a b webwork demonstration assignment 5....webwork demonstration assignment the main purpose of this...

WeBWorK demonstration assignmentThe main purpose of this WeBWorK set is to fa-

miliarize yourself with WeBWorK.Here are some hints on how to use WeBWorK ef-

fectively:• After first logging into WeBWorK change

your password.• Find out how to print a hard copy on the com-

puter system that you are going to use. Printa hard copy of this assignment.

• Get to work on this set right away and answerthese questions well before the deadline. Notonly will this give you the chance to figureout what’s wrong if an answer is not accepted,you also will avoid the likely rush and con-gestion prior to the deadline.

• The primary purpose of the WeBWorK as-signments in this class is to give you the op-portunity to learn by having instant feedbackon your active solution of relevant problems.Make the best of it!

1.(1 pt)Evaluate the expression4(9−9) = .

2.(1 pt)Evaluate the expression2/(9+5) = .Enter you answer as a decimal number listing at least4 decimal digits. (WeBWorK will reject your answerif it differs by more than one tenth of 1 percent fromwhat it thinks the answer is.)

3.(1 pt) Let r = 5.Evaluate 4/π∗ r = .

Next, enter the expression 4/(π∗r)= andlet WeBWorK compute the result.

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

1b .

Enter here the expression 1a+b .

5.(1 pt) Enter here the ex-pression

a+12+b

Enter here the expressiona+bc+d

If WeBWorK rejects your answer use the previewbutton to see what it thinks you are trying to tell it.

6.(1 pt) Enter here the ex-pression √

a+bEnter here the expression

a√a+b

Enter here the expressiona+b√a+b

7.(1 pt)Enter here the expression

√

x2 + y2

Enter here the expression

x√

x2 + y2

Enter here the expressionx+ y

√

x2 + y2

8.(1 pt)Enter here the expression

−b+√

b2 −4ac2a

Note: this is an expression that gives the solution ofa quadratic equation by the quadratic formula.

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c©UR1

Huang Hsiang-Ping WW Prob Lib1 Math course-section, semester yearWeBWorK problems. WeBWorK assignment 1 due 9/7/07 at 11:59 PM.

1.(1 pt) A line through (1, -6) with a slope of 3has a y-intercept at

2.(1 pt) An equation of a line through (-4, 5) whichis perpendicular to the line y = 3x+3 has slope:

and y intercept at:

3.(1 pt) An equation of a line through (-1, 1) whichis parallel to the line y = 3x+3 has slope:

and y intercept at:

4.(1 pt) The equation of the line with slope 4 thatgoes through the point (6,4) can be written in theform y = mx+b where m is:and where b is:

5.(3 pts)Match each graph to its equation.

(For all graphs on this page, if you are having a hardtime seeing the picture clearly, click on it. It will ex-pand to a larger picture on its own page so that youcan inspect it more closely.)

1.

2.

3.

4.

1

5.

6.

A. (x−1)2 = 2(y+1)B. x2 = −2yC. (y−1)2 = −2(x−1)D. y2 = −2xE. x2 = 2yF. (y−1)2 = 2(x+1)

6.(1 pt) The domain of the function f (x) = 303x−23 is

all real numbers x except for x where x equals7.(2 pts) For each of the following functions, de-

cide whether it is even, odd, or neither. Enter E for anEVEN function, O for an ODD function and N for afunction which is NEITHER even nor odd.

NOTE: You will only have four attempts to get thisproblem right!

1. f (x) = x8 +3x4 +2x7

2. f (x) = −5x8 −3x4 −23. f (x) = x3 + x5 + x7

4. f (x) = x8 −6x4 +3x6

8.(2 pts) Almost any kind of quantitative data can be represented by a graph and most of these graphsrepresent functions. This is why functions and graphs are the objects analyzed by calculus. The next twoproblems illustrate data which can be represented by a graph. Match the following descriptions with theirgraphs below:

1. The graph of the velocity of a car entering a superhighway vs. time.2. The graph of the distance traveled by a car as it enters a superhighway vs. time.3. The graph of the distance traveled by a car as it drives along a city street vs. time.4. The graph of the velocity of a car as it drives along a city street vs. time.

A B C D

2

9.(2.5 pts)The following questions concern the profits of firmN. The graph of the profits vs. time is given above.For each of the intervals enter the letters correspond-ing to the descriptions which describe the behavior ofthe graph on that interval. (The letters in each answermust be in alphabetical order with no spaces betweenthe letters.)

1. The interval from a to b2. The interval from b to c3. The interval from c to d4. The interval from d to e5. The interval from e to fA. The firm makes a profit on this interval.B. The firm registers a loss on this interval.C. The profit of the firm increases on this inter-

val.D. The profit of the firm decreases on this inter-

val.E. Assuming the profits are reinvested in the

firm the networth of the company is increas-ing on this interval.

F. Assuming the profits are reinvested in thefirm the networth of the company is decreas-ing on this interval.

10.(1 pt) Let f (x) = 9x3, g(x) = 12x , and h(x) =

4x2 +4.Then f ◦g◦h(6) =

11.(2 pts) Relative to the graph of

y = sin(x)

the graphs of the following equations have beenchanged in what way?

1. y = sin(x)/32. y = 3sin(x)3. y = sin(x)−104. y = sin(x−10)

A. shifted 10 units rightB. compressed vertically by the factor 3C. stretched vertically by the factor 3D. shifted 10 units down

12.(2 pts) Let g be the function below.For all graphs on this page, if you are having a hard

time seeing the picture clearly, click on it. It will ex-pand to a larger picture on its own page so that youcan inspect it more closely.

The domain of g(x) is of the form [a,b], where a isand b is .

The range of g(x) is of the form [c,d], where c isand d is .

Enter the letter of the graph which corresponds toeach new function defined below:

1. g(x−2)+2 is .2. g(2x) is .3. 2+g(−x) is .4. g(x+2)−2 is .

A B C D

E F G H

13.(2 pts) Enter a T or an F in each answer spacebelow to indicate whether the corresponding equationis true or false. An equation is true ony if it is true forall values of the variables. Disregard values that makedenominators 0.

3

You must get all of the answers correct to receivecredit.

1. 64+a64 = 1+ a

642. x+64

y+64 = xy

3. 2626−c = 1− 26

c4. 64

26+x = 6426 + 64

x

14.(1.5 pts) The equation 5x4 − 7x3 − 2x2 = 0 hasthree real solutions A, B, and C where A < B < Cand A is:

and B is:and C is:15.(1 pt) Relative to the graph of

y = x2

the graphs of the following equations have beenchanged in what way?

1. y = x2 −172. y = (x+17)2

3. y = (x−17)2

4. y = 17x2

A. shifted 17 units rightB. stretched vertically by the factor 17C. shifted 17 units downD. shifted 17 units left

16.(1 pt) Find the standard form of the parabolaequation,

y = a(x−h)2 + kwith vertex (-10,-2) and passing through the point

(-3,4).Then h is

and k is

and a is:

17.(1 pt) Find the x and y intercepts and the coor-dinates of the vertex of the parabola

f (x) =1010

(x2 −3x−8).

(1) Then the smaller one of the x intercept is

and the larger one of the x intercept is

(2) y intercept is

(3) x coordinate of the vertex (h,k):h =

and y coordinate of the vertex (h,k):k =

18.(1 pt) Find the number of units sold that pro-duces a maximum revenue from the total revenuefunction,

R = 500x−0.1x2.

(in dollars) and x is the number of units sold.Answer =

19.(1 pt) Find all the zeros of the functionf (x) = x5 +7x3 −60x

,where the three zeros,A,B,C, are real values withA < B < C.A =

B =C =20.(1 pt) Use the long division to find A,B,C so

that

x3 −19x2 +93x−30 = (Ax2 +Bx+C)(x−10)

A =B =C =


U of U Math 2201-1

Huang Hsiang-Ping.

WeBWorK assignment number 2.

due 9/14/07 at 11:59 PM.

This week let’s take a break and solve some prob-lems just for ”fun”.

1.(10 pts) This is a true story! In 2002 the MathDepartment obtained two new buildings, the LeroyCowles Building and the T Benny Rushing Mathe-matics Student Center. We had to evaluate furniturebids from various manufacturers that offered variousdiscounts. One manufacturer offered a ”30+30” per-cent discount. This means they cut 30 percent offthe list price and then took another 30 percent off thediscounted price. Other manufacturers offered var-ious single discounts. All other things being equalyou prefer the single discount if it is greater thanpercent.

2.(10 pts) You are approaching the island ofHawaii in a small boat. The highest point on Hawaiiis Mauna Loa at 13,677 feet. You see it just barelyabove the horizon. The radius of Earth is 3,963 miles.Ignoring atmospheric effects, you figure that you are

miles in a straight line from the top of MaunaLoa.

3.(10 pts) You are on a pleasure cruise through theuniverse and you crash in the ocean of an unknownplanet. Your spaceship floats on the water and its topis 40 feet above the surface of the water. You swimaway from the spaceship until you see its top on thehorizon. Your laser range meter tells you that youreyes are 2.7 miles away from the top of the spaceship. (You are a capable - if reckless and curious -swimmer.) The radius of the planet is miles.It’s a small world, but it’s all yours. You figure thatthe surface of the earth is times as large as thesurface of your planet, but still, your planet is plentybig enough for you (if only you can find land some-where).

4.(10 pts)

The reason why Mathematics is required for somany subjects is that it can be used to solve problemsoutside of mathematics, the dreaded word problems.There will be many word problems in this class, usu-ally leading to a mathematical problem of the kind weare discussing at the time. Students don’t like wordproblems because they involve the extra layer of con-verting the word problem to a math problem. Butkeep in mind that math classes are the only kind ofclasses you take where some problems are not wordproblems!This first word problem of this course can be solvedby deriving and solving an equation, but it can also besolved essentially by guessing and modifying the an-swer until it fits, without any algebraic manipulation.We will revisit it in the future in a more complicatedsetting.You buy a pot and its lid for a total of $11. The salesperson tells you that the pot by itself costs $10 morethan the lid. The price of the pot is $ andthe price of the lid is $ .

5.(10 pts) You are flying in an open plane at an al-titude of 6400 feet and you drop a Coca Cola bottleout of the window. The bottle will hit the ground after

seconds. (Note: this problem is a bit unrealisticsince it ignores air drag. The scenario described hereprovides the opening scene in the movie ”The Godsmust be crazy.”)

6.(10 pts) The repeating decimalz = 0.142857142857142857 . . .

can be written as a fraction z = / .Make sure you simplify your fraction before you en-ter it.

7.(10 pts) This is one of the more challenging prob-lems promised in your syllabus. It looks pretty bewil-dering but it can be solved by a straightforward appli-cation of one of the main principles we discuss in thisclass every day. Suppose

x =

√

√

√

√

1+

√

1+

√

1+

√

1+√

1+ . . .

where the square roots go on forever. You may as-sume (and it is true) that this expression actually de-fines a positive number x. What is it? Enter it as adecimal or radical expression:

1

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

2

U of U Math 2201-1 Fall 2007

Huang Hsiang-Ping.


due 9/21/07 at 11:59 PM.

1.(10 pts)The circle in the graph above is described by the

equation

(x−h)2 +(y− k)2 = r2

whereh = ,k = , andr = .

The graph is not very detailed, but all the answersin this question are integers. The screen version of thegraph is larger than the printed version. In this prob-lem, WeBWorK will tell you for each partial answerwhether it us correct or not.

2.(10 pts)The circle in the graph above is described by the

equation(x−h)2 +(y− k)2 = r2

whereh = ,k = , andr = .

3.(10 pts) Below, enter x if the graph of the givenequation is symmetric with respect to the x-axis, y if itis symmetric with respect to the y axis, o (lower caseO) if it is symmetric with respect to the origin, and n(for None) if it has none of these three symmetries.

• y = x4

• y = x2 +1• y = x2 + x• y = (1+ x2)3

4.(10 pts) Below, enter x if the graph of the givenequation is symmetric with respect to the x-axis, y if itis symmetric with respect to the y axis, o (lower caseO) if it is symmetric with respect to the origin, and n(for None) if it has none of these three symmetries.

• y = x3 + x• y = (x3 +1)2

• y = 11+x2

• y = x1+x2 .

5.(10 pts)Let

p(x) = 2x3 +3x2 +4x+5.1

As discussed in class, p(2) can be computed via syn-thetic division by filling in for the letters a through gin the table

2 3 4 5b d f

a c e gwhere

• = a,• = b,• = c,• = d,• = e,• = f ,• = g.

Moreover, p(2) = and p(x) can be written asp(x) = (x−2)q(x)+ r

whereq(x) = is a quadratic polynomial andr = is a real number.

6.(10 pts) Letp(x) = x3 − x2 +2x−5.

p(−3) can be computed via synthetic division by fill-ing in for the letters a through g in the table

1 −1 2 −5b d f

a c e gwhere

• = a,• = b,• = c,• = d,• = e,• = f ,• = g.

Moreover, p(−3) = and p(x) can be written asp(x) = (x+3)q(x)+ r


7.(10 pts)Let

p(x) = x3 −3x2 −2x+1.

p(−2) can be computed via synthetic division by fill-ing in for the letters a through g in the table

1 −3 −2 1b d f

a c e gwhere

• = a,• = b,• = c,• = d,• = e,• = f ,• = g.

Moreover, p(−2) = and p(x) can be written asp(x) = (x+2)q(x)+ r


8.(10 pts)Let

p(x) = 6x3 −5x2 −2x+1.

It is easy to check that p(1) = 0, i.e., 1 is a zero ofp. p has two more real zeros. The smaller is x = ,and the larger is x = .

9.(10 pts)Let

p(x) = 21x4 −11x3 −23x2 +11x+2.

It is easy to check that p(1) = p(−1) = 0, i.e., 1and −1 are zeros of p. p has two more real zeros.The smaller is x = , and the larger is x = .

10.(10 pts)The equation

x4 −9x3 +31x2 −49x+30 = 0has the conjugate complex pair of solutions x = 2± i.The equation also has two real solutions. The smalleris , and the larger is .

11.(10 pts)One solution of the the equation

p(x) = x4 −9x3 +30x2 −33x−13 = 0is x = 3−2i. The equation also has two real solutions.The smaller is , and the larger is .

2

12.(10 pts)Two zeros of the polynomial

p(x) = x4 −4x3 −24x2 +20x+7

are 1 and 7. The other two zeros are real, but irra-tional.The smaller is , and the larger is

.

13.(10 pts)Match the graphs shown above with the functionslisted below. Enter ”r” for red, ”g” for green, and”y” for yellow.

• : f (x) = xx−1 .

• : f (x) = x2

5(x+1) .

• : f (x) = xx2+1 .

14.(10 pts)

Match the graphs shown above with the functionslisted below. Enter ”r” for red, ”g” for green, ”p”for purple, ”b” for blue, and ”y” for yellow.

• : (x) = 1x .

• : f (x) = 1x2+1 .

• : f (x) = xx2−1 .

• : f (x) = x2

x2+1 .

• : f (x) = x3

x2−1 .

15.(10 pts) This is a warmup for the next problem.For the following functions, use ”x” to indicate that

the x-axis is an asymptote, ”h” to indicate a horizon-tal asymptote other than the x axis, ”v” to indicate avertical asymptote, ”s” to indicate a slanted asymp-tote, and ”n” the lack of an asymptote. If the graphof a function has several types of asymptotes indicatethem all in alphabetical order.

For example, the function

f (x) =x3

x2 −1

has a slanted asymptote since the degree of the nu-merator is one more than the degree of the denom-inator, and it also has two vertical asymptotes (atx = ±1). So you would enter ”sv” (without the dou-ble quotation marks. The graph of

f (x) =1x

has vertical asymptote (the y-axis) and the x-axis asan asymptote, so you would enter ”vx”. On the otherhand, the graph of

f (x) =x3

x2 +1

has only a slanted asymptote, so you would enter just”s”.

• f (x) = 1x .

• f (x) = x3

x2−1 .

• f (x) = x3

x2+1 .

To make this clear the following picture shows thegraphs involved:

3

It may not be clear from the picture that the greengraph (of f (x) = x3

x2+1 has a slanted asymptote, to

make this clearer the Figure also contains the (red)graph of its asymptote defined by the equation y = x.The yellow graph is the graph of f (x) = x3

x2−1 , theblue graph is the graph of f (x) = 1

x .

16.(10 pts) This delightful problem is taken from acollection of 200 ”problem solving cards” designedfor eighth grade. It’s the hardest problem of thatset, but of course you are well beyond eighth grade!Consult the original card here . The problem state-ment below does not say that the tank is lying on oneof its three sides, but it’s clear from the picture on thecard.A fuel storage tank is in the shape of an equilateraltriangular prism. The prism is 8 feet high and 10 feetlong. When the tank is half-full, the depth of the fuelis feet.


U of U Math 2201-1

Fall 2007

Huang Hsiang-Ping.


due 10/5/07 at 11:59 PM.

For your convenience, following is a list of themost basic properties of the trigonometric functions.

• DefinitionLet t be an angle in standard position. Thus

its vertex is at the origin, its initial side is onthe x-axis, and its terminal side intersects theunit circle (of radius 1 around the origin) at apoint (x,y). Then:

cos t = x, sint = y, tan t =sin tcos t =

yx .

Note: cos, sin, and tan are functions, butthe parentheses around their arguments areusually omitted whenever this causes no con-fusion. Of course, tant is undefined whencos t = 0. Otherwise the trigonometric func-tions are defined for all real numbers t.

• Key IdentityWe will use the following key identity all

the time. It follows straight from the defi-nition of sin and cos as the coordinates of apoint on the unit circle. For all real number t:

cos2 t + sin2 t = 1.

Here we use the standard notation: cos2 t =(cos t)2, and similarly for the sin function.

• Periodicity

cos(t +2π) = cost, sin(t +2π) = sint, tan(t +π) = tant.• Sign reversal

cos(t +π) = −cos t, sin(t +π) = −sin t.

• SymmetryThe cos function is even, sin and tan are

odd, i.e.,

cos t = cos(−t), sin t =−sin(−t) tan t =− tan(−t).• Right triangle

Let t be one of the acute angles of a righttriangle. Then

sint =opposite

hypotenuse

cos t =adjacent

hypotenuse

tan t =oppositeadjacent

Procrastination is hazardous!1.(10 pts) We need to be able to convert radians to

degrees and vice versa. Below you can enter radiansas decimal approximations, but I recommend that youenter arithmetic expressions like pi/2 for π

2 . I trustthat you can compute a decimal value on your calcu-lator if required. However, if you do enter a decimalapproximation compute and enter at least four dig-its. WeBWorK will consider your value correct if itis within one tenth of one percent of the answer thatit has been told.

In this problem WeBWorK will tell you for eachanswer whether it’s right or wrong, in the next prob-lem you will have to get everything right before get-ting credit.90◦ = rad60◦ = rad7◦ = rad−507◦ = rad

2.(10 pts) In this problem you are asked to convertradians to degrees.6πrad = ◦.−0.7rad = ◦.(√

2)

πrad = ◦.

3.(10 pts) Occasionally you may encounteran angle that’s given in the old fashioned de-grees/minutes/second form, as described in the boxon page 286 of the textbook. A degree is divided into

1

60 minutes (or minutes of degree), and a minute (ofdegree) is divided into 60 seconds.Let

a = 3◦5′17′′.Then a = degrees (in decimal notation)and a = radians.Again, enter your answers showing at least 4 digits.

4.(10 pts) It’s a little trickier to convert anangle that’s given in decimal notation into theminute/second notation. Let’s do it just once. Theangle a = 6.75444... degrees can be written asa = ◦ ′ ′′.

5.(10 pts) Positions on earth are defined by theirlongitude (how far east or west you are) and latitude(how far north or south you are). Both latitude andlongitude are angles.

A nautical mile is one minute of degree of longi-tude along the equator. (A nautical mile is a littlemore than a regular mile.) Traveling at x knots meanstraveling at x nautical miles per hour.

Suppose you fly in a Lockheed SR-71 (the fastestjet ever built) along the equator at a speed of 1,800knots (roughly the top speed of the SR-71).

Ignoring issues of refueling and such, it takes youhours to fly once around the world. You leave

after breakfast and are back in time for a late dinner.Some airplane!

6.(10 pts)This problem outlines how Eratosthenes of Cyrene

approximated the diameter of the earth in approxi-mately 200BC. By observing the shadow of the sun atnoon he recognized that his home town of Alexandriawas approximately 7.2◦ north of the town of Syene.He paid somebody to walk and measure the distancebetween Syene and Alexandria. Suppose it is 787 km.Based on these figures, the circumference of the earthis kilometers.

7.(10 pts) The remaining problems in this set dealwith the definitions of the basic trigonometric func-tions, sin, cos, and tan.You can answer some of these questions simply bykeying things into your calculator. However, the pur-pose of these problems is to help you get familiar withthe definitions of the basic trigonometric functions,and to improve your ability to work with those def-initions. All the questions can be answered straight

from the definitions of the trigonometric functions,perhaps after drawing a simple picture, without theaid of a calculator, and I recommend that you don’tuse one. Use ’pi’ to enter the value of π and usesqrt(...) to enter the square root of something.A line drawn from the origin and forming the angleof t = 7π

6 with the x-axis intersects the unit circle atthe point

(

−√

32 ,−1

2

)

. Complete the following equa-tions:t = degrees.

cost = .sint = .

tan t = .8.(10 pts)A line drawn from the origin and forming the angle

t with the x-axis intersects the unit circle at the point(

13 , 2

√2

3

)

. Complete the following equations:cos t = .sint = .tan t = .

9.(10 pts)A line drawn from the origin and forming the angle

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

53 ,−2

3

)

. Complete the following equations:cos t = .sint = .tan t = .

10.(10 pts)Let t be the angle between 0 and π

2 such that

sin t =14 .

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

11.(10 pts)Let again t be the angle between 0 and π

2 such that

sin t =14 .

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

2

tan(t +π) = .12.(10 pts)Here are the values of the trigonometric functions

at some basic angles. You need not memorize those,but you should be able to figure them out, for exam-ple by drawing a suitable picture involving a triangleor the unit circle.cos(0) = .sin(0) = .tan(0) = .cos(π

2) = .sin(π

2 ) = .13.(10 pts)

The angle π4 equals degrees.

Here are some more basic trig values:cos

(π4)

= .sin

(π4)

= .tan

(π4)

= .14.(10 pts)


More basic values:cos

(π3)

= .sin

(π3)

= .tan

(π3)

= .15.(10 pts)


And more basic trig values:cos

(π6)

= .sin

(π6)

= .tan

(π6)

= .16.(10 pts) You may think this first problem a little

odd. All it asks is that use your calculator to evaluatethe trigonometric functions of some angles. The rea-son for including this problem is the observation thata substantial number of mistakes in solving trig prob-lems are due to having your calculator in the wrongmode, and not noticing this fact. Whenever you useyour calculator check its mode!In this problem you will need to enter your answersas decimal approximations (with at least 4 digits).Usually WeBWorK will let you enter expressions likesin(...), but note that in that case WeBWorK alwaysassumes your angles are measured in radians.sin(15◦) =

sin(15) =cos(1.4◦) =cos(1.4) =

17.(10 pts) A question like this is unlikely to arisein an application, but we do need to learn to keepstraight the different ways of measuring angles.Suppose you have a triangle in which one angle is π

6 ,and another is 40◦. Then the third angle is de-grees, or radians.

18.(10 pts) The next few problems deal with thedefinition of the trigonometric functions in a right tri-angle. We use a standard notation: a, b, and c denotethe sides of the triangles and their lengths, and A, B,and C denote the angles opposite a, b, and c, respec-tively, as indicated in the nearby figure.

Thus a and b are the lengths of the two short sides,and c is the hypotenuse. Use the Pythagorean Theo-rem to compute any missing length. Assume also thatA denotes the angle opposite a, B the angle oppositeb, and, of course, C the right angle.

I recommend that in this problem you enter valuesof the trig functions as fractions.

Suppose a = 5 and b = 12.Then

c = ,sin(A) = ,cos(A) = , andtan(A) = .

19.(10 pts) Using the same notation as in the pre-ceding problem, suppose a = 20 and c = 29.

Thenb = ,sin(B) = ,cos(A) = , andtan(A) = .

3

20.(10 pts) You are hiking along the edge of theGreen River (which is running straight in the area ofinterest). Straight across from you on the oppositeshore there is a particularly noticeable boulder at theedge of the river. You walk 120 feet along the shoreof the river, and now the line from you to the bouldermakes an angle of 35◦ with the edge of the river. Youwhip out your trusty scientific calculator and deducethat the width of the river at this point isfeet.

21.(10 pts) You are flying a kite on a line that is 350feet long. Let’s suppose the line is perfectly straight(it never really is) and it makes an angle of 65 degreeswith the horizontal direction. The kite is flying at analtitude of feet.

22.(10 pts) Recall the definition: Let θ be an an-gle in standard position. Its reference angle is theacute angle θ′ formed by the terminal side of θ andthe horizontal axis.Below, express θ′ in the same units (degrees or radi-ans) as θ. You can enter arithmetic expressions like210-180 or 3.5-pi.

The reference angle of 100◦ is ◦.The reference angle of 350◦ is ◦.The reference angle of 4 is .

23.(10 pts)Below, express the reference angle θ′ in the sameunits (degrees or radians) as θ. You can enter arith-metic expressions like 210-180 or 3.5-pi.The reference angle of 30◦ is ◦.The reference angle of −30◦ is ◦.The reference angle of 1,000,000◦ is ◦.The reference angle of 100 is .

24.(10 pts)You approach a hill on top of which there is a tall

radio antenna. You know from your map that yourhorizontal distance from the bottom of the radio an-tenna is 600 feet. The angle of elevation to the bottomof the antenna is 10◦, and the angle of elevation to thetop of the antenna is 25◦. You figure that the heightof the hill is feet, and the height of the antennais

feet. (Enter your answers rounded to the nearestfoot.)


U of U Math 2201-1

Fall 2007

Huang Hsiang-Ping.


due 10/19/07 at 11:59 PM.1.(10 pts) This and the following three problems

cover some subtleties of the inverse trig functions. Inthese problem, WeBWorK will not accept as answersmathematical expressions that involve the trigono-metric functions or their inverses. You need to enternumbers.In this problem let’s measure angles in degrees. Asdiscussed in class, the inverse sin function returns anangle between -90 and +90 degrees. This gives per-haps surprising results when you evaluate the sinesomewhere and then the inverse sine at the result.You can just key the problems below and in the fol-lowing problems into your calculator, but try to getthem without a calculator. If you yield to temptationand do use your calculator, at least think about theanswers when they are unexpected.Complete the following equations:arcsin(sin(37◦)) = ◦,arcsin(sin(−25◦)) = ◦,arcsin(sin(100◦)) = ◦,

2.(10 pts) This is like the previous problem, exceptthat angles are measured in radians.Complete the following equations:arcsin(sin(1)) = ,arcsin(sin(−1.1)) = ,arcsin(sin(2)) = .

3.(10 pts)Complete the following equations:arccos(cos(37◦)) = ◦,arccos(cos(−25◦)) = ◦,arccos(cos(100◦)) = ◦,

4.(10 pts)Complete the following equations:arctan(tan(37◦)) = ◦,

arctan(tan(−25◦)) = ◦,arctan(tan(100◦)) = ◦,

5.(10 pts) Match the functions with their graphs.1. f (x) = cos(x)2. f (x) = sin(x)3. f (x) = tan(x)4. f (x) = arcsin(x)5. f (x) = arccos(x)6. f (x) = arctan(x)

A B C D E F(Click on image for a larger view . The small im-

ages may not show up properly on a hard copy, butthey will be fine in a browser.)

6.(10 pts) Suppose you have a right trianglewhose two short sides are of length 3 and 4 respec-tively. Then the smaller of the two acute angles is

degrees and the larger acute angle isdegrees. If you enter your angles as

decimal approximations compute at least one digitbeyond the decimal point.

7.(10 pts) Suppose you have a right triangle whosehypotenuse has length 8. One of the other sides haslength 5. Then the smaller of the two acute anglesis degrees and the larger acute angle is

degrees8.(10 pts) You are driving along the highway and

see a sign indicating that you are about to enter adownhill slope of 10%. You realize that you are go-ing to travel downward at an angle of degreesto the horizontal. (Enter your answer as a mathemat-ical expression or with at least two digits beyond thedecimal point.)

9.(10 pts) You are about to land your Cessna air-plane in Salt Lake City. You are approaching the run-way at a ground speed of 78 miles per hour and youare sinking at 390 feet per minute. (The ground speedis the speed of the point on the ground directly under-neath your plane. You can also think of it as the hor-izontal component of your current velocity.) You aregoing to hit the runway at an angle of

1

degrees. (Enter your answer as a mathematical ex-pression, or with at least three digits beyond the dec-imal point.)

10.(10 pts) Geographic directions are described indegrees, counting clockwise from a line going duenorth. Thus, for example, due north is zero degrees,east is 90 degrees, and southwest is 225 degrees.All distances in this question are measured horizon-tally as you would measure them on a map. You canalso think of flying at the altitude of the peak in ques-tion, instead of driving along the highway. The direc-tion in which you see the peak is called its bearing.So a bearing of 135 degrees means it’s southeast ofyou. Enter your answers as mathematical expressions(recommended) or as decimal approximations with atleast 4 digits total.You are driving at a constant speed of sixty miles perhour along a straight road going north. You see aprominent peak at a bearing of 45 degrees, and youknow that that peak is 10 miles east of the road. Atthis time your distance from the peak is miles.Five minutes later you see the peak at a bearing of

degrees. After another five minutes thepeak is due east of you. At that precise spot there is ahistorical marker that tells you about the peak. 7 min-utes after you pass the marker the peak is at a bearingof degrees. You sit back in your car andreflect on the pleasant fact that the trigonometry classyou are taking makes it possible for you to figure outthat t minutes after you pass the historical marker thebearing of the peak is degrees. (Enter amathematical expression involving the variable t.)

11.(10 pts) The Great Pyramid of Cheops has asquare base with a length of 756 feet. Its height is482 feet. If you walk straight up from the center ofthe north side to the top of the pyramid you have toclimb an angle of degrees.You decide to simplify your life and walk up alongone of the ridges. Thus you have to climb only at anangle of degrees.On your way up the ridge you walk a distance of

feet.12.(10 pts)

You are hiking along the west shore of a river that’sflowing due north. You notice a tree on the far shoreat a bearing of 30 degrees. You walk on for another

100 feet and you are stopped by an unclimbable cliff.You contemplate swimming across the river and won-der how wide it is. The tree on the other side now ap-pears at a bearing of 45 degrees. Remembering yourtrig class, you figure out that the river is feetwide.

13.(10 pts)Suppose

sin(u) =35

and cos(u) is negative. Thencos(u) =tan(u) =sin(−u) =cos(−u) =tan(−u) =sin(u+π) =cos(u+π) =tan(u+π) =sin

(

u+ π2)

=

cos(

u+ π2)

=

tan(

u+ π2)

=

14.(10 pts)In this problem you need to get everything right to

get credit. The idea is that you go over everythingan think about the whole context. Use the precedingproblem as a guide.Suppose

sin(u) =1213

and cos(u) is negative. Note that cosu is a rationalnumber. I recommend you enter fractions below.cos(u) =tan(u) =sin(−u) =cos(−u) =tan(−u) =sin(u+π) =cos(u+π) =tan(u+π) =sin

(

u+ π2)

=

cos(

u+ π2)

=

tan(

u+ π2)

=

15.(10 pts)2

Supposecosu =

513

and sinu is negative. Here are some small variationson the previous problems:sin(u) =sin(u−π) =cos(u−π) =sin

(

u− π2)

=

cos(

u− π2)

=

16.(10 pts) Consider two fire towers in the Cana-dian wilderness. Tower B is 40 miles due East oftower A. The guards notice a fire that appears at abearing of 70 degrees from Tower A, and a bearingof 310 degrees from Tower B.

The fire is at a distance miles from thetower B (the closer of the two towers).


HONOR 2201-1 Fall 2007

Huang Hsiang-Ping.


due 11/2/07 at 11:58 PM.Note, for questions 5-18 use the rules of differ-

entiation rather than the limit definition for findingderivatives.

1.(10 pts) Let f (x) = x2 − 3. Find the slope ofthe curve y = f (x) at the point x = 1 by calculatingf (1+h)− f (1)

h and determining what number it ap-proaches as h approaches 0.

f (1+h)− f (1)

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

2.(10 pts) Let f (x) = 2x3 − x2. Find f ′(x) by cal-

culating f (x+h)− f (x)h and determining what it ap-

proaches as h approaches 0.f (x+h)− f (x)

h = x2+ xh+ h2+ x+ hf ′(x) = x2+ x.

3.(10 pts) A ball is thrown straight up so its heightt seconds later is −16t2 +32t +6 feet.a. Find the velocity of the ball at t seconds after it isthrown.

ft/secb. At what time does the ball reach its maximumheight?t = secc. What is the value of the maximum height?Maximum height = ftd. Find the acceleration of the ball at any time t.a= ft/sec2

4.(10 pts) Let

f (x) =1

x−8Algebraically simplify the secant line slope

f (x+h)− f (x)h , and enter the numerator below:

f (x+h)− f (x)h = /((x+h−8)(x−8))

Let h −→ 0 to deduce the derivative,f ′(x) =The equation of the tangent line passing through

the point on the graph of f with x-coordinate 10 canbe written in the form y = mx+b, where

m =b =

5.(10 pts) This a simple exercise in computingderivatives of polynomials. The derivative of

p(x) = 8x2 +4x+9is p′(x) = .The derivative of

q(x) = 2x5 −2x4 +6x3 −3x2 +4x+7is q′(x) = .

6.(10 pts) More derivatives: The derivative ofp(x) = (2x−1)2

is p′(x) = .The derivative of

q(x) = (2x−1)3

is q′(x) = .7.(10 pts) If f (x) = 12x+2, then

f ′(−10)= .8.(10 pts) If f (x) = 6+7x−4x2 then

f ′(−1) = .9.(10 pts) This problem will help you practice

computing tangents in the next problem. Letf (x) = x2.

Then f ′(x) = .The tangent to the graph of f through the point (1,1)has the slope and the

y-intercept .It intercepts the x-axis at x = .

10.(10 pts) The slope of the tangent line to theparabola y = 2x2 + 2x + 4 at the point (−1,4) is:

The equation of this tangent line can be written in theform y = mx+b where m is:and where b is:

11.(10 pts) You toss a rock up vertically at an ini-tial speed of 51 feet per second and release it at an

1

initial height of 6 feet. The rock will remain in the airfor seconds.It will reach a maximum height of feet afterseconds.Note: Ignore air resistance.

12.(10 pts) If f (x) = 4+ 2x + 6

x2 , find f ′(x).

Find f ′(2).

13.(10 pts) If f (x) = 4x+44x+5 , find f ′(x).

Find f ′(5).

[NOTE: When entering functions, make sure that youput all the necessary *, (, ), etc. in your answer. ]

14.(10 pts) The rate of change of electric chargewith respect to time is called current. Suppose that13t3 + t coulombs of charge flow through a wire in tseconds. (a) Find the current in amperes (coulombsper second) after 3 seconds. (b) When will a 20-ampere fuse in the line blow?

a) Current after 3 seconds: am-peres.

b) A 20-ampere fuse will blow at: seconds.15.(10 pts) A space traveller is moving from left to

right along the curve y = x2. When she shuts off theengines, she will continue travelling along the tan-gent line at the point where she is at that time. Atwhat point should she shut off the engines in order toreach the point (4,15)?

She should shut off the engine at ( , )16.(10 pts) Let

f (x) = −9cosx+7tanx.Then

f ′(x) = .17.(10 pts) If

f (x) =3sinx

3+ cosxthenf ′(x) =

18.(10 pts)lim

x−→∞

x−4x+4 = .

limx−→∞

x−4x2 +4 = .


HONOR 2201-1 Fall 2007

Huang Hsiang-Ping.


due 11/16/07 at 11:58 PM.1.(10 pts) If

f (x) =√

4x+3then f ′(x) = .

2.(10 pts) Iff (x) = tan4x

then f ′(x) =

3.(10 pts) If

f (x) =√

3x2 +5x+7then f ′(x) =and f ′(3) = .4.(10 pts) If

f (x) = cos(2x+3)

then f ′(x) = .5.(10 pts) Let

f (x) =x

cosx2 .

f ′(x) = .6.(10 pts) Let

f (x) = xsinx2.

f ′(x) = .f ′′(x) = .

7.(10 pts) Let

f (x) = sin 1x .

f ′(x) = .

Letg(x) =

1sinx .

g′(x) = .8.(10 pts) The function f (x) = 2x3−33x2 +168x+

6 has one local minimum and one local maximum.It is helpful to make a rough sketch of the graph tosee what is happening.This function has a local minimum at x = withvalue f (x) = ,and a local maximum at x = with value f (x) =

.9.(10 pts) At what point does the normal to y =

5− 3x− 1x2 at (1,1) intersect the parabola a secondtime?( , )

The normal line is perpendicular to the tangentline. If two lines are perpendicular their slopes arenegative reciprocals – i.e. if the slope of the first lineis m then the slope of the second line is −1/m

10.(10 pts) A rancher wants to fence in an area of1000000 square feet in a rectangular field and thendivide it in half with a fence down the middle parallelto one side. What is the shortest length of fence thatthe rancher can use?

11.(10 pts) A rectangle is inscribed with its baseon the x-axis and its upper corners on the parabolay = 2−x2. What are the dimensions of such a rectan-gle with the greatest possible area?

Width = Height =12.(10 pts) A cylinder is inscribed in a right cir-

cular cone of height 5 and radius (at the base) equalto 7.5. What are the dimensions of such a cylinderwhich has maximum volume?

Radius = Height =13.(10 pts) A fence 2 feet tall runs parallel to a

tall building at a distance of 2 feet from the build-ing. What is the length of the shortest ladder that willreach from the ground over the fence to the wall ofthe building?


Math 1210-1 Spring 2007

Huang Hsiang-Ping.


due 12/7/07 at 11:58 PM.1.(10 pts) Use the Special Sum Formulas (see Sec-

tion 5.3 of Varberg, Purcell and Rigdon) to find:∑10

i=1((i−1)(4i+3)) = .2.(10 pts) Consider the integral

Z 7

3

(

3x +3

)

dx

(a) Find the Riemann sum for this integral usingright endpoints and n = 4.

(b) Find the Riemann sum for this same integral, us-ing left endpoints and n = 4

3.(10 pts) Evaluate the integral below by interpret-ing it in terms of areas. In other words, draw a pictureof the region the integral represents, and find the areausing high school geometry.

Z 1

−1

√

1− x2dx

4.(10 pts) Evaluate the sum:6∑i=1

(2− i)

5.(10 pts) IfR 1

0 f (x)dx = 4,R 2

0 f (x)dx = 2, andR 2

0 g(x)dx = −3, evaluate each integral.(a)

R 21 f (x)dx =

(b)R 0

1 f (x)dx =(c)

R 20 3 f (x)dx =

(d)R 2

0 [2g(x)−3 f (x)]dx =(e)

R −20 f (−x)dx =

6.(10 pts) If f (x) =R x

5 t7dtthen

f ′(x) =f ′(6) =

7.(10 pts) Evaluate the integral by interpreting it interms of areas. In other words, draw a picture of theregion the integral represents, and find the area usinghigh school geometry.

Z 1

0|10x−9|dx

8.(10 pts) Consider the integralZ 10

4(3x2 +4x+5)dx

(a) Find the Riemann sum for this integral usingright endpoints and n = 3.

(b) Find the Riemann sum for this same integral, us-ing left endpoints and n = 3

9.(10 pts) Use part I of the Fundamental Theoremof Calculus to find the derivative of

f (x) =Z x

5

(

15t2−1

)7dt

f ′(x) =[NOTE: Enter a function as your answer. Make surethat your syntax is correct, i.e. remember to put allthe necessary *, (, ), etc. ]


h(x) =Z sin(x)

−4(cos(t5)+ t) dt

h′(x) =[NOTE: Enter a function as your answer. Make surethat your syntax is correct, i.e. remember to put allthe necessary *, (, ), etc. ]


g(x) =Z 6x

9x

u+5u−4du

12.(10 pts) Consider the function f (x) = − x22 +8.

In this problem you will calculateR 4

0 (− x2

2 + 8)dxby using the definition

Z b

af (x)dx = lim

n→∞

[

n∑i=1

f (xi)∆x]

1

The summation inside the brackets is Rn which isthe Riemann sum where the sample points are chosento be the right-hand endpoints of each sub-interval.

Calculate Rn for f (x) = − x2

2 + 8 on the interval[0,4] and write your answer as a function of n withoutany summation signs.Rn =limn→∞ Rn =

13.(10 pts) The value ofR 1

0 (x+1)2dx is

14.(10 pts) The value ofR 7

61x2 dx is

15.(10 pts) Evaluate the definite integralZ 6

2(6x+3)dx


3(6x2 −10x+7)dx


−8(64− x2)dx


1

6x2 +5√x dx

19.(10 pts) Evaluate the definite integralZ π

06sin(x)dx

20.(10 pts) Evaluate the integral below by inter-preting it in terms of areas. In other words, draw apicture of the region the integral represents, and findthe area using high school geometry.

Z 5

−5

√

25− x2dx

21.(10 pts) Evaluate the integralZ 2

3sin(t)dt

22.(10 pts) The velocity function is v(t) = t2−5t +6 for a particle moving along a line. Find the dis-placement and the distance traveled by the particleduring the time interval [-3,6].

displacement =distance traveled =


a b webwork demonstration assignment 5....webwork demonstration assignment the main purpose of this...

Documents