Calculus II Maple T.A. Course Module Gord Clement and Jack Weiner, University of Guelph Calculus II Maple T.A. Course Module . Gord Clement and Jack Weiner, University of Guelph

Calculus II Maple T.A. Course Module

Gord Clement and Jack Weiner, University of Guelph

This course module has been designed to accompany the second semester of the introductory

honours calculus course at the University of Guelph. At Guelph, we have a twelve week

semester, with three fifty minute classes and a fifty minute lab each week.

This is a theoretical course intended primarily for students who need or expect to pursue further

studies in mathematics, physics, chemistry, engineering, or computer science. It is a continuation

of the first semester course Calculus I. These materials have been successfully used with classes

ranging in size from 15 to 600 students.

Topics:

inverse functions

inverse trigonometric functions

hyperbolic functions

L'Hôpital's Rule

techniques of integration

applications of integration to volumes and arc length

parametric equations

polar coordinates

Taylor and MacLaurin series

functions of two or more variables

partial derivatives

The course module consists of 10 question banks and 10 assignments, designed to be used

weekly, beginning in week 2. Almost all questions are algorithmically generated, with

algorithmically generated solutions provided in the question feedback. Over half are Maple

graded.

The tests are presented both as ‘practice’ and as ‘homework’. Students are encouraged to do

practice tests first, where we have configured the tests so that they can check their answers as

they proceed. In the Guelph course, the students are allowed five attempts at the homework quiz,

with only their best mark counting towards their final grade. Each is weighted out of 2%. While

the students do treat these as tests, they really constitute ‘enforced homework’.

The module was first implemented in the Winter, 2007 semester and again in every Winter

semester since. Each time, 500 or more students accessed the tests. After a few coding

adjustments in 2007, the tests have run very smoothly. Incidents where a student insists T.A.

graded a question incorrectly are rare. In each such case, so far, the student has been in error. The

tests are robust.

Following this introduction, you will find

a table of contents for the 10 tests

a T.A. Protocol Sheet

a T.A. Syntax Sheet (We strongly recommend that the students use text rather than

equation editor entry for their answers.)

This course module is copyright Gord Clement, Jack Weiner and Maplesoft. However, you are

welcome to modify and implement the course module at your institution. We encourage you to

send us your suggestions for improvements and/or new questions. If we incorporate any of the

latter, your contribution will be gratefully acknowledged!

Contact Information:

Professor Jack Weiner

Department of Mathematics and Statistics

University of Guelph

Guelph, Ontario, Canada

N1G 2W1

(519) 824-4120, extension 52157

[email protected]

Gord Clement

Westside Secondary School

Orangeville, Ontario, Canada

L9W 5A2

(519) 265-0608

[email protected]

August 22, 2012

mailto:[email protected]


Calculus II Maple TA Tests

Prepared by: Professor Jack Weiner ([email protected])

and

Gord Clement ([email protected])

Department of Mathematics and Statistics

University of Guelph

Table of Contents

Test 1: Inverse Functions and Inverse Trig

Test 2: Arctrig Derivatives and Integrals

Test 3: Hyperbolic Trig Functions

Test 4: L’Hôpital’s Rule

Test 5: Integration by Parts and Trig Products

Test 6: Integration by Trig Substitution

Test 7: Integration by Partial Fractions & Improper Integrals

Test 8: Volumes of Revolution & Arc Length

Test 9: Parametric Equations

Test 10: Polar Coordinates



TA PROTOCOL

PLEASE FOLLOW THIS TA PROTOCOL!

1) Work through the sample TA test in your course manual.

2) Do a couple of "Practice" tests. Use "How did I do?" to check your answers on the go. Use "Preview" to check your syntax.

Note: Preview does not recognize interval notation. So don't use preview to check questions requiring aninterval.

Hint: If the answer involves "complicated" math, enter it in Maple, then copy and paste this into TA. TAwill translate your answer into correct syntax. Neat. Please don't abuse this suggestion by getting Maple to DO the questions for you. By all means, use Maple to check your answers.

3) Now you are ready for prime time. You should be able to get perfect on a "Homework" quiz in one or two attempts. You will allowed FIVE attempts. Only your BEST mark will count on TA.

4) DO NOT LEAVE TA TILL THE LAST DAY THE QUIZ IS OPEN!

5) ALWAYS GRADE YOUR TEST WHEN YOU ARE FINISHED. If you didn't do all questions when you grade, TA will inform you. Then you MUST click grade again. If you click, "View Details", you will see your entire test and be given the option of printing it.

6) Always "QUIT AND SAVE" after you finish a test whether it is homework or practice.

All your homework tests are saved in the system and you can retrieve and view them at any time.

Unless indicated otherwise in class, do NOT switch math entry mode to symbolic math. Continue to use text entry.

Always include arithmetic operations. For example, don't enter xy when you mean x*y. (TA and Maple will treat xy as a single symbol.) Use brackets generously but only and unless otherwise specified in the insructions to a question. Please pay attention to those extra instructions when they are included.

TA SYNTAX (Keep this sheet with you whenever you work on at TA test.)

Math Expression TA text entry syntax

x*y; x/y

x^y

a/(b*c) or TA likes a/b/c (but I don't)

sqrt(x) or x^(1/2) Do not use x^.5!

x^(2/3)

abs(x)

ln(x); log[2](x)

; ; exp(x); e or exp(1); pi or Pi; infinity

sin(x)^2 or (sin(x))^2

TA always uses 1+ tan(x)^2 for sec(x)^2 but sec(x)^2 is fine.

TA always uses 1+ cot(x)^2 for csc(x)^2 but csc(x)^2 is fine.

TA always uses sin(x)/cos(x)^2 for sec(x)*tan(x) but sec(x)*tan(x) is fine.

TA always uses cos(x)/sin(x)^2 for csc(x)*cot(x) but csc(x)*cot(x) is fine.

Test 1: Inverse Functions and Inverse Trig

Question 1: Score 1/1

For a function to have an inverse, it must be

Correct

Your Answer: one to one

Comment:


Correct

Your Answer:

Comment: Interchange and and solve.


State the exact value of arccot . Give your answer in radian measure. Use Pi or pi for . Correct

Your Answer: 1/6*Pi

Comment: Remember arccot has range .


Find the domain of the function ( ). Correct

Your Answer:

Comment: arcsec is undefined for

arcsec is undefined for


Correct

Your Answer:

Comment:


Correct

Your Answer: -1.4

Comment:


Correct

Your Answer: 4/9*Pi

Comment: Remember arccsc has range .


Correct

Your Answer: 1/6*Pi

Comment: Remember arctan has range .


Correct

Your Answer: Does not exist!

Comment: -0.5 is not in the domain of arcsec.


Find the exact value of .

Correct

Your

Answer:

Comment: Remember that arccos has range . This means that the triangle you draw should be in the first or second

quadrant, use the CAST rule to determine which.


Find the exact value of

Correct

Your Answer:

Comment:

arccos

In this picture , by the Pythagorean Theorem .

From here , .

In this picture , , by the Pythagorean Theorem .

From this we see , .

Now we just need to use the formula


Complete the square: Correct

Your Answer:

Comment:

Test 2: Arctrig Derivatives and Integrals


Find the derivative:

Correct

Your Answer:

Comment:



Correct

Your Answer:

Comment:


Correct

Your Answer:

Comment:


Correct

Your Answer:

Comment:


Correct

Your Answer: (64-25*x^2)^(1/2)+1/5*arcsin(5/8*x)+C

Comment:


Correct

Your Answer: -1/2*ln(81+25*x^2)+1/45*arctan(5/9*x)+C

Comment:


Correct

Your Answer: 2/3*ln(3*x^2-24*x+49)+17/3*3^(1/2)*arctan((x-4)*3^(1/2))+C

Comment:


Correct

Your Answer: -2*(7-2*x^2-4*x)^(1/2)-3*2^(1/2)*arcsin(1/3*2^(1/2)*(x+1))+C

Test 3: Hyperbolic Trig Functions


Algebraically, Using this,

Correct

Your Answer:

Comment:


Which of the following is the graph of y= ? Correct

Your Answer:

Comment:



Correct

Your Answer:

Comment: Remember + + + - - -



Correct

Your Answer:




Correct

Your Answer:



Remember that If 9 then Correct

Your Answer:

Comment:


Find the integral:

Correct

Your Answer:



Find the integral:

Correct

Your Answer:



Find the integral:

Correct

Your Answer:

Comment:


Which of the following is the graph of y=arccosh(x)?

Correct

Your Answer:

Comment:


Correct

Your Answer:

Comment:


BABA works for archyperbolics!

Correct

Your Answer:

Comment:

Test 4: L’Hôpital’s Rule


Which of the following (there may be more than one!) are indeterminate forms?

Choice Selected

`1^infinity` Yes [answer withheld]

`infinity/0` No [answer withheld]

`0/0` Yes [answer withheld]

`0^1` No [answer withheld]

Correct

Number of available correct choices: 2


Which of the following conditions form the HYPOTHESIS for the basic form of L'Hopital's Rule?

Let f and g be functions defined on an open interval containing such that

(i)

(ii) and exist on

(iii) for

(iv) for

(v) exists

(vi)

Correct

Your Answer: (i), (ii), (iii), (v)

Comment:


Correct

Your Answer: -1/3

Comment: This " " limit is set up perfectly for L'Hopital's Rule.


Correct

Your Answer: 1/2

Comment:

" "

= by L'Hopital's Rule

=


Correct

Your Answer: 1/2

Comment:

" "

= "0/0" by L'Hopital's Rule

= by L'Hopital's Rule =


Correct

Your Answer: 1

Comment:

" "


=


Correct

Your Answer: 2

Comment:

" "

= "0/0" by L'Hopital's Rule

= by L'Hopitals Rule

=


Correct

Your Answer: 1/2

Comment: Make a common denominator to combine the fractions, then use L'Hopital's Rule


Evaluate:

Correct

Your Answer: -1/5

Comment: This is the indeterminant form .Start this question by dividing top and bottom by .


Evaluate:

Hint: When

Correct

Your Answer: -1/5

Comment: This is the indeterminant form " ". To start this question divide top and bottom by .


Correct

Your Answer: 1

Comment:

" "

= " "

= by L'Hopital's Rules

=


Correct

Your Answer: e^-28

Comment:

" "

= " "


=


Correct

Your

Answer: infinity

Comment: If your answer is correct, GOOD!If you applied L'Hopital's Rule here, you probably got the wrong answer. This

question does NOT involve an indeterminate form. BE CAREFUL!

Test 5: Integration by Parts and Trig Products


Find

Correct

Your Answer: 1/4*cos(2*x)+1/2*x*sin(2*x)+C

Comment:

Let


Find

Hint: Remember .

Correct

Your Answer: -1/2*x*cot(2*x)+1/4*ln(abs(sin(2*x)))+C

Comment:

Let


Find .

Hint: Let so that Substitute and use Integration by Parts on the resulting integral.

Correct

Your Answer: 2*cos(x^(1/2))+2*x^(1/2)*sin(x^(1/2))+C

Comment:

Let

Let


Correct

Your Answer:

Comment:


Correct

Your Answer:

Comment:


Evaluate .

Correct

Your Answer: 1/10*(-1+x^10)*exp(x^10+3)+C

Comment:

Let


Evaluate

Correct

Your Answer: -1/10*x^10*cos(x^10+3)+1/10*sin(x^10+3)+C

Comment:

Let

=


Evaluate . Don't forget absolute value where it is needed.

Correct

Your Answer: 1/2*sec(x)*tan(x)+1/2*ln(abs(sec(x)+tan(x)))+C

Comment:

Let


Correct

Your Answer:

Comment:


Correct

Your

Answer:

Comment: The question asked for the BEST strategy. If you let , you will end up with a question that

requires . You know the integral of and you can now use I by P to

integrate . This works but is two steps longer than the BEST strategy, which is to use I by P right away.


Correct

Your Answer:

Comment:


Correct

Your Answer:

Comment:


Correct

Your Answer: 1/2*x+1/16*sin(8*x)+C

Comment:

=

=

Test 6: Integration by Trig Substitution


An integral involves If you solve it by trigonometric substitution, you set Correct

Your Answer:

Comment:


Correct

Your Answer:

Comment:


Correct

Your Answer:

Comment:


An integral involves a power of the expression . After completing the square, we

would use the trigonometric substitution Correct

Your Answer:

Comment: After completing the square we have a power of .


An integral involves a power of the expression . After completing the square, we would

use the trigonometric substitution Correct

Your Answer:

Comment: After completing the square we have a power of .


Evaluate dx.

Correct

Your Answer: -((1-9*x^2)^(1/2)+3*arcsin(3*x)*x)/x+C

Comment:

Let .

therefore and

, Note for therefore .

Where and

Therefore, in both cases


Evaluate dx.

Correct

Your Answer: -(-x+arcsin(x)*(1-x^2)^(1/2))/(1-x^2)^(1/2)+C

Comment:

Let .

therefore and

, Note for therefore .

Where and

Therefore, in both cases


Evaluate dx.

Correct

Your Answer: 1/(1+9*x^2)^(1/2)*x+C

Comment:

Let

, Note for therefore

Where

In both cases,


Evaluate dx.

Correct

Your Answer: 1/108*(6*x+9*arctan(2/3*x)+4*arctan(2/3*x)*x^2)/(9+4*x^2)+C

Comment:

Let

Where

In both cases,


Evaluate assuming

Hint: Since only use the second quadrant triangle. Don't use absolute value anywhere in

your answer. When , and

so

Correct

Your Answer: -ln((x^2-1)^(1/2)-x)+C

Comment:

Let , Note: since we only use the second quadrant.

Note: for

from hint.

Test 7: Integration by Partial Fractions & Improper Integrals


As a partial fraction decomposition, we set equal to

Correct

Your Answer:

Comment:



Correct

Your Answer:

Comment:



Correct

Your Answer:

Comment:


True or False: When evaluating the following integral using Partial Fractions, you must first divide the

bottom into the top. dx Correct

Your Answer: False

Comment:


Evaluate dx.

Correct

Your Answer: 11/20*ln(abs(x-3))-9/5*ln(abs(x+2))+5/4*ln(abs(x+1))+C

Comment:

Let

for all

Set :

Set :

Set :

Now

=


Evaluate dx. Correct

Your Answer: x-13/6*ln(abs(x+3))+7/6*ln(abs(x-3))+C

Comment:

After performing long division,

Let

Set :

Set

Now,


Evaluate dx.

Correct

Your Answer: 1/5*arctan(1/2*x)+1/5*ln(x^2+4)+3/5*ln(abs(x-1))+C

Comment:

Let

=

Set :

Coefficient of

Coefficient of :

Therefore,

=


Evaluate the improper integral: dx. Your answer should be a finite number,

infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral

as the limit of a proper integral.

Correct

Your Answer: DNE

Comment:

dx

=

=

=

Therefore the integral does not exist.


Evaluate dx. Your answer should be a finite number, infinity, -infinity, or enter DNE if

it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.

Correct

Your Answer: -infinity

Comment:

=


Evaluate dx. Your answer should be a finite number, infinity, -infinity, or enter DNE if

it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.

Correct

Your Answer: 7

Comment:


Evaluate dx. Your answer should be a finite number, infinity, -infinity, or enter

DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper

integral.

Hint: Remember BABA and the Chain Rule in Reverse:

Correct

Your Answer: 1/12*Pi

Comment:


Evaluate the improper integral dx. Your answer should be a finite number, infinity,

-infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as the

limit of a proper integral.

Correct

Your Answer: 2

Comment:

=

=

Test 8: Volumes of Revolution & Arc Length


The integral using vertical rectangles which finds the volume obtained when the region bounded

by and is rotated about the line -4 is given by

Correct

Your Answer:

Comment:

This is a shell.

thickness


The integral using vertical rectangles which finds the volume obtained when the region bounded


Correct

Your Answer:

Comment:

This is a difference of discs.


The integral using horizontal rectangles which finds the volume obtained when the region bound


Correct

Your Answer:

Comment:

This is a difference of discs.


The integral using horizontal rectangles which finds the volume obtained when the region bounded

by and is rotated about the line 1 is given by

Correct

Your Answer:

Comment:

This is a shell.

thickness =


By rotating the semi-circle about the axis, we can find the volume of a sphere of

radius 2. The integral which gives this volume is

Correct

Your Answer:

Comment:

This is a disc.


By rotating the line about the axis from to 2, we can find the volume of a

cone of radius 2 and height 1. The integral using vertical rectangles which gives this volume is

Correct

Your Answer:

Comment:

This is a shell

thickness


By rotating the line about the axis from to 2, we can find the volume of a

cone of radius 5 and height 2. The integral using horizontal rectangles which gives this volume is

Correct

Your Answer:

Comment:

This is a disc.


?

Correct

Your Answer:

dx

Comment: The arclength of from to is given by


. Give your answer to TWO decimal places.

(HINTS: The integral you will have to evaluate is NOT HARD. To get your approximation, work out

your answer exactly using THE Fundamental Theorem of Calculus--F(b)-F(a)--and then go to Maple and

use "evalf(F(b)-F(a))".)

Correct

Your Answer: 7.8466

Comment: Arclength


Correct

Your

Answer:

Comment: How would you modify this integral so that you would find the length of the upper half of the ellipse

x^2/a^2+y^2/b^2=1?

Test 9: Parametric Equations


Which of the following pairs of parametric equations draws the circle of radius 6 from (6, 0) counter-

clockwise to (0,-6)?

Correct

Your Answer:

Comment:


Find , where and .

Correct

Your Answer: -1/16*(sin(3*t)*t+cos(3*t))/t^7

Comment:

=

=

=


Find dy/dx where and . Correct

Your Answer: 4*cos(4*t)/(4+4*tan(4*t)^2)

Comment:


Which of the following pairs of parameric equations draws the ellipse

clockwise from (0, -3) to (7,0)?

Correct

Your Answer:

Comment:


State the intercept(s) using set notation, that is, { }, for the parametric equations

and .

(Some answers involve "ln". Don't use absolute value signs in your answer if they are not necessary. Use

exp(x) for )

Correct

Your Answer: {-sin(9), sin(3)}

Comment:

To find intercepts, set and solve.

therefore or .

Corresponding to we have intercept -sin(9).

Corresponding to we have intercept sin(3).

Enter your answer as

{-sin(9), sin(3)}


State the intercept(s) using set notation, that is, { }, for the parametric equations

and .

(Some questions involve "ln". Don't use absolute value if it is not necessary. Remember to use exp(x)

for )

Correct

Your Answer: {49, 64}

Comment:

To find y intercepts, set and solve.

=

Therefore or

Corresponding to we have y intercept 64.

Corresponding to we have y intercept 49.

Enter your answer as {64, 49}.


A pair of parametric equations is defined for all real numbers . The first derivative is given

by . For what intervals is the graph INCREASING, that is,

when is increasing as increases?

Note/Hint: A function can still be increasing on an interval even if its derivative is 0 or undefined

somewhere in the interval!

Correct

Your Answer:

Comment:


A pair of parametric equations using parameter is defined for The first derivative is given

by . At and , we have Correct

Your Answer:

Comment:

Horizontal tangents occur when .


A pair of parametric equations using parameter is defined for The first derivative is given

by . At , we have Correct

Your Answer:

Comment:

Vertical tangents occur when .


Vertical asymptotes are finite values, that is , where approaches either plus or minus

infinity.

List in set notation, that is, { }, the value(s) of for the relation given by

and .

Correct

Your Answer: {0, 1/8}

Comment:

Therefore is a vertical asymptote.


Enter your answer as {0, 1/8}.


Horizontal asymptotes are finite values, that is , where approaches either plus or minus

infinity.

List in set notation, that is, { }, the value(s) of for the relation given by

and .

Correct

Your Answer: {2}

Comment:


never tends to , therefore this is our only vertical asymptote.

Enter your answer as {2}.


A pair of parametric equations is defined for all real numbers The second derivative is given

by . For what intervals is the graph concave down?

Note/Hint: A function can still be concave down on an interval even if its second derivative is 0 or

undefined somewhere in the interval!

Correct

Your Answer:

Comment:

Test 10: Polar Coordinates


The rectangular coordinates corresponding to polar coordinates ( , ) are

Correct

Your Answer: ( , )

Comment:


One pair of polar coordinates corresponding to rectangular coordinates ( , ) are

Correct

Your Answer: ( , )

Comment:

Since , is in the 3rd quadrant, choose ,

therefore one name for the point in polar coordinates is ,


Which of the following gives all possible coordinates of polar coordinates ( , )?

Correct

Your Answer: ( , ) and ( , )

Comment:


Two of the following polar equations generate this graph. Which?

Choice Selected

No [answer withheld]


Yes [answer withheld]


Correct


Partial Grading Explained

Comment:



Choice Selected





Correct



Comment:



Choice Selected





Correct



Comment:



Choice Selected





Correct



Comment:



Choice Selected





Correct



Comment:



Choice Selected





Correct



Comment:


Which of the following polar equations generates this graph?

Correct

Your Answer:

Comment:


Which of the following polar equations generates this graph?

Correct

Your Answer:

Comment:


Polar equation can also be written with parametric equations

Correct

Your Answer: and

Comment: To change to parametric equations use

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Calculus II Maple T.A. Course Module Gord Clement and Jack Weiner, University of Guelph Calculus II Maple T.A. Course Module . Gord Clement and Jack Weiner, University of Guelph