � x �

4. limx � a f � x �g � x �

5. limx � a f � x �h � x �

6. limx � a h � x �f � x �

7. limx � a � g � x �

8. limx � ag � x � 1

9. limx � a 1

g � x �� h � x �


lima � 1 a3 � aa2 � 1

15.(1 pt) Let F be the function below.

If you are having a hard time seeing the picture

clearly, click on the picture. It will expand to a larger

picture on its own page so that you can inspect it more

clearly.

Evaluate each of the following expressions.

Note: Enter ’DNE’ if the limit does not exist or is

not defined.

a) limx �� 1 � F � x � =

b) limx �� 1 � F � x � =

c) limx �� 1F � x � =

d) F �� 1 � =e) limx � 1 � F � x � =

f) limx � 1 � F � x � =g) limx � 1F � x � =

h) limx � 3F � x � =

i) F � 3 � =

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c�UR

2


1.(1 pt) (Review of section 0.4)

Simplify the expression:

x� 12

x� 52

into the form xn, where

n = .

2.(1 pt) Simplify the expression by removing all

possible factors from the radical.

3�40x8y5

You will end up with u 3�v where u and v are ex-

pressions in x and y. Then

u = , and

v = .

3.(1 pt) Simplify the expression by performing the

operation and write variables with exponents instead

of radicals.3� � 16x3y43�128y2

You will end up with � 2rxsyt with r� s � t numbers.Then

r = ,

s = , and

t = .

4.(1 pt) (Review section 0.5)

Multiply and simplify:� x 13 � x 12 �� 4x 23 � 3x 32 � Answer: .

5.(1 pt) Let f � x � � 4 if x � 9,f � x � � 0 if x � 9,f � x � � � x � 12 if � 1 � x � 9,f � x � � 13 if x �� 1.Sketch the graph of this function and find following

limits if they exist (if not, enter DNE).

1. limx � 9 � f � x �

2. limx � 9 � f � x �

3. limx � 9 f � x �

4. limx �� 1 � f � x �

5. limx �� 1 � f � x �

6. limx �� 1 f � x �

6.(1 pt) Find the value of x that makes f � x � dis-continuous. If f � x � is continuous everywhere, enterNIL.

f � x � � x2 � 9x�3

Then x = .

7.(1 pt) Find the values of x that make f � x � discon-tinuous. If f � x � is continuous everywhere, enter NIL.Enter your solutions in increasing order, i.e., x1 � x2.

f � x � � 2x � 1x2 � 4

Then x1 = , and

x2 = .

8.(1 pt) Find the values of x that make f � x � discon-tinuous. If f � x � is continuous everywhere, enter NILin all boxes. Enter your solutions in increasing order,

i.e., x1 � x2.Let f � x � � � x � 1 � 3 � 1 for x � 1,f � x � � 1

3x� 23for 1 � x � 2, and

f � x � � 2x � 4 for x � 2.Then x1 = , and

x2 = .


limx � ∞

3�3x

10 � 11x10.(1 pt) Evaluate the limit

limx � ∞

3x�11

6x2 � 8x � 911.(1 pt) Evaluate the limit

limx � ∞

10x3 � 10x2 � 11x2 � 3x � 5x3

12.(1 pt) Evaluate the limit1

limx � ∞

� 9 � x �� 7 � 6x �� 3 � 3x �� 7 � 10x �13.(1 pt) Annuities

If an annuity makes an infinite series of equal pay-

ments at the end of the interest periods, it is called a

perpetuity. If a lump sum investment of An is needed

to result in n periodic payments of Rwhen the interest

rate per period is i, then

An� R 1 �!� 1 � i � n

i " (a) Evaluate lim

n � ∞

An to find a formula for the lump

sum payment for a perpetuity. You will have an an-

swer of the formu

vwhere u and v are expressions in

R and i.

Then u = , and

v = .

(b) Find the lump sum investment needed to make

payments of 100 dollars per month in perpetuity if in-

terest is 12%, compounded monthly. Answer:

dollars.

14.(1 pt) Cost-benefit

The percentage p of particulate pollution that can

be removed from the smokestacks of an industrial

plant by spendingC dollars is given by,

p � 99C

7300�C

Find the percentage of the pollution that could be

removed if spendingCwere allowed to increase with-

out bound. Answer:

15.(1 pt) If f � x � � 2x2, find f #$� x � using the defini-tion of the derivative.

Find f # � 5 � .16.(1 pt) If f � x � � 4x2 � 4x � 10, find f # � x � using

the definition of the derivative.

Find f # � 4 � .17.(1 pt) If f � t � � 3t 1, find f # � t � using the defini-

tion of the derivative.

Find f # � 2 � .18.(1 pt) If f � x � � 4 � 3x � 3x2, find f #$� 5 � using

the definition of the derivative.

19.(1 pt) If f � x � � 4

x2, find f # � 4 � using the defini-

tion of the derivative.

20.(1 pt) Let

f � x � � �3�2x

Find f # � 2 � using the definition of the derivative.Answer:

21.(1 pt) Total cost

Suppose total cost in dollars from the production of x

printers is given by

C � x � � 0 0001x3 � 0 005x2 � 28x � 3000 Find the average rate of change of total cost when

production changes:

(a) from 100 to 300 printers. Answer: dollars

per unit

(b) from 300 to 600 printers. Answer: dollars

per unit

22.(1 pt) Consumer expenditure

Suppose that the demand x for a product is

x � 10 � 000 � 100p �where p is the price per unit. Then the consumer

expenditure for the product is

E � p � � px � 10 � 000p � 100p2 What is the instantaneous rate of change of con-

sumer expenditure with respect to price at the follow-

ing:

(a) any price p? Answer:

(b) p � 5? Answer:(c) p � 20? Answer: .

23.(1 pt) If the tangent line to y � f � x � at (-10, -4)passes through the point (-8, 2), find

A. f �%� 10 � �B. f #$�%� 10 � �24.(1 pt) The area of a square with side s is A � s � �

s2. What is the rate of change of the area of a square

with respect to its side length when s � 12?2


3


1.(1 pt) If f � x � � 18x � 5, find f # �� 11 � .2.(1 pt) If f � x � � 6 � 6x � 3x2, find f # �%� 5 � .3.(1 pt) If f � x � � 3x2 � 12x � 10, find f # � x � .Find f # � 3 � .4.(1 pt) If f � x � � 12x20 � 8x10 � 2x7 � 17x � 9, findf # � x � .5.(1 pt) If f � t � � 11

t6, find f #$� t � .

6.(1 pt) If f � x � � 6 � 4x

� 7

x2, find f # � x � .

7.(1 pt) If f � x � � 5

x10� 4

4�x3

�x5 � 4, find the

equation of the tangent line to f � x � at x � 1.Simplify your answer to the form y � mx � b, wherem = , and

b = .

8.(1 pt) If f � x � � � 3x2 � 4 �� 2x � 6 � , find f # � x � byfirst distributing using the FOIL method.

9.(1 pt) If f � x � � � 3x2 � 3 �� 6x � 6 � , find f # � x � byusing the Product Rule.

10.(1 pt) If y � � 2x2 � x � 3 �� 3� x � 2 � x � 1 � , thendy

dx= .

Hint: You do not need to simplify your answer.

11.(1 pt)

Let f � x � � 4

6x � 7 . Findf # � x � �12.(1 pt) If

f � x � � 5 � x27�x2

find f # � x � .Find f # � 1 � .

13.(1 pt) If z � x2 � x3

2 � x � x2 , thendz

dx= .


14.(1 pt) The population of a slowly growing bac-

terial colony after t hours is given by

p � t � � 5t2 � 29t � 150Find the growth rate after 3 hours. Answer:

15.(1 pt) Revenue

Suppose the revenue (in dollars) from the sale of x

units of a product is given by

R � x � � 20x2 � 58x3x � 1

Find the marginal revenue when 25 units are sold.

Answer: dollars per unit (round to two decimal

places).

16.(1 pt) Let

f � x � � �4�5x

f # � 1 � �17.(1 pt) Let

f � x � � � x3 � 5x � 6 � 3f #$� x � �f #$� 1 � �18.(1 pt) Let

k � x � � 57� 2x3 � x � 6 � 14

Then k # � x � �19.(1 pt) If p � 1� 2q4 � 3q � 1 � 34 , thendp

dq= .


20.(1 pt) Let

f � x � � �3x � 1 � �

4x

2

Then f #$� x � �21.(1 pt) Pricing and sales

1

Suppose the weekly sales volume y (in thousands

of units sold) depends on the price per unit of the

product, p, according to

y � 20 � 4p �1 � 3

5 � p � 0where p is in dollars.

Find the rate of change in sales volume when the

price is 23 dollars. Answer: (round your answer

to three decimal places)


2


1.(1 pt) Let

f � x � � � x3 � 3x � 6 � 3f # � x � �f #$� 3 � �2.(1 pt) If f � x � � 6x5 4x4 4x3

x4, find f # � x � .

3.(1 pt) If f � x � � � 6x 6 � 2x4, find f # � x � .

4.(1 pt) Let h � x � � 5x2 � 2x � 1. Then h #$� x � isand h #&# � x � is5.(1 pt) Let f � x � � � 5x � 2 � 5. Then f # � x � is

and f #'# � x � is6.(1 pt) Let f � x � � �

x�1. Then f #$� x � is

and f #'# � x � is7.(1 pt) If f � x � � � 5x � 9

x2 9 � 8, find f # � x � .8.(1 pt) If f � x � �)( 6x � 5

6x 5 , find f # � x � .9.(1 pt) If f � x � � x � x2 � 14x � 1, find f # � x � .10.(1 pt) Let f � x � � 8x7 � 4

x6.

Then f #$� x � isf #'# � x � is11.(1 pt) Let f � 4 � � x � � �

x6 � 11.Then f � 5 � � x � isf � 6 � � x � is12.(1 pt) Suppose profit function for a product is

P � x � � � 0 8x2 � 530x � 4388. Find the marginalprofit function.

MP � x � is13.(1 pt) Suppose cost function for a product is

C � x � � x3 � 8x2 � 66x � 8400. Find the marginal costat x � 160.14.(1 pt) If a particle travels as a function of time

according to the formula

s � t � � 0 01t3 � 15t � 100What is the acceleration of the particle when t � 2?

Answer: .

15.(1 pt) The demand q for a product at price p is

given by

q � 10 � 000 � 50 � 0 02p2 � 500What is the rate of change of demand with respect to

price?

Answer: .

16.(1 pt) If the revenue function for a product is

R � x � � 40x2

3x � 5What is the marginal revenue? Answer: .

17.(1 pt) The function f � x � � 2x3 � 27x2 � 108x �4 has two critical values. The smaller one equals

and the larger one equals

18.(1 pt) The function f � x � � 2x3 � 24x2 � 72x � 1has one relative minimum and one relative maximum.

This function has a relative minimum at x equals

with value

and a relative maximum at x equals with value

19.(1 pt) The function f � x � � � 2x3 � 21x2 � 36x �5 has one relative minimum and one relative maxi-

mum.


with value


20.(1 pt) The function f � x � � 6x � 5x 1 has onerelative minimum and one relative maximum.


with value


21.(1 pt) For x *,+-� 10 � 13 . the function f is definedby

f � x � � x7 � x � 8 � 8On which two intervals is the function increasing (en-

ter intervals in ascending order)?

to

and

to1

Find the region in which the function is positive:

to

Where does the function achieve its minimum?

22.(1 pt) Consider the function f � x � � 6x � 5x 1.For this function there are four important intervals:�� ∞ � A . , + A � B � , � B � C � , and +C � ∞ � where A, and C arethe critical numbers and the function is not defined at

B.

Find A

and B

and C

For each of the following intervals, tell whether f � x �is increasing (type in INC) or decreasing (type in

DEC).�%� ∞ � A . :+ A � B � :� B � C . :+C � ∞ � :


2


1.(1 pt) Consider the function f � x � � � 6x2 � 10x �7. f � x � is increasing on the interval �%� ∞ � A . and de-creasing on the interval + A � ∞ � where A is the criticalnumber.

Find A

At x � A, does f � x � have a local min, a local max, orneither? Type in your answer as LMIN, LMAX, or

NEITHER.

2.(1 pt) Consider the function f � x � � 12x5 �30x4 � 160x3 �

3. For this function there are four

important intervals: �%� ∞ � A . , + A � B . , + B � C . , and +C � ∞ �where A, B, andC are the critical numbers.

Find A

and B

and C

At each critical number A, B, and C does f � x � havea local min, a local max, or neither? Type in your

answer as LMIN, LMAX, or NEITHER.

At A

At B

At C

3.(1 pt) Answer the following questions for the

function

f � x � � x � x2 � 16defined on the interval +/� 5 � 5 . .A. f � x � is concave down on the region to

B. f � x � is concave up on the region to

C. The inflection point for this function is at

D. The minimum for this function occurs at

E. The maximum for this function occurs at

4.(1 pt) Consider the function f � x � � 1

12x4

� 66x3

�5

2x2

�10x

�7.

f � x � has two inflection points (keep in mind thatthe Second Derviative is real handy in determining

these!) at x = C and x = D withC � DwhereC is

and D is

Finally for each of the following intervals, tell

whether f � x � is concave up (type in CU) or concavedown (type in CD).�� ∞ � C . :+C � D . :

+D � ∞ �5.(1 pt) Consider the function f � x � � 12x5 �

45x4 � 200x3 � 6.f � x � has inflection points at (reading from left toright) x � D, E, and Fwhere D is

and E is

and F is

For each of the following intervals, tell whether f � x �is concave up (type in CU) or concave down (type in

CD).�%� ∞ � D . :+D � E . :+ E � F . :+ F � ∞ � :6.(1 pt) Consider the function f � x � � 3x � 5

6x � 2 . For thisfunction there are two important intervals: �%� ∞ � A �and � A � ∞ � where the function is not defined at A.Find A

For each of the following intervals, tell whether f � x �is increasing (type in INC) or decreasing (type in

DEC).�%� ∞ � A � :� A � ∞ �Note that this function has no inflection points, but

we can still consider its concavity. For each of the

following intervals, tell whether f � x � is concave up(type in CU) or concave down (type in CD).�%� ∞ � A � :� A � ∞ �7.(1 pt) For the given cost function

C � x � � 16900 �700x

�x2 find:

a) The cost at the production level 1000

b) The average cost at the production level 1000

c) The marginal cost at the production level 1000

d) The production level that will minimize the aver-

age cost

e) The minimal average cost

8.(1 pt) For the given cost function C � x � �250

�x� x2

1000find





age cost.

e) The minimal average cost.

9.(1 pt) For the given cost function

C � x � � 72900 �500x

�x2 find:





age cost

e) The minimal average cost

10.(1 pt) If 2300 square centimeters of material is

available to make a box with a square base and an

open top, find the largest possible volume of the box.

Volume = cubic centimeters.

11.(1 pt) The manager of a large apartment com-

plex knows from experience that 120 units will be oc-

cupied if the rent is 490 dollars per month. A market

survey suggests that, on the average, one additional

unit will remain vacant for each 7 dollar increase in

rent. Similarly, one additional unit will be occupied

for each 7 dollar decrease in rent.

What rent should the manager charge to maximize

revenue?

Answer = dollars per month

12.(1 pt) An agency charges 10 dollars per person

for a trip to a concert if 30 people travel in a group.

For each person above the initial 30, the charge will

be reduced by 20 cents. How many people will max-

imize the total revenue for the agency if the trip is

limited to at most 50 people?

Answer: people

13.(1 pt) If the total cost function for a product is

C � x � � � x �5 � 3

where x is the number of hundreds of units produced,

how many units should you produce to minimize av-

erage cost?

Answer: hundreds of units

14.(1 pt) A small business has weekly average

costs, in dollars, of

C � 100x

�30

� x10

where x is the number of units produced each week.

The competitive market price for the product is 46

dollars per unit. If production is limited to 150 units

per week, find the level of production that yields max-

imum profit, and find the maximum profit.

Production level: units per week

Maximum profit: dollars

15.(1 pt) A time study showed that, on average, the

productivity of a worker after t hours on the job can

be modeled by

P � t � � 27t �6t2 � t3

where 0 � t � 8 and P is the number of units pro-duced per hour.

(a) After how many hours will productivity be

maximized?

Answer: hours (round to the nearest tenth of an

hour).

(b) What is the maximum productivity?

Answer: units per hour (round to the nearest

unit).

16.(1 pt) The running yard for a dog kennel must

contain at least 900 square feet. If a 20-foot side of

the kennel is used as part of one side of a rectangu-

lar yard with 900 square feet, what dimensions will

require the least amount of fencing?

Enter with width less than or equal to length.

Width = feet.

Length = feet.

Minimum amount of fencing required = feet.

17.(1 pt) From a tract of land a developer plans

to fence a rectangular region and then divide it into

two identical rectangular lots by putting a fence down

the middle. Suppose that the fence for the outside

boundary costs 5 dollars per foot and the fence for

the middle costs 2 dollars per foot. If each lot con-

tains 13,500 square feet, find the dimensions of each

lot that yield the minimum cost for the fence.

Round your answers to the nearest foot, and enter

the smaller dimension of each lot first.

Then width = feet, and

length = feet, and

the minimum fence cost (using your rounded an-

swers) =

dollars.

18.(1 pt) Suppose that a company needs 60,000

items during a year and that preparation for each pro-

duction run costs 400 dollars. Suppose further that2

it costs 4 dollars to produce each item and 75 cents

to store an item for one year. Use the inventory cost

model to find the number of items in each production

run that will minimize the total costs of production

and storage.

Answer =

What is the minimum cost of production and storage?

Answer = dollars


3


1.(1 pt) Let

f � x � � 5x � 15x�2

Then f � x � has a horizontal asymptote aty = ,

and a vertical asymptote at

x = .

2.(1 pt) Let

f � x � � 3x2

4x2 � 1Then f � x � has a horizontal asymptote at

y = ,

and vertical asymptotes at

x = and x = .

Enter your vertical asymptotes in increasing order.

3.(1 pt) An entrepreneur starts new companies and

sells them when their growth is maximized. Suppose

that the annual profit for a new company is given by

P � x � � 22 � 12x � 18

x�1

where P is in thousands of dollars and x is the number

of years after the company is founded. If she wants to

sell the company before profits begin to decline, after

how many years should she sell it?

Answer: years

4.(1 pt) Let

f � x � � 2ln � 4x �f # � x � �f # � 3 � �5.(1 pt) Let

f � x � � + lnx . 3f #$� x � �f # � e3 � �6.(1 pt) Let

f � x � � log2 � 1 � x � x2 �Then f # � 1 � � (Enter an exact answer using

logs.)

7.(1 pt) Between the years 1976 and 1998, the per-

cent of moms who return to the work force within one

year after they had a child is given by

w � x � � 1 11 � 16 94lnxwhere x is the number of years past 1976.

If this model is accurate beyond 1998, at what rate

will the percent be changing in 2005?

Answer: percent per year.

Enter an exact answer. Hint: the function above out-

puts percents, so no conversion is necessary.

8.(1 pt)

Let f � x � � � 4ex � 3 � e 5.f # � 0 � �9.(1 pt) Find the derivative of the function

g � x � � � 4x2 � 3x � 4 � exg # � x � �10.(1 pt) Find the derivative of the function

g � x � � ex

1�5x

g # � x � �11.(1 pt) Let

h � x � � e2x2 ln � 4x � 2 �Then h # � 0 � �12.(1 pt) Find the slope of the tangent line to the

curve

� 1x2 � 2xy � 2y3 � � 4at the point �%� 2 � 0 � .13.(1 pt) Use implicit differentiation to find the

slope of the tangent line to the curve

4x2�3xy � 3y3 � 4

at the point � 2 � 2 � .m �14.(1 pt) Use implicit differentiation to find the

slope of the tangent line to the curve

y

x � 7y � x7 � 91

at the point � 1 � 8 55 � .m �15.(1 pt) Suppose that the number of mosquitoesN

(in thousands) in a certain swampy area near an ex-

pensive resort area is related to the number of pounds

of insecticide x sprayed on the nesting areas accord-

ing to

Nx � 10x �N � 300

What is the rate of change of N with respect to x

when 49 pounds of insecticide is used?

Answer: thousands of mosquitoes per pound

16.(1 pt) The area of a circle is changing at a rate

of 1 square inch per second. At what rate is its radius

changing when the radius is 2 inches?

Answer: inches per second

17.(1 pt) Two boats leave the same port at the same

time, with boat A traveling north at 15 knots and boat

B traveling east at 20 knots. How fast is the distance

between them changing when boat A is 30 nautical

miles from port? (a knot is a unit of speed equal to

nautical miles per hour)

Hint: you will need to use both the Pythagorean the-

orem and the equation distance = rate * time.

Answer: knots


2


1.(1 pt) Write the equation of the tangent line to

the curve

5x lny�6xy � 48 at point � 8 � 1 � .

The line can be written in the form y � mx �b,

where m= and b = .

2.(1 pt) Each edge of a cube is increasing at a rate

of 0 7 inches per second. How fast is the volume ofthe cube increasing when an edge is 12 inches long?

Answer

cubic inches per second.

3.(1 pt) Suppose that the demand for a product is

given by

2p2q � 10 � 000 �9 � 000p2

(a) Find the elasticity when p � 50 and q � 1212.Answer: . (Round answer to the nearest

hundredth.)

(b) Tell what type of elasticity this is: .

Type U for unitary, E for elastic, or I for inelastic.


given by � p �1 � �q

�1 � 1000

(a) Find the elasticity when p � 39.Answer: . (Round answer to the nearest

hundredth.)




given by

q � 5000

1�e2p

� 2(a) Find the elasticity when p � 10 and q � 9.

Answer: . (Round your answer to the nearest

hundredth.)



6.(1 pt) If the demand function for a fixed period of

time is given by p � 38 � 2q and the supply functionbefore taxation is p � 8 �

3q, what tax per item will

maximize the total tax revenue?

Answer: t =

7.(1 pt) Consider the function f � x � � 3x3 � 7x2 �4x � 9.An antiderivative of f � x � isF � x � � Ax4 �

Bx3�Cx2

�Dx

where A is and B is andC is and D is

8.(1 pt) Consider the function f � x � � 8x3 � 12x2 �4x � 10. Enter an antiderivative of f � x �9.(1 pt) Consider the function f � x � � 40x3 �

15x2�16x � 1. Enter an antiderivative of f � x �

10.(1 pt) Consider the function f � x � � 5x10 �4x5 �

8x3 � 9.Enter an antiderivative of f � x �11.(1 pt) Consider the function f � x � � 9x8 �

6x5 �9x2 � 3.An antiderivative of f � x � is F � x � � Axn �

Bxm�

Cxp�Dxq where

A is and n is

and B is and m is

and C is and p is

and D is and q is

12.(1 pt) Consider the function f � x � � 10

x3� 10

x7.

Let F � x � be the antiderivative of f � x � with F � 1 � � 0.Hint: F(1)=0 just gives us that coordinate point we

need to determine the constant, C.

Then F � x � �13.(1 pt) Consider the function f � x � � 2

x2� 3

x6.

Let F � x � be the antiderivative of f � x � with F � 1 � � 0.Then F � 5 � equals14.(1 pt) Note: You can get full credit for this

problem by just answering the last question correctly.

The initial questions are meant as hints towards the fi-

nal answer and also allow you the opportunity to get

partial credit.

Consider the indefinite integral 0 x6 1 4 �14x7 2 11 dx

Then the most appropriate substitution to simplify

this integral is

u =

Then dx � f � x � du where1

f � x � =After making the substitution we obtain the inte-

gral 0 g � u � du whereg � u � =This last integral is: =

�C

(Leave out constant of integration from your answer.)

After substituting back for u we obtain the follow-

ing final form of the answer:

=�C

(Leave out constant of integration from your answer.)

15.(1 pt) Evaluate the integral by making the given

substitution. 0 dx� 4x �2 � 3

u � 4x �2

16.(1 pt) Find

F � x � � 0 x � x2 �4 � 4 dx

Give a specific function for F � x � .F(x) =

17.(1 pt) Evaluate the indefinite integral.0 x5 �5

�x6dx

18.(1 pt) Evaluate the indefinite integral.

0 7� t �7 � 6dt

19.(1 pt) (NOTE: This problem is from section

12.3 - note that this has the form similar to the power

rule case, except that n � � 1 and so must be treateddifferently.)

Evaluate the indefinite integral, choosing any inte-

gration constant (c-value) you wish.0 x5

x6�7dx

20.(1 pt) Evaluate the integral by making the given

substitution. 0 dx� 3x �14 � 2

u � 3x �14

21.(1 pt) Find

F � x � � 0 x � x2 �3 � 2 dx

Give a specific function for F � x � .F(x) =

22.(1 pt) Evaluate the indefinite integral.0 x5 �7

�x6dx


2

WeBWorK demonstration assignment

The main purpose of this WeBWorK set is to fa-

miliarize yourself with WeBWorK.

Here are some hints on how to use WeBWorK ef-

fectively:3 After first logging into WeBWorK changeyour password.3 Find out how to print a hard copy on the com-puter system that you are going to use. Print

a hard copy of this assignment.3 Get to work on this set right away and answerthese questions well before the deadline. Not

only will this give you the chance to figure

out what’s wrong if an answer is not accepted,

you also will avoid the likely rush and con-

gestion prior to the deadline.3 The primary purpose of the WeBWorK as-signments in this class is to give you the op-

portunity to learn by having instant feedback

on your active solution of relevant problems.

Make the best of it!

1.(1 pt)

Evaluate the expression

8 � 5 � 1 � = .

2.(1 pt)

Evaluate the expression

2 4�� 2 � 5 � = .

Enter you answer as a decimal number listing at least

4 decimal digits. (WeBWorK will reject your answer

if it differs by more than one tenth of 1 percent from

what it thinks the answer is.)

3.(1 pt) Let r � 3 Evaluate 4 4 π � r � .

Next, enter the expression 4 4�� π � r � � and let

WeBWorK compute the result.

4.(1 pt) Enter here the expression 1a

� 1b.

Enter here the expression 1

a � b .5.(1 pt) Enter here the expression

a�1

2�b

Enter here the expression

a�b

c�d

If WeBWorK rejects your answer use the preview

button to see what it thinks you are trying to tell it.

6.(1 pt) Enter here the expression�a�b


a�a�b


a�b�

a�b

7.(1 pt)

Enter here the expression�x2�y2


x�x2�y2


x�y�

x2�y2

8.(1 pt)

Enter here the expression� b � �b2 � 4ac2a

Note: this is an expression that gives the solution of

a quadratic equation by the quadratic formula.


1


1.(1 pt) 0 9�ex dx = +C

2.(1 pt) Evaluate the indefinite integral.0 x5ex6dx+C

3.(1 pt) Evaluate the indefinite integral.0 x�1

x2�2x

�2dx

+C

4.(1 pt) Evaluate5 � 7� x � 17x� dx

+C

5.(1 pt) Evaluate0 8x3 � 6x2 � 50x � 36x2

�6

dx

+C

6.(1 pt) Evaluate 0 e12x

e12x�6dx

+C

7.(1 pt)

If the marginal cost function is

MC � 5x � 5and the cost of producing 8 units is 980, find the cost

function.

C � x � �8.(1 pt)

If the marginal revenue function is

MR � 11 � 4

x � 5 �find the revenue function. Hint: To find c, think about

how much revenue is created by selling zero units.

R � x � �9.(1 pt)

If the marginal profit function isMP � � 1 4x � 7and the company breaks even when they produce 10

units, find the profit function.

P � x � �10.(1 pt) Given

f #&# � x � � 1x � 0and f # �%� 1 � � 0 and f �%� 1 � � 1.

Hint: start with integrating the second derivative

function, and use the ”fixed point” f # �%� 1 � � 0. Thensolve for C, giving you the first derivative function.

Now integrate this function, and use the second fixed

point given to solve the second part of the problem.

Find f # � x � �and find f � 1 � �11.(1 pt) A particle is moving with acceleration

a � t � � 18t � 2. its position at time t � 0 is s � 0 � � 13and its velocity at time t � 0 is v � 0 � � 9. Hint: this isthe same concept as the previous problem. Treat ac-

celeration as the second derivative and velocity as the

first derivative, with the distance being the original

function.

What is its position at time t � 6?12.(1 pt) The functions

y � x2 � cx2

are all solutions of equation:

xy # � 2y � 4x2 �6� x � 0 �Find the constant c which produces a solution which

also satisfies the initial condition y � 6 � � 3.c �13.(1 pt) Find the particular solution of the differ-

ential equation

dy

dx� � x � 7 � e 2y

satisfying the initial condition y � 7 � � ln � 7 � .Answer: y=

Your answer should be a function of x.

14.(1 pt) Find the particular solution of the differ-

ential equation

x2

y2 � 8 dydx � 12y

satisfying the initial condition y � 1 � � �9.

Answer: y=

Your answer should be a function of x.

15.(1 pt) Solve the separable differential equation

7x � 2y � x2 � 1dydx

� 0Subject to the initial condition: y � 0 � � 7y �

1

(function of x only)

16.(1 pt) Find f � x � if y � f � x � satisfiesdydx

� 48yx5and the y intercept of the curve y � f � x � is 2.f � x � �17.(1 pt) Find a function y of x such that

9yy # � x and y � 9 � � 10 y � (function of x)

18.(1 pt) Solve the differential equation� y3x � dydx

� 1 �x

Use the initial condition y � 1 � � 2 Express y4 in terms of x y4 �( function of x)

19.(1 pt) Find the function y � y � x � (for x � 0 )which satisfies the separable differential equationdydx

� 2 � 16xxy2;x � 0

with the initial condition: y � 1 � � 3y �

( function of x only)


2


1.(1 pt) Evaluate the definite integral 0 86

1

x2dx is

2.(1 pt) Evaluate the definite integral0 84

� 9x2 � 4x �9 � dx

3.(1 pt) Evaluate the definite integral0 6 6 � 36 � x2 � dx4.(1 pt) Evaluate the definite integral.0 2

0

dx

3x�3

5.(1 pt) Evaluate the definite integral

0 73

2x2�10�xdx

6.(1 pt) Evaluate the definite integral0 81

6�xdx

7.(1 pt) Evaluate the definite integral

0 21

e3 7 xx2dx

Answer:

8.(1 pt) Find area below the curve

y � 4 �x

�90

from � 54 to 54.Answer:

9.(1 pt) Find area below the curve

y � � 5x2 �10x

�175

from � 3 to 4.Answer:

10.(1 pt) Sketch the region enclosed by the given

curves. Don’t forget to find the points of intersection,

they are important in setting up your integral. Then

find the area of the region.

y � 7x � y � 5x2


curves. Then find the area of the region.

y � 3x2 � y � x2 �1

12.(1 pt)

Find area enclosed by f � x � � x4 �8 and g � x � �

40 � x4.Answer:

13.(1 pt)

Find area enclosed by f � x � � �x

�19 and g � x � �

1

13x

� 59

13.

Answer:


curves. Then find the area of the region.

y � e5x � y � e8x � x � 1

15.(1 pt)

Find the area enclosed between

f � x � � 0 5x2 �4

and

g � x � � xFrom x � � 8 to x � 716.(1 pt) Farmer Jones, and his wife, Dr. Jones,

decide to build a fence in their field, to keep the

sheep safe. Since Dr. Jones is a mathematician, she

suggests building fences described by y � 6x2 andy � x2 �

11. Farmer Jones thinks this would be much

harder than just building an enclosure with straight1

sides, but he wants to please his wife. What is the

area of the enclosed region?

17.(1 pt)

Find the average value of f � x � � 8x � 1

4x � 6 on+ 2 � 9 . .Answer:

18.(1 pt)

The cost function for a product is C � x � � 0 2x2 �100x

�110.

Find average cost over + 0 � 750 . .Answer:

19.(1 pt)

Find consumer’s surplus at the market equilibrium

point given that the demand function is

p � �196 � 44x and the supply function is p �

x�5.

Answer:

20.(1 pt)

Find producer’s surplus at the market equilibrium

point if supply function is

p � 0 8x �16 and the demand function is p � 453 8 6

x � 14 .Answer:

21.(1 pt) Determine whether the integral is diver-

gent or convergent. If it is convergent, evaluate it. If

not, state your answer as ”divergent.”0 ∞

0

8e xdx22.(1 pt) Determine whether the integral is diver-

gent or convergent. If it is convergent, evaluate it. If

not, state your answer as ”divergent.”0 ∞

2

2� x �3 � 3 7 2dx

23.(1 pt)

Find the value of c that satisfies the following equa-

tion:5∞

3

cx3dx � 1 ?

Answer:


2


1.(1 pt) Evaluate the function at the given values:

z � x2 � xyx � y

at x � 3 and y � 2.Answer:

2.(1 pt) If z � x � y � � x lny � y lnx, then� thenz(1,1) = .

3.(1 pt) If f � w � x � y � z � � wx � yz2xy � wz , then

f � 2 � 3 � 1 �9� 1 � = .

4.(1 pt) If f � x � y � � � xy3 � y � 2, then∂ f

∂x= , and

∂ f

∂y= .

5.(1 pt) If f � x � y � � ln � xy � 1 � , then∂ f

∂x= , and

∂ f

∂y= .

6.(1 pt) Find the slope of the tangent in the positive

y-direction to the surface

z � x3 � 5xy at the point � 2 � 1 �9� 2 � .Answer:

7.(1 pt) If f � x � y � � 4xy � x3y2 � x3 � 2y, then(a)

∂2 f

∂x2= ,

(b)∂2 f

∂y2= ,

(c)∂2 f

∂x∂y= ,

(d)∂2 f

∂y∂x= .

8.(1 pt) The total cost of producing 1 unit of a prod-

uct is given by

C � x � y � � 30 � 10x2 � 20y � xywhere x is the hourly labor rate and y is the cost per

pound of raw materials. The current hourly rate is 15

dollars and the rawmaterials cost 6 dollars per pound.

How will an increase of

(a) 1 dollar per pound for the raw materials affect

the total cost?

Answer:

(b) 1 dollar in the hourly labor rate affect the total

cost?

Answer:

9.(1 pt) Test for relative maxima and minima of the

function z � 4x2 � y2 � 4x � 1:(a) The critical point has x-coordinate and

y-coordinate ,

(b) Evaluate D at the critical point to get: ,

(c) Then the critical point is a relative maxi-

mum (MAX), relative minumum (MIN), or neither

(NONE): .

10.(1 pt) Test for relative maxima and minima of

the function z � x2 � 4xy � y2 � 6y:(a) The critical point has x-coordinate and

y-coordinate ,




(NONE): .

11.(1 pt) Test for relative maxima and minima of

the function z � 46 � x2 � 2xy � 4y2:(a) The critical point has x-coordinate and

y-coordinate ,




(NONE): .

12.(1 pt) Find the values for each of the dimensions

of a closed-top box of length x, width y, and height z

if the volume equals 27,000 cubic inches and the box

requires the least amount of material to make.

Hint: You want to minimize surface area of the box

so you need to have a surface area in terms of x and

y.

Then the dimensions that produce minimal surface

area are:

x = inches

y = inches

z = inches1


2

young-seonlee math1100-4,spring2004 - university of utah · young-seonlee math1100-4,spring2004...

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