Peter Alfeld Math 1210-3, Fall 2005

Peter Alfeld Math 1210-3, Fall 2005WeBWorK problems. WeBWorK assignment 1 due 9/12/05 at 11:59 PM.

1.(1 pt) Evaluate the expression 33

3−4 .

2.(1 pt) Evaluate the expression 64−4/3.

[NOTE: Your answer can be an algebraic expressionor fraction. ]

3.(1 pt) The expression(

x3y3z3x−2

x5y3z5y4

)−3equals xryszt

where r, the exponent of x, is:and s, the exponent of y, is:and finally t, the exponent of z, is:[NOTE: Your answers can be algebraic expressionsor fraction.]

4.(1 pt) Find the distance between (4, 7) and (-1,-1).

5.(1 pt) The equation of the line with slope 4 thatgoes through the point (8,5) can be written in theform y = mx+b where m is:and where b is:

6.(1 pt) The equation of the line that goes throughthe point (1,9) and is parallel to the line 5x + 4y = 2can be written in the form y = mx + b where m is:

and where b is:7.(1 pt) The equation of the line that goes through

the points (−3,−7) and (2,10) can be written in theform y = mx+b where m is:and where b is:

8.(1 pt) You arrive in Paris and the forcast isfor a low of 17 and a high of 25 degrees Celsius.What is the forcasted low temperature in Fahrenheit?

What is the forcasted high temperature in Fahrenheit?

9.(1 pt) Consider the inequalityx2 < 4x+5.

The solution of this inequality consists of one ormore of the following intervals: (−∞,A), (A,B), and(B,∞) where A < B.

Find AFind B

For each interval, answer YES or NO to whetherthe interval is included in the solution.

(−∞,A)(A,B)(B,∞)

10.(1 pt) By completing the square, the expressionx2 +18x+188 equals (x+A)2 +Bwhere A is:and B is:

11.(1 pt) Let f (x) = x2 − 3. Find the slope ofthe curve y = f (x) at the point x = 1 by calculatingf (x+h)− f (x)

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

f (x+h)− f (x)h = Slope of f (x)

at x = 1: .12.(1 pt) Let f (x) = x3. Find the slope of the

curve y = f (x) at the point x = 1 by calculatingf (x+h)− f (x)

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

f (x+h)− f (x)h = Slope of f (x)

at x = 1: .13.(1 pt) Let f (x) = 2x + 5. 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 = f ′(x)= .

14.(1 pt) Let f (x) = x3 + 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 = f ′(x) =.

15.(1 pt) Suppose a pebble is dropped from the topof a four story building which is 64 feet high, and thatits position x(t) is given by x(t) = 64−16t2 feet.

Find the instantaneous velocityv(t) = x′(t) =

and the time when the pebble hits the groundt = .

1

16.(1 pt) Suppose you throw a baseball 15 feetstraight up and then catch it at the height you let go.

What is the net displacement of the baseball?feet.

What is the total distance traveled?feet.

17.(1 pt) A steam catapult aboard an aircraft car-rier can accelerate an F-18 Hornet so that its velocityis given by

v(t) = 104.85tfeet/second.

If the jet reaches its take-off velocity of 173 milesper hour at the end of the runway, how long does ittake for the jet to take off?

seconds.How long is the runway?

feet.18.(1 pt) Consider the interval [1,2]. Find the areas

under the following graphs on this interval.f (x) = x+1

.g(x) = x2

.

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c©UR2


1.(1 pt) The point P(2,14) lies on the curve y =x2 + x + 8. If Q is the point (x,x2 + x + 8), find theslope of the secant line PQ for the following valuesof x.If x = 2.1, the slope of PQ is:and if x = 2.01, the slope of PQ is:and if x = 1.9, the slope of PQ is:and if x = 1.99, the slope of PQ is:Based on the above results, guess the slope of the tan-gent line to the curve at P(2,14).

2.(1 pt) If a ball is thrown straight up into the airwith an initial velocity of 85 ft/s, its height in feetafter t seconds is given by y = 85t − 16t2. Find theaverage velocity for the time period begining whent = 1 and lasting(i) 0.5 seconds(ii) 0.1 seconds(iii) 0.01 secondsFinally based on the above results, guess what theinstantaneous velocity of the ball is when t = 1.

3.(1 pt) The slope of the tangent line to theparabola y = 2x2 + 4x + 3 at the point (1,9) is:

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

4.(1 pt) The slope of the tangent line to the curvey = 2

√x at the point (4,4.0000) is:

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

5.(1 pt) If an arrow is shot straight upward on themoon with a velocity of 50 m/s, its height (in meters)after t seconds is given by h = 50t −0.83t2.What is the velocity of the arrow (in m/s) after 5 sec-onds?After how many seconds will the arrow hit the moon?

With what velocity (in m/s) will the arrow hit themoon?

6.(1 pt) If f (x) = 4x2 −2x−29, find f ′(x).

Find f ′(4).

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

7.(1 pt) If f (x) = (5x2 −7)(5x+2), find f ′(x).

[NOTE: Your answer should be a function in terms ofthe variable ’x’ and not a number! ]

8.(1 pt) For what values of x does the curve y =4x2 − 8x + 7 have positive slope? Negative slope?Zero slope?

Positive slope: xNegative slope: xZero slope: x

Instructions: For each line, enter a relational sign(e.g. =, <, >, etc.) in the first answer box and anumber in the second.

9.(1 pt) If a ball is thrown straight up in such a waythat its height t seconds later is

s(t) = −16t2 +32t +6,find the velocity of the ball at t seconds after it is

thrown. At what time t does the ball reach its maxi-mum height? (Hint: the velocity will be positive be-fore this time and negative after it).

Velocity = .Time at which maximum height is reached =

.10.(1 pt) Find

R

(7x2 −2x+6)dx.R

(7x2 − 2x + 6)dx =+C, where C is the integration constant.

11.(1 pt) Find the antiderivative of 4x3−5x that hasthe value 4 when x = 1.

The desired antiderivative is: .12.(1 pt) Find

R 20 (x3 +2)dx.

R 20 (x3 +2)dx = +C.

13.(1 pt) A particle travels along a horizontal lineso that its velocity at time t is v(t) = 2t +3t2 +1 feetper second. Suppose that at time t = 1 the particle isat the origin. What is the location of the particle attime t = 3?

1

Particle’s location at t = 3 is: feetfrom the origin.

14.(1 pt) The domain of the function f (x) =√3x−32 is all real numbers in the interval [A,∞)

where A equals15.(1 pt) 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) = x3 + x5 + x7

2. f (x) = x8 +3x6 +2x7

3. f (x) = x8 −6x6 +3x6

4. f (x) = x−2

16.(1 pt) This problem gives you some practiceidentifying how more complicated functions can bebuilt from simpler functions.

Let f (x) = x3 −27and let g(x) = x−3. Match thefunctions defined below with the letters labeling theirequivalent expressions.

1. ( f (x))2

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

4. f (x2)

A. 729−54x3 + x6

B. −27+ x6

C. 9−6x+ x2

D. −30+ x3

17.(1 pt) Relative to the graph ofy = x2

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

1. y = (x+13)2

2. y = (x−13)2

3. y = x2 +134. y = (x2)/3A. shifted 13 units leftB. compressed vertically by the factor 3C. shifted 13 units rightD. shifted 13 units up



1.(1 pt) Let f be the linear function (in blue) and letg be the parabolic function (in red) below.

If you are having a hard time seeing the pictureclearly, click on the picture. It will expand to a largerpicture on its own page so that you can inspect it moreclosely.

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 ) =2.(1 pt) If f is one-to-one and f (10) = 2, then

f−1(2) =and ( f (10))−1 = .If g is one-to-one and g(7) = 13, theng−1(13) =and (g(7))−1 = .If h is one-to-one and h(−12) = 2, thenh−1(2) =and (h(−12))−1 =

3.(1 pt) If f (x) = 4x−13, thenf−1(y) =f−1(−5) =

4.(1 pt) If f (x) = x2, x ≥ 0,then f−1(10) =

5.(1 pt) Let

f (x) =x+3

x+10f−1(−4) =

6.(1 pt) Let

f (x) =15x+10, 1 ≤ x ≤ 2

The domain of f−1 is the interval [A,B]where A = and where B =7.(1 pt) For each of the following angles, find the

degree measure of the angle with the given radianmeasure:5π6

3π4

5π3

5π2

3π8.(1 pt) For each of the following angles, find the

radian measure of the angle with the given degreemeasure (you can enter π as ’pi’ in your answers):40360−1604030

9.(1 pt) For each of the followings angles (in ra-dian measure), find the sin of the angle (your answercannot contain trig functions, it must be an arithmeticexpression or number):π6π4π3π2π2π

10.(1 pt) For each of the followings angles (in ra-dian measure), find the cos of the angle (your answercannot contain trig functions, it must be an arithmeticexpression or number):π6π4π3

1

π2π2π

11.(1 pt) If θ = 3π4 , then

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

12.(1 pt) If θ = −5π6 , then

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

13.(1 pt) The angle of elevation to the top of abuilding is found to be 13◦ from the ground at a dis-tance of 6000 feet from the base of the building. Findthe height of the building.

14.(1 pt) A survey team is trying to estimate theheight of a mountain above a level plain. From onepoint on the plain, they observe that the angle of ele-vation to the top of the mountain is 32◦. From a point1000 feet closer to the mountain along the plain, theyfind that the angle of elevation is 35◦.How high (in feet) is the mountain?

15.(1 pt) Let F be the function below.If you are having a hard time seeing the picture

clearly, click on the picture. It will expand to a largerpicture on its own page so that you can inspect it moreclearly.

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) =

16.(1 pt) Evaluate the limit

limx→1

x−1x2 +7x−8


lims→1

s3 −1s2 −1

18.(1 pt) If limx→a

f (x) = 3 and limx→a

g(x) = −1 then

limx→a

2 f (x)−3g(x)f (x)+g(x) =


limw→−2

√

−3w3 +7w2

20.(1 pt) Evaluate the right-hand limit

limx→−π+

√π3 + x3

x =

2

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

3


1.(1 pt) If

f (x) =

√x−5√x+5

find f ′(x).

Find f ′(4).

2.(1 pt) If f (x) = 6+ 6x + 2

x2 , find f ′(x).

Find f ′(4).


limx→0

sin3x6x


limx→0

sin8xsin3x


limx→0

tanx4x


limx→∞

2+3x4−6x


limx→∞

√10+10x2

(11+4x)

8.(1 pt) Evaluate

limx→∞

√

x2 +1x+1− x

9.(1 pt) For what value of the constant c is the func-tion f continuous on (−∞,∞) where

f (x) =

{

cx+2 if x ∈ (−∞,4]

cx2 −2 if x ∈ (4,∞)

10.(1 pt) If f (x) = (4x2 −4)(3x+4), find f ′(x).

[NOTE: Your answer should be a function in terms ofthe variable ’x’ and not a number! ]

11.(1 pt) If f (t) = 13t5 , find f ′(t).

[NOTE: Your answer should be a function in terms ofthe variable ’t’ and not a number! ]

12.(1 pt) If f (x) = 7x+55x+6 , find f ′(x).

Find f ′(4).


13.(1 pt)Let f (x) = −7x4√x + 6

x3√x .f ′(x) =

[NOTE: Your answer should be a function in termsof the variable ’x’ and not a number! When enteringfunctions, make sure that you put all the necessary *,(, ), etc. in your answer. ]



1.(1 pt) If

f (x) =

√x−5√x+5

find f ′(x).

Find f ′(5).

2.(1 pt) If f (x) = 7x+53x+5 , find f ′(x).

Find f ′(2).


3.(1 pt) Iff (x) =

3sinx2+ cosx

find f ′(x).

Find f ′(2).

4.(1 pt) If f (x) = 3tanxx , find f ′(x).

Find f ′(4).

5.(1 pt) Iff (x) =

tanx−4secx

find f ′(x).

Find f ′(5).

6.(1 pt) Letf (x) = 4xsinxcosx

f ′(−π2 ) =

7.(1 pt) If f (x) = (x2 +4x+3)3, find f ′(x).

Find f ′(4).

8.(1 pt) If f (x) = (5x+3)−4, find f ′(x).

Find f ′(1).

9.(1 pt) If f (x) = sin(x5), find f ′(x).

Find f ′(4).

10.(1 pt) If f (x) = sin2 x, find f ′(x).

Find f ′(5).

11.(1 pt) If f (x) = tan5x, find f ′(x).

Find f ′(2).

12.(1 pt) Letf (x) = sin(cos(x3))

f ′(x) =

13.(1 pt) Let

f (x) = (−6x2 +2)6(8x2 −4)14

f ′(x) =

14.(1 pt) Let f (x) = 1−3x1+3x . Then f ′(3) is

and f ′′(3) isand f ′′′(3) is

15.(1 pt) Find the 86 th derivative of the functionf (x) = cos(x).The answer is function

16.(1 pt) Let

f (x) =−9x1− x

f (4)(x) =

17.(1 pt) The rate of change of electric charge withrespect to time is called current. Suppose that 1

3t3 + tcoulombs of charge flow through a wire in t seconds.(a) Find the current in amperes (coulombs per sec-ond) after 3 seconds. (b) When will a 20-ampere fusein the line blow?

a) Current after 3 seconds: am-peres.b) A 20-ampere fuse will blow at: seconds.

1

18.(1 pt) The radius of a spherical balloon is in-creasing at the rate of 0.25 inch per second. If theradius is 0 at time t = 0, find the rate of change in thevolume at time t = 3.

Rate of change in volume at t = 3:inch3/second.

19.(1 pt) Find all points on the graph of y = 13x3 +

x2 − x where the tangent line has slope 1.( , )

( , )Instruction: Enter the points in order of increasing

x-coordinate.20.(1 pt) 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 ( , )

21.(1 pt) Find all points on the graph of y =9sinxcosx where the tangent line has horizontal.

The tangent of graph is horizontal whenx = + k ,where k is an integer.Instruction: There are many ways to express the

answere here. However, WeBWorK is expecting thatyou choose positive values for both answer boxes andthe smallest possible value for the first one.

22.(1 pt) At time t seconds, the center of a bob-bing cork is 2sint centimeters above (or below) waterlevel. What is the velocity of the 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.



1.(1 pt) Use implicit differentiation to find theslope of the tangent line to the curve

yx−7y = x8 −2

at the point (1, −1−6).

m =

2.(1 pt) A street light is at the top of a 16.000 ft.tall pole. A man 5.600 ft tall walks away from thepole with a speed of 3.500 feet/sec along a straightpath. How fast is the tip of his shadow moving whenhe is 45.000 feet from the pole?

3.(1 pt) A spherical snowball is melting in such away that its diameter is decreasing at the rate of 0.4cm/min. At what rate is the volume of the snowballdecreasing when the diameter is 14 cm? (Note theanswer is a positive number).

4.(1 pt) Gravel is being dumped from a conveyorbelt at a rate of 10 cubic feet per minute.

It forms a pile in the shape of a right circular cone,such that the ratio of the base diameter to the heightis always equal to 1.

How fast is the height of the pile increasing whenthe pile is 21 feet high?Recall that the volume of a right circular cone withheight h and radius of the base r is given by

V =13πr2h

5.(1 pt) Use linear approximation to estimate theamount of paint in cubic centimeters needed to applya coat of paint 0.030000 cm thick to a hemisphericaldome with a diameter of 65.000 meters.

6.(1 pt) Let y = 5x2.Find the change in y, ∆y when x = 1 and ∆x = 0.3

Find the differential dy when x = 1 and dx = 0.3

7.(1 pt) Use linear approximation, i.e. the tangentline, to approximate 3√125.4 as follows:

Let f (x) = 3√x. The equation of the tangent line tof (x) at x = 125 can be written in the form y = mx+bwhere m is: and where b is:Using this, we find our approximation for 3√125.4 is

8.(1 pt) The functionf (x) = 4x3 −30x2 +0x−3

is decreasing on the interval ( , ).It is increasing on the interval ( −∞, ) and the

interval ( , ∞ ).The function has a local maximum at .9.(1 pt) For x ∈ [−12,10] the function f is defined

byf (x) = x6(x−4)5

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?

10.(1 pt) The hands on a clock are of lengths 5inches (minute hand) and 4 inches (hour hand). Howfast is the distance between the tips of the handschanging at 3:00.

Rate of change of distance at 3:00 between the tipsof the hands: inch(es) per minute.

11.(1 pt) Einstein’s Special Theory of Relativitysays that mass m is related to velocity v by the for-mula

m =m0

√

1− v2/c2= m0

(

1− v2

c2

)−1/2.

Here, m0 is the rest mass and c is the velocityof light. Use differentials to determine the percentincrease in mass of an object when its velocity in-creases from 0.8c to 0.84c.

Approximate percent increase:12.(1 pt) Identify the critical points and find the

maximum value and minimum value of the followingfunction on the given interval.

f (x) = x3 −3x+1, over [−3/2,3].1

Critical Points: , .Maximum: .Minimum: .

Instructions:1) When entering the critical points, please enterthem in the order that they appear on the real line.2) If the function has no critical points, enter thestring NONE in all answer boxes for critical points.

13.(1 pt) Identify the critical points and find themaximum value and minimum value of the followingfunction on the given interval.

f (x) = 3√x, over [−1,27].Critical Points: , .

Maximum: .Minimum: .

Instructions:1) When entering the critical points, please enterthem in the order that they appear on the real line.2) If the function has no critical points, enter thestring NONE in all answer boxes for critical points.

14.(1 pt) What number exceeds its square by themaximum amount? Begin by convincing yourselfthat this number is on the interval [0,1].

Answer: .15.(1 pt) A rectangle is to be inscribed in a semi-

circle of radius r with its base touching that of thesemicircle. What are the dimensions of rectangle ifits area is to be maximized?

Dimensions: r×r.



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

f (x) =x3

x2 −36defined on the interval [−16,15].

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

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

at andB. f (x) is concave up on the region to

and toC. The inflection points for this function are

, and2.(1 pt) Answer the following questions for the

functionf (x) = sin2(x/2)

defined on the interval [−5.4831852,1.3707963].Enter points, such as inflection points in ascending

order, i.e. smallest x values first.Rememer that you can enter ”pi” for π as part of youranswer.

A. f (x) is concave down on the region to

B. A global minimum for this function occurs at

C. A local maximum for this function which isnot a global maximum occurs at

D. The function is increasing on toand on to .

3.(1 pt) The function f (x) = 7x+9x−1 has one lo-cal 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 equals

with valueand a local maximum at x equals withvalue

4.(1 pt) Consider the function f (x) = 4(x− 4)2/3.For this function there are two important intervals:(−∞,A) and (A,∞) where A is a critical number.

Find AFor each of the following intervals, tell whether f (x)is increasing (type in INC) or decreasing (type inDEC).(−∞,A):(A,∞):For each of the following intervals, tell whether f (x)is concave up (type in CU) or concave down (type inCD).(−∞,A):(A,∞):

5.(1 pt) Consider the function f (x) = 4x + 8x−1.For this function there are four important intervals:(−∞,A], [A,B),(B,C], and [C,∞) where A, and C arethe critical numbers and the function is not defined atB.Find Aand Band CFor each of the following intervals, tell whether f (x)is increasing (type in INC) or decreasing (type inDEC).(−∞,A]:[A,B):(B,C]:[C,∞)Note that this function has no inflection points, butwe can still consider its concavity. For each of thefollowing intervals, tell whether f (x) is concave up(type in CU) or concave down (type in CD).(−∞,B):(B,∞):

6.(1 pt) A Norman window has the shape of a semi-circle atop a rectangle so that the diameter of thesemicircle is equal to the width of the rectangle. Whatis the area of the largest possible Norman windowwith a perimeter of 34 feet?

7.(1 pt) Consider the function

f (x) = 1x3 +2x2 +4x+3

Find the average slope of this function on the interval(4,12).

1

By the Mean Value Theorem, we know there exists ac in the open interval (4,12) such that f ′(c) is equalto this mean slope. Find the value of c in the intervalwhich works

8.(1 pt) Consider the function f (x) = 6√

x + 4 onthe interval [3,7]. Find the average or mean slope ofthe function on this interval.

By the Mean Value Theorem, we know there exists ac in the open interval (3,7) such that f ′(c) is equal tothis mean slope. For this problem, there is only one cthat works. Find it.

9.(1 pt) An object thrown from the edge of a 42-

foot cliff follows the path given by y =−2x2

25 +x+42.An observer stands 2.6656 feet from the bottom of thecliff.

(a) Find the position of the object when it is closestto the observer.

(b) Find the position of the object when it is far-thest from the observer.

Answers:(a) ( , ).(b) ( , ).10.(1 pt) The illumination at a point is inversely

proportional to the square of the distance of the pointfrom the light source and directly proportional to theintensity of the light source. If two light sources ares feet apart and their intensities are I and J respec-tively, at what point between them will the sum oftheir illuminations be a minimum?

Solution:Let x be the distance from I at which the sum of

the illuminations be minimum. Thenx = .Instruction: Give your answer in terms of s, I and

J.11.(1 pt) Find the equation of the line that is tan-

gent to the ellipse b2x2 +a2y2 = a2b2 in the first quad-rant and forms with the coordinate axes the triangle

with smallest possible area (a and b are positive con-stants.)

The equation of the required line is:x + y + = 0.

12.(1 pt) Find the indicated limit. Make sure thatyou have an indeterminate form before you applyl’Hopital’s Rule.

limx→π/2cosxπ2 − x = .

Instruction: If your answer is ∞, enter ”Infinity”; ifit is −∞, enter ”-Infinity”.

13.(1 pt) Find the indicated limit. Make sure thatyou have an indeterminate form before you applyl’Hopital’s Rule.

limx→0x3 −3x2 + x

x3 −2x = .Instruction: If your answer is ∞, enter ”Infinity”; if

it is −∞, enter ”-Infinity”.14.(1 pt) Find the indicated limit. Make sure that

you have an indeterminate form before you applyl’Hopital’s Rule.

limx→0sinx−tanx

x2 sinx = .Instruction: If your answer is ∞, enter ”Infinity”; if

it is −∞, enter ”-Infinity”.15.(1 pt) Find the indicated limit. Make sure that

you have an indeterminate form before you applyl’Hopital’s Rule.

limx→0+x2

sinx− x = .Instruction: If your answer is ∞, enter ”Infinity”; if

it is −∞, enter ”-Infinity”.16.(1 pt) Find

limx→0x2 sin(1/x)

tanx = .Instruction: If your answer is ∞, enter ”Infinity”; if

it is −∞, enter ”-Infinity”.17.(1 pt) Use a CAS (Computer Algebra System)

to evaluate the following limit:

limx→0cosx−1+ x2/2

x4 = .Instruction: If your answer is ∞, enter ”Infinity”; if

it is −∞, enter ”-Infinity”.



1.(1 pt) Evaluate the integral:R s(s+1)2

√s ds.

Answer: + C.2.(1 pt) Evaluate the indefinite integral:

R 3y√

2y2 +5dy.

Answer: + C.3.(1 pt) Find:

R

sin2 xdx.Answer: + C.4.(1 pt) A car traveling at 47 ft/sec decelerates at a

constant 6 feet per second squared. How many feetdoes the car travel before coming to a complete stop?

5.(1 pt) A ball is shot at an angle of 45 degrees intothe air with initial velocity of 44 ft/sec. Assuming noair resistance, how high does it go?

How far away does it land?

Hint: The acceleration due to gravity is 32 ft persecond squared.

6.(1 pt) Consider the function f (t) = 8sec2(t)−2t2. Let F(t) be the antiderivative of f (t) withF(0) = 0.Then F(5) =

7.(1 pt) Consider the function f (x) whose secondderivative is f ′′(x) = 2x + 6sin(x). If f (0) = 2 andf ′(0) = 4, what is f (5)?

8.(1 pt) Consider the function f (x) = 9x9 + 5x7 −8x3 −9.Enter an antiderivative of f (x)

9.(1 pt) A particle is moving with accelerationa(t) = 24t + 4. Its position at time t = 0 is s(0) = 1and its velocity at time t = 0 is v(0) = 4. What is itsposition at time t = 14?

10.(1 pt) A stone is dropped from the edge of aroof, and hits the ground with a velocity of -185feet per second. How high (in feet) is the roof?

11.(1 pt) Consider the differential equation:

dydx =

√xy .

a) Find the general solution to the above differen-tial equation. (Instruction: Call your integration con-stant C.)

Answer: y = .b) Find the particular solution of the above differ-

ential equation that satisfies the condition y = 4 atx = 1.

Answer: y = .12.(1 pt) Consider the differential equation:dudt = u3(t3− t).a) Find the general solution to the above differen-

tial equation. (Instruction: Write the answer in a formsuch that its numerator is 1 and its integration con-stant is C — rename your constant if necessary.)

Answer: u = .b) Find the particular solution of the above differ-

ential equation that satisfies the condition u = 4 att = 0.

Answer: u = .13.(1 pt) An object is moving along a coordinate

line subject to acceleration a (in centimeters per sec-ond per second) as follows

a = (1+ t)−4

with initial velocity v0 = 0 (in centimeters per sec-ond) and directed distance s0 = 10 (in centimeters).Find both the velocity v and the directed distance safter 2 seconds.

Velocity after 2 seconds: centimeter(s)per second.Directed distance after 2 seconds: cen-timeter(s).

14.(1 pt) The wolf population P in a certain statehas been growing at a rate proportional to the cuberoot of the population size. The population was esti-mated at 1000 in 1980 and at 1700 in 1990.

a) Find the differential equation for P(t) and thecorresponding conditions. (Instruction: Use C for theconstant of proportionality.)

dPdt =

P( ) = and P( ) =1

b) Solve your differential equation.P = .c) When will the wolf population reach 4000?The population will reach 4000 by the year .

15.(1 pt) Find ∑7k=3

(−1)k2k

k +1= .

16.(1 pt) Find ∑6k=−1 k sin(kπ/2) = .

17.(1 pt) Find the value of the following collapsingsum:

∑10k=1(2k −2k−1) = .

18.(1 pt) Use the Special Sum Formulas (see Sec-tion 5.3 of Varberg, Purcell and Rigdon) to find:

∑10i=1((i−1)(4i+3)) = .

19.(1 pt) In statistics, we define the meanx and the variance s2 of a sequence of numbersx1, . . . ,xn by

x =1n ∑n

i=1 xi.

s2 =1n ∑n

i=1(xi− x)2.Find x and s2 for the sequence of numbers 2, 5, 7,

8, 9, 10, 14.x = .s2 = .



1.(1 pt) Consider the integralZ 11

5(3x2 +4x+4)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

2.(1 pt) Evaluate the integral below by interpretingit in terms of areas. In other words, draw a pictureof the region the integral represents, and find the areausing high school geometry.

Z 6

−6

√

36− x2dx

3.(1 pt) 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 7

0|5x−5|dx

4.(1 pt)Z 8

1f (x)−

Z 3

1f (x) =

Z b

af (x)

where a= and b=

5.(1 pt) Consider the integralZ 5

1

(

4x +4

)

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

6.(1 pt) Consider the function f (x) = x2

3 +8.In this problem you will calculate

R 30 ( x2

3 +8)dx byusing the definition

Z b

af (x)dx = lim

n→∞

[

n∑i=1

f (xi)∆x]

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) = x23 +8 on the interval [0,3]

and write your answer as a function of n without anysummation signs. You will need the summation for-mulas in Section 5.3 of your textbook.Rn =limn→∞ Rn =

7.(1 pt) Compute the indefinite integralZ

(6x7 +3sec(x) tan(x))dx



1.(1 pt) Evaluate the definite integralZ 5

3

8x2 +5√x dx

2.(1 pt) If f (x) =R x

0 (t3 +5t2 +5)dtthenf ′′(x) =

3.(1 pt) If f(x) =R x3

2 t3dtthen

f ′(x) =f ′(6) =

4.(1 pt) Given

f (x) =Z x

0

t2−251+ cos2(t)dt

At what value of x does the local max of f (x) occur?x =5.(1 pt) Let f be an odd function and g be an even

function, and suppose thatZ 1

0| f (x)|dx =

Z 1

0g(x)dx = 3.

Use geometric reasoning to calculate each of thefollowing:

(a)R 1−1 f (x)dx = .

(b)R 1−1 g(x)dx = .

(c)R 1−1 | f (x)|dx = .

(d)R 1−1 xg(x)dx = .

6.(1 pt) Suppose thatZ 1

0f (x)dx = 2,

Z 2

1f (x)dx = 3,

Z 1

0g(x)dx = −1, and

Z 2

0g(x)dx = 4.

Use properties of definite integrals (linearity, inter-val additivity, and so on) to calculate the followingintegral:

Z 2

0(√

3 f (t)+√

2g(t)+π)dt.Answer: .

7.(1 pt) Let

G(x) =Z x

1xt dt.

FindG′(x) =

8.(1 pt) Find

limx→1

1x−1

Z x

1

1+ t2+ t dt.

Answer: .9.(1 pt) Use the Second Fundamental Theorem of

Calculus combined with the Generalized Power Ruleto evaluate the following integrals:

(a)R π/2

0 sin2 3xcos3xdx = .(b)

R x−1(t + |t|)dt = for x < 0, and

R x−1(t +

|t|)dt = for x ≥ 0.10.(1 pt) Find the average value of the following

function on the given interval:

f (x) =x√

x2 +16, on [0,3].

The average value of f on [0,3] is .11.(1 pt) View the following limit as a definite in-

tegral and then evaluate that integral by the SecondFundamental Theorem of Calculus.

limn→∞

n∑i=1

2n

[

1+2in +

(

2in

)2]

.

The above limit is equal to .12.(1 pt) Use the method of substitution to find the

following indefinite integral:Z zcos( 3√z2 +3)

(3√z2 +3)2

dz.

Answer: + C.13.(1 pt) Use the method of substitution to find the

following definite integral:Z π/2

−π/2cosθcos(πsinθ)dθ.

Answer: .

1

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

2


1.(1 pt) Sketch the region enclosed by the givencurves. Decide whether to integrate with respect to xor y. Then find the area of the region.x+ y2 = 30,x+ y = 0

2.(1 pt) Sketch the region enclosed by the givencurves. Decide whether to integrate with respect to xor y. Then find the area of the region.y = 9cosx,y = (6sec(x))2,x = −π/4,x = π/4

3.(1 pt) An object moves along a line so that itsvelocity at time t is v(t) = 1

2 + sin2t feet per second.Find the displacement and the total distance traveledby the object for 0 ≤ t ≤ 3π/2.

Displacement: feet.Total distance traveled: feet.4.(1 pt) Find the volume of the solid obtained by

rotating the region bounded by the given curves aboutthe specified axis.y = x6,y = 1; about y = 3

5.(1 pt) You wake up one morning, and find your-self wearing a toga and scarab ring. Always a logi-cal person, you conclude that you must have becomean Egyptian pharoah. You decide to honor yourselfwith a pyramid of your own design. You decide itshould have height h = 3360 and a square base withside s = 1980To impress your Egyptian subjects, find the volumeof the pyramid.

6.(1 pt) A ball of radius 13 has a round hole of ra-dius 7 drilled through its center. Find the volume ofthe resulting solid.

7.(1 pt) Find the volume of the solid formed by ro-tating the region inside the first quadrant enclosed byy = x4

y = 27xabout the x-axis.

8.(1 pt) Find the volume of the solid generated byrevolving about the x-axis the region bounded by theupper half of the ellipse

x2

a2 +y2

b2 = 1and the x-axis, and thus find the volume of a pro-

late spheroid. Here a and b are positive constants,with a > b.

Volume of the solid of revolution: .9.(1 pt) The region bounded by y = 2+sinx, y = 0,

x = 0 and 2π is revolved about the y-axis. Find thevolume that results.

Hint:Z

xsinx dx = sinx− xcosx+C.

Volume of the solid of revolution: .10.(1 pt) Find the length of the curve defined by

y = 4x3/2 +3from x = 2 to x = 6.

11.(1 pt) Consider the parametric curve given bythe equations

x(t) = t2 +36t +43y(t) = t2 +36t −12

How many units of distance are covered by thepoint P(t) = (x(t),y(t)) between t=0, and t=10 ?

12.(1 pt) Find the length of the following curve:

y =

Z x

π/6

√

64sin2 ucos4 u−1 du,π6 ≤ x ≤ π

3 .

Length of the curve: .13.(1 pt) Find the area of the surface generated by

revolving the following curve about the axis:

x = r cos t,y = r sin t,0 ≤ t ≤ π.

Area of the surface: .14.(1 pt) The circle x = acost,y = asin t,0 ≤ t ≤

2π is revolved about the line x = b,0 < a < b, thusgenerating a torus (doughnut). Find its surface area.

1

Area of the torus: .15.(1 pt) A force of 3 pounds is required to hold

a spring stretched 0.6 feet beyond its natural length.How much work (in foot-pounds) is done in stretch-ing the spring from its natural length to 0.8 feet be-yond its natural length?

16.(1 pt) For a certain type of nonlinear spring, theforce required to keep the spring stretched a distances is given by the formula

F = ks4/3.

If the force required to keep it stretched 8 inches is2 pounds, how much work is done in stretching thisspring 27 inches?

Amound of work done: inch-pound(s).17.(1 pt) The masses and coordinates of a system

of particles are given by the following:

5,(−3,2); 6,(−2,−2); 2,(3,5); 7,(4,3); 1,(7,−1).

Find the moments of this system with respect tothe coordinate axes, and find the coordinates of thecenter of mass.

Moment with respect to the x-axis: .Moment with respect to the y-axis: .Center of mass: ( , ).

18.(1 pt) Find the centroid of the region boundedby the following curves:

y = x2,y = x+3.

Hint: Make a sketch and use symmetry where pos-sible.

Centroid: ( , ).

19.(1 pt) Use Pappus’s Theorem to find the volumeof the torus obtained when the region inside the circlex2 + y2 = a2 is revolved about the line x = 2a.

Volume of torus: .


Documents

Peter Alfeld Math 1210-3, Fall 2005