webwork demonstration assignment - home - math · webwork demonstration assignment the main purpose...

WeBWorK demonstration assignmentThe main purpose of this WeBWorK set is to familiarize

yourself with WeBWorK.Here are some hints on how to use WeBWorK effectively:• After first logging into WeBWorK change your pass-

word.• Find out how to print a hard copy on the computer sys-

tem that you are going to use. Print a hard copy of thisassignment.

• Get to work on this set right away and answer thesequestions well before the deadline. Not only will thisgive you the chance to figure out what’s wrong if an an-swer is not accepted, you also will avoid the likely rushand congestion prior to the deadline.

• The primary purpose of the WeBWorK assignments inthis class is to give you the opportunity to learn by hav-ing instant feedback on your active solution of relevantproblems. Make the best of it!

1. (1 pt) setDemo/demo pr1.pg

Evaluate the expression5(6−8) = .

Correct Answers:

• -10


Evaluate the expression8/(7+1) = .Enter you answer as a decimal number listing at least 4 decimaldigits. (WeBWorK will reject your answer if it differs by morethan one tenth of 1 percent from what it thinks the answer is.)

Correct Answers:

• 1

3. (1 pt) setDemo/demo pr3.pgLet r = 5.

Evaluate 4/π∗ r = .Next, enter the expression 4/(π∗ r) = and let WeB-WorK compute the result.

Correct Answers:

• 6.36619772367581• 0.254647908947033

4. (1 pt) setDemo/demo pr4.pgEnter here the expression 1

a + 1b .

Enter here the expression 1a+b .

Correct Answers:

• 1/a+1/b• 1/(a+b)

5. (1 pt) setDemo/demo pr5.pgEnter here the expression

a+12+b

Enter here the expressiona+bc+d

If WeBWorK rejects your answer use the preview button tosee what it thinks you are trying to tell it.

Correct Answers:• (a+1)/(2+b)• (a+b)/(c+d)

6. (1 pt) setDemo/demo pr6.pgEnter here the expression

√a+b

Enter here the expressiona√

a+bEnter here the expression

a+b√a+b

Correct Answers:• sqrt(a+b)• a/sqrt(a+b)• (a+b)/sqrt(a+b)


Enter here the expression√x2 + y2

Enter here the expression

x√

x2 + y2

Enter here the expressionx+ y√x2 + y2

Correct Answers:• sqrt(x**2+y**2)• x*sqrt(x**2+y**2)• (x+y)/sqrt(x**2+y**2)


Enter here the expression

−b+√

b2−4ac2a

Note: this is an expression that gives the solution of a quadraticequation by the quadratic formula.

Correct Answers:• (-b+sqrt(b**2-4*a*c))/(2a)

1

Generated by the WeBWorK system c©WeBWorK Team, Department of Mathematics, University of Rochester

2

WeBWorK Assignment ZeroThe first assignment is about common errors made by algebra

students. You will be shown the error and asked to correct it. Ifthe expression does not simplify, rewrite the original expression.

Additionally, the purpose of this WeBWorK set is to famil-iarize yourself with WeBWorK.

Here are some hints on how to use WeBWorK effectively:• After first logging into WeBWorK change your pass-

word.• Find out how to print a hard copy on the computer sys-

tem that you are going to use. Print a hard copy of thisassignment.

• Get to work on this set right away and answer thesequestions well before the deadline. Not only will thisgive you the chance to figure out what’s wrong if an an-swer is not accepted, you also will avoid the likely rushand congestion prior to the deadline.

• The primary purpose of the WeBWorK assignments inthis class is to give you the opportunity to learn by hav-ing instant feedback on your active solution of relevantproblems. Make the best of it!

If the given expression does not simplify, re-enter the originalexpression.

Thought of the day: Well begun is half done.

1. (1 pt) set0/CAM pr1.pg

The following statement is false |−3|=−3.

Correct the statement by entering your answer below|−3|= .

Correct Answers:• 3


The following statement is false 3233 = 95.

Correct the statement by entering your answer below3233 = .



The following statement is false a2b5 = (ab)7.

Correct the statement by entering your answer belowa2b5 = .

Correct Answers:• a**2 * b**5


The following statement is false x + y− 3(z + w) = x + y−3z+w.

Correctly expand the expression belowx+ y−3(z+w) = .

Correct Answers:

• x+y-3*z-3*w


The following statement is false r4 −

(6−s)2 = r−12−2s

4 .

Correctly combine the fractions belowr4 −

(6−s)2 = .

Correct Answers:

• (r-12+2*s)/4


The following statement is false 3a+4b = 7ab.

Correct the statement by entering your answer below3a+4b = .

Correct Answers:

• 3*a + 4*b


The following statement is false 3x−1 = 13x .

Correctly write the expression as a fraction below3x−1 = .

Correct Answers:

• 3/x


The following statement is false√

x2 + y2 = x+ y.

Correct the statement by entering your answer below√x2 + y2 = .

Correct Answers:

• sqrt(x**2 + y**2)

1


The following statement is false x+yx+z = y

z .

Reduce the fraction belowx+yx+z = .

Correct Answers:

• (x+y)/(x+z)


The following statement is false 1x−y = −1

x+y .

Correctly factor −1 from the expression below1

x−y = .

Correct Answers:

• -1/(y-x)


The following statement is false xy + r

s = x+ry+s .

Correctly combine the fractions belowxy + r

s = .

Correct Answers:

• (x*s + y*r)/(y*s)


The following statement is false x ab = ax

bx .

Write the expression as a single fraction belowx a

b = .

Correct Answers:

• a*x/b


The following statement is false xa+xbx+xd = a+b

d .

Correctly reduce the fraction belowxa+xbx+xd = .

Correct Answers:

• (a+b)/(1+d)


The following statement is false: If 2(2− z) < 12 then z <−4.

Correct the inequality below:if 2(2− z) < 12 then z > .

Correct Answers:

• -4


The following statement is false 11− x

y= y

1−x .

Correctly simplify fraction below1

1− xy

= .

Correct Answers:

• y/(y-x)


The following statement is false a2a5 = a10.

Correct the statement by entering your answer belowa2a5 = .

Correct Answers:

• a**7


The following statement is false (3a)4 = 3a4.

Correct the statement by entering your answer below(3a)4 = .

Correct Answers:

• 81*aˆ4


The following statement is false ab −

ba = a−b

ab .

Correctly combine the fractions belowab −

ba = .

Correct Answers:

• (a**2 - b**2)/(a*b)

2


The following statement is false (x+4)2 = x2 +16.

Correctly expand the expression below(x+4)2 = .

Correct Answers:

• x**2 + 8*x + 16


The following statement is false r4 −

6−s4 = r−6−s

4 .

Correctly combine the fractions belowr4 −

6−s4 = .

Correct Answers:

• (r-6+s)/4


The following statement is false (a2)5 = a7.

Correct the expression below(a2)5 = .

Correct Answers:

• aˆ10


The following statement is false −24 = 16.

Correct the statement by entering your answer below−24 = .

Correct Answers:

• -16


The statement tanA = 4/3 is false given the right triangle

Correct the statement by entering your answer belowtanA = .

Correct Answers:

• 0.75


The statement sinB = 3/5 is false given the right triangle3

Correct the statement by entering your answer belowsinB = .

Correct Answers:

• 0.8


The statement cosC = 1 is false given the right triangle

Correct the statement by entering your answer belowcosC = .


26. (1 pt) set0/demo pr3.pgLet r = 5.

Evaluate 4/π∗ r = .Enter you answer as a decimal number listing at least 4 decimaldigits. (WeBWorK will reject your answer if it differs by morethan one tenth of 1 percent from what it thinks the answer is.)

Next, enter the expression 4/(π∗ r) = and let WeB-WorK compute the result.

Correct Answers:• 6.36619772367581• 0.254647908947033


4

Hsiang-Ping Huang math1210fall2008-2WeBWorK assignment number 1 is due : 09/10/2008 at 11:59pm MDT.The

(* replace with url for the course home page *)for the course contains the syllabus, grading policy and other information.

This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.

The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are makingsome kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you arehaving trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor forhelp. Don’t spend a lot of time guessing – it’s not very efficient or effective.

Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,you can if you wish enter elementary expressions such as 2∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e∧ (ln(2)) instead of 2,(2+ tan(3))∗ (4− sin(5))∧6−7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.

You can use the Feedback button on each problem page to send e-mail to the professors.

1. (1 pt) set1/Gross 1210 summer200 prob7.pgThe equation of the line that goes through the point (4,6) andis parallel to the line 4x + 4y = 4 can be written in the formy = mx+b where m is:and where b is:

Correct Answers:• -1• 10

2. (1 pt) set1/Gross 1210 summer200 prob8.pgThe equation of the line that goes through the points (−5,−5)and (4,5) can be written in the form y = mx + b where m is:

and where b is:

Correct Answers:• 1.11111111111111• 0.555555555555555

3. (1 pt) set1/Golden 1210 fall2001 prob12.pgLet f (x) = x3. Find the slope of the curve y = f (x) at the

point x = 1 by calculatingf (x+h)− f (x)

hand determining

what number it approaches as h approaches 0.f (x+h)− f (x)

h= Slope of f (x) at x = 1:

.Correct Answers:

• 3*x**2 + 3*x*h+ h**2• 3

4. (1 pt) set1/Golden 1210 fall2001 prob13.pg

Let f (x) = 2x + 5. Find f ′(x) by calculatingf (x+h)− f (x)

hand determining what it approaches as h approaches 0.

f (x+h)− f (x)h

= f ′(x) = .Correct Answers:

• 2

• 2

5. (2 pts) set1/Gross 1210 summer2000 set3 prob1.pgThe point P(3,17) lies on the curve y = x2 + x + 5. If Q is thepoint (x,x2 + x +5), find the slope of the secant line PQ for thefollowing values of x.If x = 3.1, the slope of PQ is:and if x = 3.01, the slope of PQ is:and if x = 2.9, the slope of PQ is:and if x = 2.99, the slope of PQ is:Based on the above results, guess the slope of the tangent lineto the curve at P(3,17).

Correct Answers:

• 7.1• 7.01• 6.9• 6.99• 7

6. (2 pts) set1/Gross 1210 summer2000 set3 prob2.pgIf a ball is thrown straight up into the air with an initial ve-locity of 85 ft/s, its height in feet after t seconds is given byy = 85t − 16t2. Find the average velocity for the time periodbegining when t = 2 and lasting(i) 0.5 seconds

(ii) 0.1 seconds

(iii) 0.01 seconds

Finally based on the above results, guess what the instanta-neous velocity of the ball is when t = 2.

Correct Answers:

• 13• 19.4• 20.84• 21

1

7. (1 pt) set1/Gross 1210 summer2000 set3 prob5.pgIf an arrow is shot straight upward on the moon with a veloc-ity of 79 m/s, its height (in meters) after t seconds is given byh = 79t−0.83t2.What is the velocity of the arrow (in m/s) after 5 seconds?

After how many seconds will the arrow hit the moon?With what velocity (in m/s) will the arrow hit the moon?

Correct Answers:

• 70.7• 95.1807228915663• -79

8. (1 pt) set1/Gross 1210 summer2000 set3 prob6.pgIf f (x) = 7x2−3x−15, find f ′(x).

Find f ′(3).

[NOTE: When entering functions, make sure that you put allthe necessary *, (, ), etc. in your answer. ]

Correct Answers:

• 2*7*x-3• 39

9. (1 pt) set1/Golden 1210 fall2001 set2 prob12.pgFind the antiderivative of 3x3 − 3x that has the value 7 whenx = 2.

The desired antiderivative is: .

Correct Answers:

• 1/4*3*x**4 - 1/2*3*x**2 - 1/4*3*2**4 + 1/2*3*2**2 + 7

10. (1 pt) set1/Golden 1210 fall2001 set2 prob13.pgFind

R 20 (x3 +2)dx.

R 20 (x3 +2)dx = .

Correct Answers:

• 8

11. (2 pts) set1/s0 1 77-82.pgFor each of the following functions, decide whether it is even,odd, or neither. Enter E for an EVEN function, O for an ODDfunction and N for a function which is NEITHER even nor odd.

NOTE: You will only have four attempts to get this problemright!

1. f (x) = x3 + x7 + x5

2. f (x) =−5x4−3x6−23. f (x) = x4−6x6 +3x6

4. f (x) = x4 +3x6 +2x5

Correct Answers:

• O• E• E• N

12. (2 pts) set1/c0s1p9.pgThis problem gives you some practice identifying how morecomplicated functions can be built from simpler functions.

Let f (x) = x3 + 1and let g(x) = x + 1. Match the functionsdefined below with the letters labeling their equivalent expres-sions.

1. f (x)/g(x)2. g( f (x))3. ( f (x))2

4. f (x2)

A. 1+ x6

B. 1+2x3 + x6

C. 1− x+ x2

D. 2+ x3

Correct Answers:

• C• D• B• A

13. (2 pts) set1/c0s2p1.pgRelative to the graph of

y = x2

the graphs of the following equations have been changed in whatway?

1. y = (10x)2

2. y = (x/10)2

3. y = (x+5)2

4. y = (x−5)2

A. stretched horizontally by the factor 10B. compressed horizontally by the factor 10C. shifted 5 units leftD. shifted 5 units right

Correct Answers:

• B• A• C• D

2


3

WeBWorK assignment 2Thought of the day: ”It’s not that I’m so smart; it’s just that I

stay with problems longer.” Albert Einstein

1. (3 pts) set2/composition.pgLet f be the linear function (in blue) and let g be the parabolicfunction (in red) below.

If you are having a hard time seeing the picture clearly, clickon the picture. It will expand to a larger picture on its own pageso that you can inspect it more closely.

Note: If the answer does not exist, enter ’DNE’:1. (f o g)( 2 ) =2. (g o f)( 2 ) =3. (f o f)( 2 ) =4. (g o g)( 2 ) =5. (f + g)( 4 ) =6. (f / g)( 2 ) =Correct Answers:

• -2• 4• -2• 4• 6• DNE

2. (3 pts) set2/srw5 1 a.pgFor each of the following angles, find the degree measure of theangle with the given radian measure:

3π

65π

44π

37π

25π

Correct Answers:

• 90

• 225• 240• 630• 900

3. (2 pts) set2/srw5 1 c.pgFor each of the following angles, find the radian measure of theangle with the given degree measure (you can enter π as ’pi’ inyour answers):

029020310170

Correct Answers:• 0• 5.06146666666667• 0.349066666666667• 5.41053333333333• 2.96706666666667

4. (3 pts) set2/srw5 1 d.pgFor each of the followings angles (in radian measure), find thesin of the angle (your answer cannot contain trig functions, itmust be an arithmetic expression or number):

π

6π

4π

3π

2π

2π

Correct Answers:• 0.5• 0.707106781186548• 0.866025403784439• 1• 0• 0

5. (3 pts) set2/srw5 1 e.pgFor each of the followings angles (in radian measure), find thecos of the angle (your answer cannot contain trig functions, itmust be an arithmetic expression or number):

π

6π

4π

3π

2π

2π

Correct Answers:• 0.866025403784439

1

• 0.707106781186548• 0.5• 0• -1• 1

6. (2 pts) set2/srwD 23b.pgIf θ = 7π

4 , then

sin(θ) equalscos(θ) equalstan(θ) equalssec(θ) equals

Correct Answers:

• -0.70710678562869• 0.707106776744405• -1.00000001256428• 1.41421357125738

7. (1 pt) set2/srw6 2 41.pgThe angle of elevation to the top of a building is found to be 12◦

from the ground at a distance of 4500 feet from the base of thebuilding. Find the height of the building.

HINT: Did you convert degrees to radians?Correct Answers:

• 956.504526389505

8. (4 pts) set2/limits.pgLet F be the function below.

If you are having a hard time seeing the picture clearly, clickon the picture. It will expand to a larger picture on its own pageso 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) =Correct Answers:

• -2• -2• -2• -1• 2• 3• DNE• 0• DNE

9. (1 pt) set2/prob1.pgEvaluate the limit

limx→4

x−4x2 +3x−28

Correct Answers:• 0.0909090909090909

10. (1 pt) set2/prob2.pgEvaluate the limit

limt→1

t3−1t2−1


11. (1 pt) 1060Library/set4 Trigonometry/s4p13.pgThe remaining problems in this set deal with the definitions ofthe basic trigonometric functions, sin, cos, and tan.

You can answer some of these questions simply by keying thingsinto your calculator. However, the purpose of these problems isto help you get familiar with the definitions of the basic trigono-metric functions, and to improve your ability to work with thosedefinitions. All the questions can be answered straight from thedefinitions of the trigonometric functions, perhaps after drawinga simple picture, without the aid of a calculator, and I recom-mend that you don’t use one. Use ’pi’ to enter the value of π

and use sqrt(...) to enter the square root of something.

A line drawn from the origin and forming the angle of t = 7π

6

with the x-axis intersects the unit circle at the point(−√

32 ,− 1

2

).

Complete the following equations:t = degrees.

2

cos t = .

sin t = .

tan t = .Correct Answers:

• 210• -0.866025403784439• -0.5• 0.577350269189626

12. (1 pt) 1060Library/set4 Trigonometry/s4p14.pg

A line drawn from the origin and forming the angle t with thex-axis intersects the unit circle at the point

(13 , 2

√2

3

). Complete

the following equations:

cos t = .sin t = .tan t = .

Correct Answers:• 0.333333333333333• 0.942809041582063• 2.82842712474619


The angle π

3 equals degrees.

More basic values:

cos(

π

3

)= .

sin(

π

3

)= .

tan(

π

3

)= .

Correct Answers:

• 60• 0.5• 0.866025403784439• 1.73205080756888


Let t be the angle between 0 and π

2 such that

sin t =14.

Then

cos t = .sin(−t) = .cos(−t) = .tan t = .

Correct Answers:

• 0.968245836551854• -0.25• 0.968245836551854• 0.258198889747161


Let again t be the angle between 0 and π

2 such that

sin t =14.

Then

cos(t +π) = .sin(t−π) = .cos(t +2π) = .tan(t +π) = .

Correct Answers:

• -0.968245836551854• -0.25• 0.968245836551854• 0.258198889747161


3

Hsiang-Ping Huang math1210fall2008-2WeBWorK assignment number 3 is due : 10/01/2008 at 11:59pm MDT.The

(* replace with url for the course home page *)for the course contains the syllabus, grading policy and other information.

This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.

The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are makingsome kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you arehaving trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor forhelp. Don’t spend a lot of time guessing – it’s not very efficient or effective.

Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,you can if you wish enter elementary expressions such as 2∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e∧ (ln(2)) instead of 2,(2+ tan(3))∗ (4− sin(5))∧6−7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.

You can use the Feedback button on each problem page to send e-mail to the professors.

1. (2 pts) 1210Library/set math1210spring2006-3/set3/p3-8.pgLet F be the function below.

If you are having a hard time seeing the picture clearly, clickon the picture. It will expand to a larger picture on its own pageso that you can inspect it more clearly.

At each given value of c, determine which of these is true aboutF(x):A - Not continuous because F(c) does not exits (DNE)B - Not continuous because limx→c F(c) does not exist (DNE)C - Not continuous beacuse limx→c F(c) 6= F(c)D - It is continuous

1. at c =−12. at c = 13. at c = 24. at c = 3

Correct Answers:

• A• C• D• B

2. (1 pt) set4/math1210spring2003 korevaar p3 13.pgLet

f (x) =1

x−4

Algebraically simplify the secant line slopef (x+h)− f (x)

h,

and enter the numerator below:

f (x+h)− f (x)h

= /((x+h−4)(x−4))

Let h−→ 0 to deduce the derivative,

f ′(x) =

The equation of the tangent line passing through the point onthe graph of f with x-coordinate 6 can be written in the formy = mx+b, where

m =b =

Correct Answers:

• -1• -1/(x-4)**2• -0.25• 2

3. (1 pt) set4/p4-1.pgIf f (x) = 5+ 2

x + 4x2 , find f ′(x).

Find f ′(4).1

Correct Answers:

• -2*x**(-2) -2*4*x**(-3)• -0.25

4. (1 pt) set4/p4-4.pgEvaluate the limit

limx→0

sin7xsin3x

Correct Answers:

• 2.33333333333333

5. (1 pt) set4/p4-5.pgIf f (x) = (3x2−4)(6x+4), find f ′(x).

[NOTE: Your answer should be a function in terms of the vari-able ’x’ and not a number! ]

Correct Answers:

• (3*2*x)*(6*x + 4) + (3*x**2 - 4 )*(6)

6. (1 pt) set4/math1210spring2003 korevaar p3 11.pgUsing your calculator to test values of h approaching zero, youshould be able to guess the following 3 limits. (We will seegeometric reasons for all of these in class.)

Remember, our trig functions eat radians unless otherwisenoted.

(a) limh→01− cos(h)

h=

(b) limh→0sin(h)

h=

(c) limh→01− cos(h)

h2 =

Correct Answers:

• 0• 1• 0.5

7. (1 pt) set4/p4-8.pgFind the equation of the line tangent to the curve

y = 3x2−10x+9

at the point(7,86) .

y =Correct Answers:

• 32*x+-138

8. (2 pts) set4/p4-9.pgFind all points on the graph of y = 1

3 x3 + x2− x where the tan-gent line has slope 1.

( , )( , )

Instruction: Enter the points in order of increasing x-coordinate.

Correct Answers:

• -2.73205080756888• 3.39871747423554• 0.732050807568877• -0.0653841409022105

9. (2 pts) set4/p4-10.pgGiven this graph, which describes the position of a particle wan-dering about on a vertical line, answer the questions using thevalues on the graph.

(Note that, the BLUE curve depicts the position of the parti-cle and the RED line is the tangent line to the graph.)

(a) What is the average velocity from t = 1 to t = 5?(b) What is the instantaneous velocity at t = 3?(c) When is it going the fastest?(d) At what time is it stopped (in order)? t = andt =

Correct Answers:

• 1.5• 1• 4• 2• 5

2

10. (2 pts) set4/p4-11.pgAssume that the function f (t) = t3− 4t2 + 3t + 3 describes theposition of a particle wandering about on a vertical line. Answerthe questions using calculus.

(a) What is the average velocity from t = 1 to t = 4?(b) What is the instantaneous velocity at t = 3?(c) At what time is it stopped (in order)? t = andt =

Correct Answers:

• 4• 6• 0.451416229645136• 2.21525043702153

11. (1 pt) set4/ps2prob5.pgEvaluate the limit

limx→0

tanx4x

Correct Answers:

• 0.25

12. (1 pt) set4/p4-7.pgIf f (x) = 6x+7

7x+5 , find f ′(x).

Find f ′(3).

[NOTE: When entering functions, make sure that you put allthe necessary *, (, ), etc. in your answer. ]

Correct Answers:

• (6*5 - 7*7)/(7*x +5)**2• -0.0281065088757396

13. (3 pts) set math1210spring2006-3/set4/math1210spring2003 korevaar p3 12 1.pg(a) Using the angle addition formulas for sin and cos (which youhave memorized!!) Fill in the blanks with appropriate functionsof h so that

sin(x+h)− sin(x)h

= sin(x) + cos(x)

cos(x+h)− cos(x)h

= sin(x) +

cos(x)

Hint: in each blank you will be entering expressions relatedto the functions of h.

(b) Use your work and part (a), to find the derivative func-tions for sin(x) and cos(x):

limh→0sin(x+h)− sin(x)

h=

limh→0cos(x+h)− cos(x)

h=

Correct Answers:• (cos(h)-1)/h• sin(h)/h• -sin(h)/h• (cos(h)-1)/h• cos(x)• -sin(x)

14. (1 pt) 1210Library/set math1210spring2007-90/set5/c1s5p5.pgThe function f is given by the formula

f (x) =−1x3 +9x2−19x+6

x−6when x < 6 and by the formula

f (x) =−6x2 +1x+awhen 6≤ x.What value must be chosen for a in order to make this functioncontinuous at 6?

a =Correct Answers:

• 191

15. (1 pt) 1210Library/set math1210spring2007-90/set6/p2-6.pgLet f (x) = 1−4x

1+4x . Then f ′(4) isand f ′′(4) isand f ′′′(4) is

Correct Answers:• -0.027681660899654• 0.0130266639527783• -0.00919529220196118

16. (1 pt) 1210Library/set math1210spring2007-90/set6/p3-10.pgFind the equation of the tangent line to the curve y = (x +1)(x2−1) at the point (1,0).y =

Correct Answers:• 4*x-4


3

Hsiang-Ping Huang math1210fall2008-2WeBWorK assignment number 4 is due : 10/08/2008 at 11:59pm MDT.THOUGHT: Don’t drink and derive!

Here’s the list of the functions which WeBWorK understands.You can use the Feedback button on each problem page to send e-mail to the professors.

1. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S05-Continuity/2-5-39.pgFor what value of c is the function defined below continuous on(−∞,∞)?

f (x) =

{cx+3, x < 2,

cx2−3, x≥ 2.

c =Correct Answers:

• 3

2. (1 pt) Library/Rochester/setLimitsRates5Continuity/ur lr 5 1.pg

A function f (x) is said to have a removable discontinuityat x = a if:1. f is either not defined or not continuous at x = a.2. f (a) could either be defined or redefined so that the newfunction IS continuous at x = a.

Let f (x) = 2x2+3x−27x−3

Show that f (x) has a removable discontinuity at x = 3 and deter-mine what value for f (3) would make f (x) continuous at x = 3.Must define f (3) = .

Correct Answers:

• 15

3. (1 pt) set5/golden math1210fall2002 p5 22.pgAt time t seconds, the center of a bobbing cork is 2sin t cen-timeters above (or below) water level. What is the velocity ofthe cork at t = 0,π/2,π?

Velocity at t = 0: cm/s.Velocity at t = π/2: cm/s.Velocity at t = π: cm/s.

Correct Answers:

• 2• 0• -2

4. (1 pt) set5/korevaar math1210spring2003 p4 4.pg

A city is hit by a flu epidemic. Officials estimate that t daysafter the beginning of the epidemic the number of persons sickwith the flu is given by

p(t) = 30t2−2t3, when 0≤ t ≤ 15.

At what rate is the flu spreading at time t = 5 days?people/day

At what rate is the flu spreading at time t = 12 days?people/day

How many days after the outbreak are the most people sick?t = days.

Correct Answers:

• 150• -144• 10

5. (1 pt) set5/s2 2 16.pgIf

f (x) =√

x−6√x+6

find f ′(x).

Find f ′(3).

Correct Answers:

• (6/sqrt(x))/((sqrt(x) +6)**2)• 0.0579430330490105

6. (1 pt) set5/s2 4 21.pgIf f (x) = 6sinx+6cosx, thenf ′(x) =

Correct Answers:

• 6*cos(x) - 6*sin(x)

7. (1 pt) set5/s2 4 24.pgIf

f (x) =5sinx

2+ cosxfind f ′(x).

1

Correct Answers:

• (5*cos(x)*2 +5)/(2+cos(x))**2

8. (1 pt) set5/s2 4 26.pgIf

f (x) =tanx−2

secxfind f ′(x).

Find f ′(4).

Correct Answers:

• cos(x) + 2*sin(x)• -2.16724861147947

9. (1 pt) set5/s2 4 30a.pgLet

f (x) =−4xsinxcosx

f ′( 3π

2 ) =Correct Answers:

• 18.8495559215388

10. (1 pt) set5/s2 5 1.pgIf f (x) = (x2 +2x+8)4, find f ′(x).

Find f ′(5).

Correct Answers:

• (4*(x**2+2*x+8)**(4-1))*(2*x+2)• 3816336

11. (1 pt) set5/s2 5 4.pgIf f (x) = sin(x4), find f ′(x).

Find f ′(1).

Correct Answers:

• (cos(x**4))*(4*x**(4-1))• 2.16120922347256

12. (1 pt) set5/s2 5 5.pgIf f (x) = sin5 x, find f ′(x).

Find f ′(1).

Correct Answers:• (5*sin(x)**(5-1))*(cos(x))• 1.35445133968776

13. (1 pt) set5/s2 5 12a.pgLet

f (x) = sin(cos(x5))f ′(x) =

Correct Answers:• -cos(cos(x**5))*sin(x**5)*5*x**(5-1)

14. (1 pt) 1210Library/set5 The Derivative/1210s5p3.pgLet

f (x) = xsinx2.

f ′(x) = .f ′′(x) = .

Correct Answers:• sin(x*x) +2*x*x*cos(x*x)• 6*x*cos(x*x)-4*x**3*sin(x**2)


f (x) = sin1x.

f ′(x) = .Let

g(x) =1

sinx.

g′(x) = .Correct Answers:

• -cos(1/x)/(x*x)• -cos(x)/sin(x)**2


f (x) = tanx2.

f ′(x) = .Let

g(x) = tan2 x.

g′(x) = .Correct Answers:

• 2*x/cos(x*x)**2• 2tan(x)/cos(x)/cos(x)

2


3

Hsiang-Ping Huang math1210fall2008-2WeBWorK assignment number 5 is due : 10/22/2008 at 11:59pm MDT.THOUGHT: The early bird may get the worm, but the second mouse gets the cheese.

1. (1 pt) 1210Library/set5 The Derivative/1210s5p9.pgsuppose 4x2 + 5x + xy = 5 and y(5) = −24. Find y′(5) by im-plicit differentiation.Your answer:

Hint: You’ll also have to solve for y.Correct Answers:

• -4.2

2. (1 pt) 1210Library/set5 The Derivative/1210s5p11.pgLet A be the area of a circle with radius r. If dr

dt = 5, find dAdt

when r = 3.Your answer:

Correct Answers:

• 94.2477795

3. (1 pt) 1210Library/set5 The Derivative/1210s5p12.pgUse implicit differentiation to find the equation of the tangentline to the curve xy3 + xy = 8 at the point (4,1). The equationof this tangent line can be written in the form y = mx+b wherem is:and where b is:

Correct Answers:

• -0.125• 1.5

4. (1 pt) 1210Library/set5 The Derivative/1210s5p15.pgSuppose

√x +

√y = 11 and y(9) = 64. Find y′(9) by implicit

differentiation.

Correct Answers:

• -2.66666666666667

5. (2 pts) 1210Library/set5 The Derivative/1210s5p17.pgThe graph of the equation

x2 + xy+ y2 = 9

is a slanted ellipse illustrated in this figure:

Think of y as a function of x. Differentiating implicitly andsolving for y′ gives:y′ = . (Your answer will depend on x and y.)

The ellipse has two horizontal tangents. The upper one hasthe equation

y = .The right most vertical tangent has the equationx = .That tangent touches the ellipse wherey = .Hint: The horizontal tangent is of course characterized byy′ = 0. To find the vertical tangent use symmetry, or think ofx as a function of y, differentiate implicitly, solve for x′ and thenset x′ = 0.

Correct Answers:

• -(2*x+y)/(x+2*y)• 3.46410161513775• 3.46410161513775• -1.73205080756888

6. (1 pt) 1210Library/set5 The Derivative/1210s5p19.pgThe altitude of a triangle is increasing at a rate of 3.0 centime-ters/minute while the area of the triangle is increasing at a rateof 2.0 square centimeters/minute. At what rate is the base of thetriangle changing when the altitude is 12.0 centimeters and thearea is 86.0 square centimeters?Your answer:Hint: The area A of a triangle with base b and height h is givenby

A =12

bh.

Differentiate implicitly.Correct Answers:

1

• -3.25

7. (1 pt) 1210Library/set5 The Derivative/1210s5p20.pgGravel is being dumped from a conveyor belt at a rate of 20 cu-bic feet per minute. It forms a pile in the shape of a right circularcone whose base diameter and height are always the same. Howfast is the height of the pile increasing when the pile is 16 feethigh? Recall that the volume of a right circular cone with heighth and radius of the base r is given by V = π

3 r2h.Your answer: feet per minute.

Correct Answers:

• 0.0994718394324346

8. (1 pt) 1210Library/set5 The Derivative/1210s5p23.pgA spherical snowball is melting in such a way that its diameteris decreasing at rate of 0.3 cm/min. At what rate is the volumeof the snowball decreasing when the diameter is 10 cm.Your answer (cubic centimeters per minute) shouldbe a positive number.Hint: The volume of a sphere of radius r is

V =4πr3

3.

The diameter is twice the radius.Correct Answers:

• 47.12388975

9. (1 pt) set6/set6 pr4.pgLet

y =x+10x+8

,x 6=−8.

Find the equation of the line tangent to the curve at the point

(2,6/5) .

y =Correct Answers:

• (8-10)/(8+2)ˆ2*x+2*(10-8)/(8+2)ˆ2+(2+10)/(8+2)

10. (1 pt) set6/set6 pr8.pg

Let f (x) =(x2−3

)2. For what values of x is f ′′(x) = 0? Writethe answers in increasing order.

, .Correct Answers:

• -1• 1

11. (1 pt) set6/set6 pr13.pgFind the slope of the tangent line to the curve given by the equa-tion √

y+7xy2 =−2at the point (−0.428571428571429,1).

y′ =Correct Answers:

• 1.27272727272727

12. (1 pt) set6/set6 pr14.pgIf the variables s and t are related by the equation

st +7t3 = 12

find dsdt .

dsdt =Correct Answers:

• -(3*7*tˆ2+s)/t

13. (1 pt) set6/set6 pr7.pgFor y2 + xy− x2 = 11,(a) find dy

dx = ,as a function of x and y.(b) find the slope of the tangent at (2,3)

Correct Answers:• (2*x - y)/(2*y + x)• 0.125

14. (1 pt) set6/set6 pr9.pgA street light is at the top of a 16.000 ft. tall pole. A man 5.000ft tall walks away from the pole with a speed of 4.000 feet/secalong a straight path. How fast is the tip of his shadow movingwhen he is 34.000 feet from the pole?


15. (1 pt) 1210Library/set math1210fall2002-1/set5/pr6.pgIf

f (x) =tanx−3

secxfind f ′(x).

Find f ′(3).

Correct Answers:• cos(x) + 3*sin(x)• -0.566632472420844

16. (1 pt) 1210Library/set math1210fall2002-1/set5/pr7.pgIf f (x) = sin(x5), find f ′(x).

Find f ′(5).

Correct Answers:• (cos(x**5))*(5*x**(5-1))• -1979.77854901275


2

Hsiang-Ping Huang

Math 1210-2, Fall 2008

Assignment 6 due November 5

The early bird may get the worm, but the second mouse gets thecheese.

1. (3 pts) Library/UVA-Stew5e/setUVA-Stew5e-C04S05-CurveSketch-/4-5-06.pgSuppose that

f (x) = 7x5−2x4.

(A) Find all critical values of f . If there are no critical values,enter -1000. If there are more than one, enter them separated bycommas.Critical value(s) =

(B) Use interval notation to indicate where f (x) is increasing.Note: When using interval notation in WeBWorK, you use

I for ∞, -I for −∞, and U for the union symbol. If there are novalues that satisfy the required condition, then enter ”” withoutthe quotation marks.

Increasing:

(C) Use interval notation to indicate where f (x) is decreasing.Decreasing:

(D) Find the x-coordinates of all local maxima of f . If there areno local maxima, enter -1000. If there are more than one, enterthem separated by commas.

Local maxima at x =

(E) Find the x-coordinates of all local minima of f . If there areno local minima, enter -1000. If there are more than one, enterthem separated by commas.

Local minima at x =

(F) Use interval notation to indicate where f (x) is concave up.Concave up:

(G) Use interval notation to indicate where f (x) is concavedown.Concave down:

(H) Find all inflection points of f . If there are no inflectionpoints, enter -1000. If there are more than one, enter them sep-arated by commas.Inflection point(s) at x =

(I) Find all horizontal asymptotes of f . If there are no horizontalasymptotes, enter -1000. If there are more than one, enter themseparated by commas.Horizontal asymptote(s): y =

(J) Find all vertical asymptotes of f . If there are no verticalasymptotes, enter -1000. If there are more than one, enter themseparated by commas.Vertical asymptote(s): x =

(K) Use all of the preceding information to sketch a graph of f .When you’re finished, enter a ”1” in the box below.

Graph Complete:Correct Answers:

• 0, 0.228571428571429• (-infinity,0) U (0.228571428571429,infinity)• (0,0.228571428571429)• 0• 0.228571428571429• (0.171428571428571,infinity)• (-infinity,0.171428571428571)• 0.171428571428571• -1000• -1000• 1

2. (2 pts) Library/maCalcDB/setDerivatives1/ur dr 1 11.pgAnswer the following True-False quiz. (Enter ”T” or ”F”.)

1. If f ′(c) = 0, then c is either a local maximum or a localminimum.

2. If f (x) and g(x) are increasing on an interval I, thenf (x)g(x) is increasing on I.

3. A continuous function on a closed interval always at-tains a maximum and a minimum value.

4. If a function has a local maximum at c, then f ′(c) existsand is equal to 0.

5. Continuous functions are always differentiable.6. Differentiable functions are always continuous.7. If f (x) = e2, then f ′(x) = 2e.

Correct Answers:

• F• F• T• F• F• T• F

3. (2 pts) set7/c3s3p4.pgFor x ∈ [−11,13] the function f is defined by

f (x) = x4(x−2)3

1

On which two intervals is the function increasing?to

andto

Find the region in which the function is positive: to

Where does the function achieve its minimum?Correct Answers:

• -11• 0• 1.14285714285714• 13• 2• 13• -11

4. (1 pt) set7/c3s8p2.pgA rectangle is inscribed with its base on the x-axis and its uppercorners on the parabola y = 11− x2. What are the dimensionsof such a rectangle with the greatest possible area?

Width = Height =Correct Answers:

• 3.82970843102535• 7.33333333333333

5. (2 pts) set7/golden-math1210fall2001-ps6-q13.pgIdentify the critical points and find the maximum value and min-imum value of the following function on the given interval. Re-call, critical points are either endpoints, stationary points, or sin-gular points.

f (x) = 3√

x, over [−1,27].Critical Points: , , .

Maximum: .Minimum: .

Instructions: When entering the critical points, please enterthem in the order that they appear on the real line.

Correct Answers:

• -1• 0• 27• 3• -1

6. (1 pt) set7/nsc4 6 3.pgIf 2000 square centimeters of material is available to make a boxwith a square base and an open top, find the largest possible vol-ume of the box.Volume = cubic centimeters.

Correct Answers:

• 8606.62965823871

7. (1 pt) set7/nsc4 6 16.pgA fence 2 feet tall runs parallel to a tall building at a distanceof 3 feet from the building. What is the length of the shortestladder that will reach from the ground over the fence to the wallof the building?

Correct Answers:

• 7.02348237921997

8. (1 pt) set7/s3 8 6.pgA rancher wants to fence in an area of 1500000 square feet in arectangular field and then divide it in half with a fence down themiddle parallel to one side. What is the shortest length of fencethat the rancher can use?

Correct Answers:

• 6000

9. (2 pts) set6/set6 pr1.pgLet f (x) = x2 + 1

x −√

x+π. Answer the following.f ′(x) =

f ′′(x) =f (1) =f ′(1) =f ′′(1) =Correct Answers:

• 2*x - 1/xˆ2 - 1/(2*sqrt(x))• 2 + 2/xˆ3 + 1/(4*sqrt(xˆ3))• 4.14159265358979• 0.5• 4.25

10. (2 pts) set7/c3s3p1.pgThe function

f (x) = 2x3 +6x2−288x−5

is decreasing on the interval ( , ).It is increasing on the interval ( −∞, ) and the interval (, ∞ ).The function has a local maximum at .Correct Answers:

• -8• 6• -8• 6• -8

11. (1 pt) Library/ma122DB/set7/s4 1 5.pgLet f (x) be the function shown in the graph below.

2

Click on the graph to enlarge it.

Determine the absolute minimum of the function shown inthe graph.Answer: x = , f (x) = .

Determine the absolute maximum of the function shown inthe graph.

Answer: x = , f (x) = .

Select maximum or minimum for the following:The function attains a local ? at x = 2.The function attains a local ? at x = 4.The function attains a local ? at x = 5.

Correct Answers:• 6• 0• 3• 4• minimum• minimum• maximum

12. (1 pt) Library/Indiana/Indiana setDerivatives10 5Optim-/c3s8p1.pgFind the point on the line 4x+4y+6 = 0 which is closest to thepoint (4,3).

( , )Correct Answers:

• -0.25• -1.25


3

Hsiang-Ping HuangMath 1210-2, Fall 2008


Thought: Those who won’t have no advantage over those whocan’t.

1. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S06-InfLimits/2-6-37f.pgLet

f (x) =x2 +1x−6

3x2 +7x−6.

Find the horizontal and vertical asymptotes of f (x). If there areno asymptotes of a given type, enter 1000. If there are morethan one of a given type, list them separated by commas.

Horizontal asymptote(s): y =

Vertical asymptote(s): x =

Correct Answers:

• 0.333333333333333• 0.666666666666667

2. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S06-InfLimits/2-6-19.pgEvaluate

limx→∞

1−10√

x9+10

√x.

Enter I for ∞, -I for −∞, and DNE if the limit does not exist.Limit =

Correct Answers:

• -1

3. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S06-InfLimits/2-6-40.pgLet

f (x) =x−1√

6x2 +5x+5.

Find the horizontal and vertical asymptotes of f (x). If there areno asymptotes of a given type, enter None . If there are morethan one of a given type, list them separated by commas.


Vertical asymptote(s): x =Correct Answers:

• 0.408248290463863, -0.408248290463863• None

4. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S06-InfLimits/2-6-15c.pgEvaluate the limit

limx→∞

2x3−6x2−2x6−10x−5x3

Enter I for ∞, -I for −∞, and DNE if the limit does not exist.Limit =

Correct Answers:

• -0.4

5. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S06-InfLimits/2-6-39.pgLet

f (x) =x

4√x4 +2.

Find the horizontal and vertical asymptotes of f (x). If there areno asymptotes of a given type, enter 1000. If there are morethan one of a given type, list them separated by commas.


Vertical asymptote(s): x =

Correct Answers:

• 1• 1000

6. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S06-InfLimits/2-6-44.pgSuppose that

f (x) = (6+ x)6(4− x)(2− x).

Evaluate the following limits. Enter I for ∞, -I for −∞, andDNE if the limit does not exist.

limx→∞

f (x) =

limx→−∞

f (x) =

Correct Answers:

• I• I

1

7. (2 pts) Library/UVA-Stew5e/setUVA-Stew5e-C04S05-CurveSketch-/4-5-07kmca.pgSuppose that

f (x) =7x−5x+6

.

(A) Use interval notation to indicate where f (x) is concaveup.Note: When using interval notation in WeBWorK, you use Ifor ∞, -I for −∞, and U for the union symbol. If there are novalues that satisfy the required condition, then enter ”” withoutthe quotation marks.

Concave up:

(B) Use interval notation to indicate where f (x) is concavedown.Concave down:

(C) Find all horizontal asymptotes of f . If there are no horizon-tal asymptotes, enter -1000. If there are more than one, enterthem separated by commas.Horizontal asymptote(s): y =

(D) Find all vertical asymptotes of f . If there are no verticalasymptotes, enter -1000. If there are more than one, enter themseparated by commas.Vertical asymptote(s): x =

Correct Answers:

• (-infinity,-6)• (-6,infinity)• 7• -6


f (x) = x4−5x3.



I for ∞, -I for −∞, and U for the union symbol. If there are novalues that satisfy the required condition, then enter ”” withoutthe quotation marks.

Increasing:

(C) Use interval notation to indicate where f (x) is decreasing.

Decreasing:


Local maxima at x =


Local minima at x =




(I) Use all of the preceding information to sketch a graph of f .When you’re finished, enter a ”1” in the box below.


• 0, 3.75• (3.75,infinity)• (-infinity,3.75)• -1000• 3.75• (-infinity,0) U (2.5,infinity)• (0,2.5)• 0, 2.5• 1


f (x) =4x

x2−49.



I for ∞, -I for −∞, and U for the union symbol. If there are no2

values that satisfy the required condition, then enter ”” withoutthe quotation marks.

Increasing:

(C) Use interval notation to indicate where f (x) is decreasing.Decreasing:


Local maxima at x =


Local minima at x =




(I) Find all horizontal asymptotes of f . If there are no horizontalasymptotes, enter -1000. If there are more than one, enter themseparated by commas.Horizontal asymptote(s): y =

(J) Find all vertical asymptotes of f . If there are no verticalasymptotes, enter -1000. If there are more than one, enter themseparated by commas.Vertical asymptote(s): x =

(K) Use all of the preceding information to sketch a graph of f .When you’re finished, enter a ”1” in the box below.


• -1000• {}• (-infinity,-7) U (-7,7) U (7,infinity)• -1000• -1000• (-7,0) U (7,infinity)

• (-infinity,-7) U (0,7)• 0• 0• -7, 7• 1

10. (4 pts) set8/c3s4p3.pgAnswer the following questions for the function

f (x) =x3

x2−16defined on the interval [−17,16].

Enter points, such as inflection points in ascending order, i.e.smallest x values first.

Enter intervals in ascending order also.A. The function f (x) has vertical asympototes at and

B. f (x) is concave up on the region to andto

C. The inflection points for this function are ,and

Correct Answers:• -4• 4• -4• 0• 4• 16• -4• 0• 4

11. (4 pts) set8/s3 4 6a.pgConsider the function f (x) = 2x+5x−1. For this function thereare four important intervals: (−∞,A], [A,B),(B,C], and [C,∞)where A, and C are the critical numbers and the function is notdefined at B.Find Aand Band CFor each of the following intervals, tell whether f (x) is increas-ing (type in INC) or decreasing (type in DEC).(−∞,A]:[A,B):(B,C]:[C,∞)Note that this function has no inflection points, but we can stillconsider its concavity. For each of the following intervals, tellwhether f (x) is concave up (type in CU) or concave down (typein CD).(−∞,B):(B,∞):

Correct Answers:• -1.58113883008419• 0• 1.58113883008419• INC

3

• DEC• DEC• INC

• CD• CU


4

Hsiang-Ping HuangMath 1210-2, Fall 2008


Once your mind has been stretched, it will never resume its orig-inal shape.

— A. Einstein1. (7 pts) set8/math1210-90 fall2002 rossi-p8-8.pg

Here’s an interesting example in which the graph crosses itshorizontal asymptote.

If you think an answer below is ∞, enter INF. If you thinkan answer is −∞, enter -INF. If the function does not have therequested attribute, answer ”no”. If more intervals are giventhan are needed, enter ”no” in all extra blanks.

Let

y =x+2

(x−6)2

a. The graph has a vertical asymptote x = a fora= .b. The horizontal asymptote isy= .c. As x approaches a from the left, y approaches

.d. As x approaches a from the right, y approaches

.e. The graph has a local maximum atx= .f. The graph has a local minimum atx= .g. The graph is increasing in the intervals( , ) and( , ).h. The graph is concave down in the intervals( , ) and( , ).

Correct Answers:

• 6• 0• INF• INF• NO• -10• -10• 6• NO• NO• -INF• -18• NO

• NO

2. (1 pt) set8/set8 pr6.pgEvaluate the limit

limx→∞

2+4x11−9x

Correct Answers:

• -0.444444444444444

3. (1 pt) set8/set8 pr7.pgEvaluate the limit

limx→∞

√11+2x2

(8+7x)

Correct Answers:

• 0.202030508910442

4. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C04S02-MeanValThm-/4-2-13a.pgConsider the function f (x) = 1

x on the interval [2,9].(A) Find the average or mean slope of the function on this inter-val.

Average Slope =(B) By the Mean Value Theorem, we know there exists a c in theopen interval (2,9) such that f ′(c) is equal to this mean slope.Find all values of c that work and list them (separated by com-mas) in the box below.

List of values:Correct Answers:

• -0.0555555555555556• 4.24264068711928

5. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C04S02-MeanValThm-/4-2-12.pgConsider the function f (x) = 3x3−4x on the interval [−4,4].(A) Find the average or mean slope of the function on this inter-val.

Average Slope =(B) By the Mean Value Theorem, we know there exists at leastone c in the open interval (−4,4) such that f ′(c) is equal to thismean slope. Find all values of c that work and list them (sepa-rated by commas) in the box below.

List of values:Correct Answers:

• 44• -2.3094010767585, 2.3094010767585

1

6. (1 pt) Library/Rochester/setDerivatives12MVT/c3s2p1.pgConsider the function

f (x) = 2x3 +1x2−3x+1

Find the average slope of this function on the interval (−2,5).

By the Mean Value Theorem, we know there exists a c in theopen interval (−2,5) such that f ′(c) is equal to this mean slope.Find the value of c in the interval which works

Correct Answers:

• 38• 2.45270560758362

7. (1 pt) Library/ma122DB/set7/s4 2 25.pgDoes there exist a continuous function f (x) such that f (0) =6, f (2) = 8 and f ′(x)≤−3 for all x in (0,2)?

Answer: (yes or no )Note: You only have one chance to input your answer.

Correct Answers:

• No

8. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C04S09-NewtonsMethod/4-9-05.pgUse Newton’s method to approximate a root of the equationx3 + x+3 = 0 as follows:Let x1 =−1 be the initial approximation.The second approximation x2 isand the third approximation x3 is

Correct Answers:

• -1.25• -1.21428571428571

9. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C04S09-NewtonsMethod/4-9-09.pgUse Newton’s method to approximate the value of

3√40

as follows:Let x1 = 3 be the initial approximation.The second approximation x2 isand the third approximation x3 is

Correct Answers:

• 3.48148148148148• 3.42103292367364

10. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C05S04-IndefInts/5-4-06.pgEvaluate the indefinite integral:Z (

33√

x−6 3√

x2

)dx = + C.

Correct Answers:

• (3*3/2)*x**(2/3) - (3*6/5)*x**(5/3)

11. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C05S04-IndefInts/5-4-05c.pgEvaluate the indefinite integral:Z (

7x4− 3x5 −3

)dx = + C.

Correct Answers:• 7*(x**5)/5 - 3*(x**(-5+1))/(-5+1) - 3*x

12. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C05S04-IndefInts/5-4-05a.pgEvaluate the indefinite integral:Z (

7s4−6s5)

ds = + C.

Correct Answers:• (7/5)*s**5 - (6/6)*s**6

13. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C05S04-IndefInts/5-4-07.pgEvaluate the indefinite integral:Z (

3x2 +6x−6)

dx = + C.

Correct Answers:• (3/3)*x**3 + (6/2)*x**2 - 6*x

14. (1 pt) set9/s3 10 2func.pgConsider the function f (x) = 8x3−24x2 +12x−2. Enter an an-tiderivative of f (x)

Correct Answers:• 2*xˆ4-8*xˆ3+6*xˆ2-2*x

15. (1 pt) set9/set9 pr7.pgConsider the differential equation:

dydx

=√

xy

.

a) Find the general solution to the above differential equation.(Instruction: Call your integration constant C.)

Answer: y = .

b) Find the particular solution of the above differential equa-tion that satisfies the condition y = 4 at x = 1.

2

Answer: y = .Correct Answers:

• (x**(3/2)+C)**(2/3)• (x**(3/2)+7)**(2/3)

16. (1 pt) set9/set9 pr8.pgConsider the differential equation:

dudt

= u3(t3− t).

a) Find the general solution to the above differential equation.(Instruction: Write the answer in a form such that its numeratoris 1 and its integration constant is C — rename your constant ifnecessary.)

Answer: u = .

b) Find the particular solution of the above differential equa-tion that satisfies the condition u = 4 at t = 0.

Answer: u = .Correct Answers:

• 1/sqrt(t**2-(t**4)/2+C)• 1/sqrt(t**2-(t**4)/2+1/16)

17. (1 pt) set9/set9 pr11.pgA rectangle is inscribed with its base on the x-axis and its uppercorners on the parabola y = 8− x2. What are the dimensions ofsuch a rectangle with the greatest possible area?

Width = Height =Correct Answers:

• 3.2659863237109• 5.33333333333333


3

Hsiang-Ping Huang

Math 1210-3

Assignment 9

Thought: Education is what you get when you read the fineprint. Experience is what you get when you don’t.

1. (1 pt) set10/set10 pr1.pgUse the Special Sum Formulas (see Section 4.1 of Varberg, Pur-cell and Rigdon) to find:

∑10i=1((i−1)(4i+3)) = .

Correct Answers:

• 1455

2. (1 pt) set10/set10 pr2.pgConsider the integral Z 9

5

(3x

+1)

dx

(a) Find the Riemann sum for this integral using right end-points and n = 4.

(b) Find the Riemann sum for this same integral, using left end-points and n = 4

Correct Answers:

• 5.63690476190476• 5.90357142857143

3. (1 pt) set10/set10 pr3.pgEvaluate the integral below by interpreting it in terms of areas.In other words, draw a picture of the region the integral repre-sents, and find the area using high school geometry.Z 2

−2

√4− x2dx

Correct Answers:

• 6.283185308

4. (1 pt) set10/set10 pr4.pgEvaluate the sum:

6

∑i=1

(2− i)

Correct Answers:

• -9

5. (1 pt) set10/set10 pr5.pgIf

R 10 f (x)dx = 4,

R 20 f (x)dx = 2, and

R 20 g(x)dx = −3, evaluate

each integral.

(a)R 2

1 f (x)dx =

(b)R 0

1 f (x)dx =

(c)R 2

0 3 f (x)dx =

(d)R 2

0 [2g(x)−3 f (x)]dx =

(e)R −2

0 f (−x)dx =

Correct Answers:• -2• -4• 6• -12• 2

6. (1 pt) set10/set10 pr6.pgSketch the region under the curve y = 16−x2 between x = 0 andx = 3, showing the inscribed polygon corresponding to a regularpartition of [0,3] into 6 subintervals. Using this partition, whatis the approximate area under the curve?

Area ≈


7. (1 pt) set10/set10 pr8.pgEvaluate the definite integralZ 9

3

6x2 +5√x

dx


8. (1 pt) set10/set10 pr9.pgUse part I of the Fundamental Theorem of Calculus to find thederivative of

f (x) =Z x

4

(14

t2−1)15

dt

f ′(x) =[NOTE: Enter a function as your answer. Make sure that your

1

syntax is correct, i.e. remember to put all the necessary *, (, ),etc. ]

Correct Answers:• (1/4*xˆ2-1)ˆ15

9. (1 pt) set10/set10 pr7.pgEvaluate the definite integralZ 8

3(2x+4)dx


10. (1 pt) Indiana/Indiana setIntegrals4FTC/sc5 4 19a.pgUse part I of the Fundamental Theorem of Calculus to find thederivative of

g(x) =Z 2x

4x

u+1u−3

du

[NOTE: Enter a function as your answer. Make sure that yoursyntax is correct, i.e. remember to put all the necessary *, (, ),etc. ] For more help see: WeBWorK functions

Correct Answers:• 2*(2*x+1)/(2*x-3)-4*(4*x+1)/(4*x-3)

11. (1 pt) Library/Indiana/Indiana setIntegrals4FTC/c4s4p4.pg

If f (x) =Z x2

xt2dt

thenf ′(x) =f ′(−2) =

Correct Answers:• 2*x*xˆ4 - xˆ2• -68

12. (1 pt) 1210Library/set math1210fall2006-2/set9/sc5 2 3.pgConsider the integral Z 5

2(3x2 +4x+6)dx

(a) Find the Riemann sum for this integral using right end-points and n = 3.

(b) Find the Riemann sum for this same integral, using left end-points and n = 3

Correct Answers:• 216• 141

13. (1 pt) 1210Library/set math1210fall2006-2/set9/sc5 2 30.pgZ 18

4f (x)−

Z 8

4f (x) =

Z b

af (x)

where a= and b=Correct Answers:

• 8• 18

14. (1 pt) Library/Indiana/Indiana setIntegrals4FTC/sc5 4 12.pgUse part I of the Fundamental Theorem of Calculus to find thederivative of

f (x) =Z x

−1

√t3 +1dt

f ′(x) =[NOTE: Enter a function as your answer. Make sure that yoursyntax is correct, i.e. remember to put all the necessary (, ), etc.]

Correct Answers:• sqrt(xˆ3+1)

15. (1 pt) Library/Indiana/Indiana setIntegrals4FTC/sc5 4 13.pgUse part I of the Fundamental Theorem of Calculus to find thederivative of

f (x) =Z x

2

11+ t1 dt

f ′(x) =[NOTE: Enter a function as your answer. Make sure that yoursyntax is correct, i.e. remember to put all the necessary *, (, ),etc. ]

Correct Answers:• 1/(1+xˆ(1))

16. (1 pt) Indiana/Indiana setIntegrals4FTC/sc5 4 14.pgUse part I of the Fundamental Theorem of Calculus to find thederivative of

F(x) =Z 6

xsin(t4)dt

F ′(x) =

[NOTE: Enter a function as your answer.]Correct Answers:

• -sin(xˆ4)


2

Hsiang-Ping HuangMath 1210-2

Assignment 10

THE END IS NEAR! Procrastination may have its own naturalconsequences.

1. (1 pt) set10/math1210-90 fall2002 rossi set10 p7.pgSketch the region enclosed by the given curves. Decide whetherto integrate with respect to x or y. Then find the area of the re-gion.y = 2x,y = 7x2

Correct Answers:

• 0.0272108843537415

2. (1 pt) set10/math1210-90 fall2002 rossi set10 p8.pgSketch the region enclosed by the given curves. Decide whetherto integrate with respect to x or y. Then find the area of theregion.y = 6x2,y = x2 +2

Correct Answers:

• 1.68654808542314

3. (1 pt) Indiana/Indiana setIntegrals3Definite/sc5 3 17.pgEvaluate the integral Z 2

3sin(t)dt

Correct Answers:

• -0.573845660053303

4. (1 pt) Indiana/Indiana setIntegrals3Definite/c4s4p6.pg

The value ofZ 6

4

1x2 dx is

Correct Answers:

• 0.0833333333333333

5. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C06S02-VolumeSlicing-/6-2-16.pgAs a hardworking student, plagued by too much homework, youspend all night doing math homework. By 6am, you imagineyourself to be a region bounded by

y = 11x2, x = 0, x = 1, y = 0

As you grow more and more tired, the world begins to spinaround you. However, according to Newton, there is no differ-ence between the world spinning around you, and you spinningaround the world. Unfortunately, you are so tired that you thinkthe world is the x-axis. What is the volume of the solid you (the

region) create by spinning about the x-axis?Volume =

Correct Answers:

• 76.026542216873

6. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C06S05-AveValue/6-5-06.pgFind the average value of : f (x) = 4sinx+8cosxon the interval [0,13π/6]

Average value =

Correct Answers:

• 0.666379060904651

7. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C06S05-AveValue/6-5-02.pgA car drives down a road in such a way that its velocity ( in m/s)at time t (seconds) is

v(t) = 1t1/2 +5

.Find the car’s average velocity (in m/s) between t = 2 and t = 7.

Correct Answers:

• 7.09224427369413

8. (1 pt) Library/Rochester/setIntegrals22Average/csuf in 22 1.pg

Find the average value of f (x) =6x3 +9x on the interval [1,2].

Average value =

Correct Answers:

• 15.75

9. (1 pt) Indiana/Indiana setIntegrals3Definite/s4 4 17.pgEvaluate the definite integralZ 8

2(10x+8)dx

Correct Answers:

• 348

10. (1 pt) set10/korevaar math1210spring2003-set12-p8.pgFind the volume of the solid obtained by rotating the triangularregion bounded by the x-axis, the line y = 4x and the line x = 9about the x-axis.

Volume =Correct Answers:

• 12214.512238752

1

11. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C06S02-VolumeSlicing/6-2-14.pgFind the volume of the solid obtained by rotating the regionbounded by the given curves about the specified line.

y = 1/x4, y = 0, x = 2, x = 8;

about y =−3.

Volume =

Correct Answers:

• 0.776632344891228

12. (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C05S05-Substitution-/5-5-52.pgEvaluate

Z 2√π

0x1 cos(x2)dx.

Definite Integral =

Correct Answers:

• 0

13. (1 pt) Library/Rochester/setIntegrals17Approximations/sc5 8 6.pgUse Simpson’s Rule and all the data in the following table to

estimate the value of the integralZ −4

−10ydx.

x -10 -9 -8 -7 -6 -5 -4y 3 9 9 4 3 -8 0


14. (2 pts) Library/Rochester/setIntegrals17Approximations-/osu in 17 3.pgApproximate the definite integralZ 11

3|4− x|dx

using 4 subintervals of equal length and(a) midpoint rule

(b) trapezoidal rule

(c) Simpson’s rule

Correct Answers:• 24• 26• 25.3333333333333


2

webwork demonstration assignment - home - math · webwork demonstration assignment the main purpose...

Documents