Engineering Mathematics TI-92

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International Journal of Mechanical Engineering Education Vol 27 No 1

Engineering mathematicsteaching via the TI-92Part I: Enhancing basic skillsL. SHORT, Mathematics Department, Napier University, Edinburgh,Scotland, UK

Received 28th July 1997

Revised 12th January 1998

We discuss the use of the TI-92 Supercalculator in enhancing the teaching and learning of

first-year engineering mathematics. Several illustrative examples are discussed which high-light some of the essential features of the technology, and which can be used to develop basicmathematical skills. In addition some of the educational implications of the existence of such

powerful hand-held technology is considered.

The TI-92 is the first example of another, and much more powerful, generation of super

calculators. In fact, it is difficult not to call the TI-92 a true computer. It is small,

inexpensive (150), hand-held, user-friendly, and yet has a large screen display (240 by

128 pixels), and a QWERTY keyboard and, most importantly, contains a collection ofvery useful and powerful integrated software packages for doing algebra, geometry,

calculus, statistics, and matrix algebra.

The built-in software on the TI-92 includes five graphing packages (including an

improved sequence grapher and a wonderful 3D grapher), computer interactive geometry

co-developed with the authors of CABRI Geometry, and a powerful computer algebra

1. INTRODUCTION

The developments in computer technology are beginning, and will inevitably continue, to

alter the emphasis in the teaching of many scientific disciplines. In a mechanical engineering

context Perez and Blake [1] discuss the use of Maple as a teaching tool in an introductory

course on control systems. But it is not only mechanical engineers that are concerned with

computer-enhanced teaching of engineering students. Mathematicians are also keen to

embrace current technology, not only in the teaching of their own mathematics students, but

also in their service teaching for science courses in general, and engineering courses inparticular.

An alternative to using sophisticated software packages on PCs in Engineering/

Mathematics/Computer Laboratories is now available in the form of the Texas Instruments

TI-92 calculator. The best overview we can give of the technology is to paraphrase Waits

and Demana [2]:


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system co-developed with the authors of DERIVE, as well as advanced programming

software. Computer PC or MAC versions of the built-in software on the TI-92 could cost

four or five times that of the TI-92.

14 L. Short

In addition we may note the presence of a text editor, extensive toolbar facilities, and

around 200K of available memory; the latter feature is soon to be standard, rather than

requiring an enhanced memory chip.

Before continuing we wish to stress the following. Our aim is not to promote the TI-92 in

any commercial sense; we keep referring to the TI-92 specifically because there is no other

calculator on the market with anything like the same facilities. Indeed it is our hope that, by

bringing the inherent potential of such technology to a wider audience, other manufacturers

will be encouraged to produce similar (and hopefully superior) machines for student use.

Such competition can, in the long term, only be beneficial for all concerned.

For the past academic year the TI-92 has been used in a pilot project as an integral part of

the first-year mathematics course for a combined class of Energy with Environmental, and

Energy with Management, Engineers. This has been made possible by a substantial subsidy

from the Mathematics and MMSE (Mechanical, Manufacturing and Systems Engineering)

departments, allowing all the students to purchase a TI-92. It is intended that the machine

will be used extensively by the students for all three years of mathematics they are required

to study. (For the next academic session the class will further be combined with Mechanical

Engineering students.)

We may observe that one major reason for our use of the TI-92 is its portability and the

consequent need not to use any laboratory facilities. The use of the latter is a very significant

problem at Napier. But there is one feature in common. An overhead attachment on a TI-92

allows the students to see the calculator screen and follow the calculations on their own

machine. This allows a very interactive form of teaching, with the traditional distinction

between lectures and tutorials inevitably becoming somewhat blurred. In addition every

TI-92 can be connected to the overhead attachment, so that students can come to the front

and explain a computation on their own calculator. This is a very useful feature, and helps

build student confidence.

Reports on various mathematical aspects of the project can be found in references [35].

Here we consider some of the mathematical skills that the TI-92 can be used to enhance, and

attempt to ascertain the effectiveness of the technology in a teaching environment.

2. LEARNING THE SYSTEM VERSUS IMMEDIATE APPLICATIONS

There are two possible initial approaches to using the TI-92:

(i) Consider (engineering) applications as soon as possible.

(ii) Learn to use the TI-92 as effectively as possible before looking at applications.

We decided to employ (ii) for several reasons:

(a) Since the TI-92 is to be used by the class for three years, it is important that the basics

are well understood. With technology as sophisticated as this, there are a lot of

operating features that need to be appreciated if efficient and effective use is to be

attained.


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Engineering mathematics teaching via the TI-92 15

(b) Applications are not fully appreciated unless they tie in with material in the appropri-

ate engineering courses. Applications must be taught, but they must not be rushed

into.

(c) Basic mathematical ideas mustbe understood.

Few mathematicians or engineers would argue with this last general sentiment, although

there may be differences in what concepts are considered basic. Lawson [6] gives what many

would consider a reasonable set of basic requirements, together with a discussion of student

weaknesses in these areas. In addition he very briefly considers the impact symbolic

calculators can be expected to have, together with the following warning: Finally some

conclusions are drawn from the analysis and a warning note is given about relying too much

on new technology for an answer to some of the problems uncovered.

We believe that, for example, the naive assignment

Use of TI-92 Loss of algebraic skills

is hopelessly simplistic. Indeed, as we hope to show, a more realistic appraisal is

Correct use of TI-92 Enhancementof algebraic skills,

a conclusion in line with that of Perez and Blake [1]. Indeed, not only can algebraic skills be

enhanced, but also number skills, graphical skills, geometric skills, .

The present paper can thus be viewed as a reply to Lawsons rather dire warning.

Applications of the TI-92 will be considered in a future application [7].

4. EXPLORING SOME BASIC MATHEMATICAL IDEAS

One very notable feature of the TI-92 is that it permits students to investigate, and explore,

3. A PRELIMINARY INDICATION OF STUDENT SKILLS

An initial diagnostic test was conducted, a portion of which is given in the Appendix at the

end of this paper. Because the class size was small (roughly 20), and the test consisted of

only a small number of questions (30 in fact), we do not attach any great statistical signifi-

cance to the results. However, the following general features were very apparent:

(i) Algebraic manipulation was poor.

(ii) Arithmetic skills, especially fractions, were minimal.

(iii) Graphical/geometrical knowledge was very limited.

(iv) Calculus appeared uncharted territory to many.

In addition we did not attempt to run a diagnostic test at the end of the course to try and

assess any improvement in skills. This was felt to place too big a burden on students revising

for end-of-year exams. In addition, our major goal was to have the students actually using

the technology, hopefully with some proficiency, rather than overly concerning ourselves

with educational issues of secondary importance (at least to the students). We hope to

address the latter in future years of the course.

One should also note that the Scottish educational system has some significant

differences to that in England. Roughly speaking, A-level material will be covered in the

first year of a degree course.


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16 L. Short

mathematical ideas for themselves, coupled, of course, with guidance from the lecturer. This

can be achieved even at a very elementary level, as we seek to show in this section. To do

this we consider some of the questions from the diagnostic test, and their various extensions.

For details of each of the following examples refer to the appropriate question in theAppendix.


(a) The Pretty Print feature of the TI-92 ensures expressions appear in mathematical

format in the History area (where computations are displayed, evaluated and stored),

even though they may not do so on the entry line (where computations are input).

(b) The TI-92 will operate symbolically as well as numerically.(c) By pressing Enter, rather than just Enter, in (1a) an answer in decimal form is

obtained. The TI-92 will operate in a variety of fashions, depending on the Mode

settings.

(d) The factor command, on the F2: Algebra toolbar, allows factorization of expres-

sions. The precise effect of the command depends on both the (optional) factoring

argument used (see Fig. 2), and on whether any key simplification occurs.

(e) The expand command, again on the F2 : Algebra toolbar, functions roughly as the

inverse of the factor command.

(f) The With operator , accessed directly from the keyboard, allows temporary substi-

tutions in an expression. The logical operator and is found on the Math Test menu.(g) Once an expression has been entered, and stored in the History area, it can be retrieved

in one of two ways, either by highlighting the entry using the appropriate cursor and

autopasting via Enter, or using 2nd ANS, termed the last answer facility. See the

entry line of Fig. 1. In this way complicated expressions need only be entered once,

and recalled if necessary.

Example 1 (Question 2)

The manipulations of fractions seems to cause great difficulties for many students. The

TI-92 will perform such computations in a variety of ways. Most simply

1 2 + 2 3 Enter

gives

12 + 23 76 (1a)

as shown in the top line of Fig. 1. Of course, used in this direct fashion, students may not

learn anything about how fractions are manipulated but, nevertheless, will be able to obtain

the correct answer. However, one can proceed less directly and seek to obtain some of the

intermediate calculations in order to illustrate the method used. In the present case we would

like to produce the complete computation

12

23

1 3 2 22 3

76

+ = + =. ..

(1b)

We cannot quite do this, but Fig. 1 depicts some possibilities which all but demonstrate (1b).

The difficulty lies in the automatic simplification rules the TI-92 employs: the expression

1.3 + 2.2 will always be simplified to 7. We may note the following:


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Fig.

2.

Transpositio

nofformulae.

Fig.

1.

Somecomputation

sinvolvingfractions.


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18 L. Short


Transportation of formulae, i.e. the manipulation of algebraic expressions, is an important

ability for students to master. Here a solution is obtained as in Fig. 2, and we observe the

following:

(a) If a calculation does not lead to a simplification, the TI-92 indicates this by produc-

ing a more complicated-looking formula for the result. In this way one is led to

choosing, at each stage, the correct way of proceeding. For example, the next to last

line in Fig. 2 reflects the computation

1 1 1

f v u

v u

vu= + =

+(2)

and does not help in solving for v.

(b) The solve command, also on the F2: Algebra toolbar, allows formulae to be trans-

posed automatically, as depicted in the last line of Fig. 2. This feature can be used in

one of two ways:

(i) as a check on the computations, and

(ii) as a shortcut, one the underlying principles are understood.

(c) The availability of the last answer facility is crucial in allowing previous results to

be easily re-used. For example, ans(1) 1u produces the second line of Fig. 2.

(d) The left-hand entry in each computation in the History area shows precisely the

computation being performed by the TI-92. This is invaluable for student learning, as

is the Pretty Print feature. In addition the right-hand side contains the answer to the

computation, and this gives rise to the terminology entryanswer pair. Thus, with

reference to the computation in (c), the l.h.s. of line 2 shows that 1u is being

subtracted from both sides of the previous equation. The resulting equation is then

displayed at the r.h.s. of line 2. How are lines 3 and 4 obtained?

(e) Further computations based on (2) are considered in [4]. In addition the formula (2)

appears in a variety of contexts [8].


Graphical analysis on the TI-92 is particularly impressive and powerful, and this is awonderful feature for the many students who are more comfortable with problems they can

visualize. The graphical solution of Question 9 is depicted in Fig. 3(a), and we note the

following points:

(a) We have used a split-screen to show both the curves themselves and the associated

scales (window variables). Tick marks along either axis can be shown, and their

spacing is controlled by the values of xscl and yscl. It is possible to suppress all this

information if desired.

(b) The value of xres (x-resolution) controls the smoothness of the curve by specifying

at which pixels points are plotted. This in turn affects how quickly the plots areproduced. If one wishes to obtain a quick impression of the shape of a curve, a large

value of xres is used.

(c) Different curves can easily by distinguished by drawing them in different stylesthe

options Line and Thick are used in Fig. 3(a). The F6 : Style toolbar is available for this

purpose when using the Y = Editor to specify the curves to be plotted.



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Fig.

3.

Intersectionofcurves.


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20 L. Short

(d) The labelling of the curves in Fig. 3(a) can be done directly on the screen using the

Text option of the F7 toolbar depicted at the top of the screen.

Students should be familiar with the graphical distinction between even and odd powers

both as an aid to appreciating the shapes polynomial curves can take, but also for later work

on Fourier series. For sketching purposes a sequence of powers can be very compactly

entered using lists. On the TI-92 lists have many applications, but this is probably the

simplest.

Since the location of the intersection point in Fig. 3(a) is easily guessed, we now modify

the problem to that of determining the intersection point(s) of the curves

y = (x + 3)2 10 and y = x3 (3)

The graphs of Fig. 3(b) indicate three such points, and we can proceed to locate them

precisely in two distinct ways:

(i) Graphically: Using the F3 : Trace toolbar allows us to move along a curve to obtain

approximate intersection points. To obtain more precise information we use the

F2 : Zoom toolbar which permits cursor movement anywhere on the Graph screen.

Using 2 : ZoomIn, repeatedly if necessary, can then be used to locate the intersection

as accurately as required.

(ii) Algebraically: There are various possibilities here using the F2 : Algebra toolbar and

the solve, zeros and factor options. See Fig. 3(c) for details. Here the two

functions in (3) have been entered in the Y = Editor as y10 and y11 respectively.

Then, for example, the intersection points in (3) are computed via

solve(y10(x) = y11(x), x)

Before continuing there are two further important points to be made:

(1) To appreciate the great ease with which the results of Fig. 3 can be obtained really

requires one to see the TI-92 in action. The very user-friendly nature of the technology

is a major asset in encouraging students to explore material for themselves.

(2) There are many options available on the TI-92, and this in turn allows many problems

to be solved in a variety of ways. For example, the intersections in Fig. 3(c) can also

be located using the F5 : Math toolbar with 5 : Intersections. Not only does this enable

checks to be made on computations, it also helps give students a flavour of theinterconnected nature of mathematical ideas and techniques.



We can use the Geometry menu to investigate this as depicted in Fig. 4. The sequence of

steps is the following:

(a) Draw a circle, as in Fig. 4(a), using the F3 toolbar and 1 : Circle. Use the appropriate

cursors to move the + screen cursor to the desired centre of the circle and pressEnter. The across cursor is then used to control the radius of the circle.(b) Construct a diameter of the circle using the F2 toolbar and 5 : Segment. Use or

to locate the end points of the segment, and Enter to define them.

(c) Repeat (b) twice more to construct the remaining two sides of the triangle, as in Fig.

4(b). An alternative is to use the F3 toolbar and 3 : Triangle.


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Fig.

4.

Angleina

semicircle.


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22 L. Short

(d) Measure the appropriate angle using the F6 toolbar and 3 : Angle. The three points

defining the angle are selected in turn, using the cursors and pressing Enter, and the

required value of the angle is returned. This leads to Fig. 4(c), where the angle is

given in degrees since this was the current angle mode setting of the TI-92. (See alsoQuestion 10 of the Appendix in this connection.)

(e) Repeat the computations of (d) for another triangle. This is easily accomplished by

dragging a vertex of the original triangle, which entails using the F1 toolbar and

1 : Pointer to point to the vertex to be dragged. Then hold down the dragging hand

on the keyboard and use the cursors to drag the vertex to the desired position.

(f) In both (d) and (e) we obtain an angle of 90.00. We can use the F6 toolbar and8 : Check Property to formally verify that we have a right-angle. This is achieved by

showing the two sides enclosing the suspected right-angle are perpendicular, and

3 : Perpendicular accomplishes this as in Fig. 4(d). Each segment is chosen in turn via

the cursors and Enter. After the second is selected we obtain the statementPerpendicular to this segment to establish the result.

We note the following:


(i) There is such a large variety of features available on the Geometry menu that a good

deal of practice is required to obtain familiarity with them. However, many of the

features operate in a straightforward fashion, and there is usually little difficulty in

implementing a construction once the basics have been assimilated.

(ii) The Geometry menu allows a very dynamic, and interactive, view of elementary

geometry to be developed by students. This is in stark contrast to the traditional, rather

arid, approach which fails to motivate many students.(iii) The result of Question 12 is a special case of the (Euclid) theorem that the angles at

the circumference of a circle, standing on the same arc, are equal. This in turn follows

from the theorem that the angle at the centre of a circle is twice any angle at the

circumference, standing on the same arc. These two results can also be investigated in

a similar way to Fig. 4, as the reader may investigate.

(iv) Do engineering students still need to learn any geometry? The answer to this must

surely be Yes. As well as the countless applications of geometrical techniques to

practical problems, such ideas often provide more insight into a problem and can lead

to quick and elegant solutions. The use of the TI-92 provides a, potentially very

powerful, means of reversing the current sad decline in geometrical thinking at alllevels of mathematics teaching.


We can solve this with the push of a couple of buttons on the TI-92. See the top line of

Fig. 5(a). Further standard derivatives are just as easily calculated, together with derivatives

involving the standard rules of differentiation, i.e. Chain, Product and Quotient rulessee

Fig. 5(a). Integration can also be accomplished easily, as depicted in Fig. 5(b).Of course such calculations are, by themselves, notsuitable as a teaching aid. However,

as in Example 2, we can break up the computations to produce intermediate results which

are of great importance in a teaching situation. Fig. 6(a) provides a simple example of this in

the context of differentiation from first principles, a process usually found rather difficult by

most students. We note the following:


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Fig.

5.

Somecalculu

scomputations.

Fig.

6.

Differentiationfromf

irstprinciples.


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24 L. Short

(a) We can use the TI-92s ability to define functions to make the computation quite

compact. In addition the usual textbook convention, of using x to denote a small

increment in x, can be adhered to since the TI-92 is equipped with a Greek alphabet

(as well as many further character types).(b) The calculation can be further streamlined as in the top two lines of Fig. 6(b) and,

once sufficient proficiency has been attained, use can be made of 3 : Limit on the

F3 : Calc toolbar. This is shown in the third entry in Fig. 6(b) and, although somewhat

unnecessary, the limit command can also be used to complete the calculation in Fig.

6(a).

(c) Derivatives of trigonometric functions in degree mode, together with the differentia-

tion of log10 x, usually cause difficulties for many students. The TI-92 yields these

directly, although again first principles can be invoked for teaching purposes. See the

last entry in Fig. 6(b).

As well as aiding on the technical side of calculus, the TI-92 also provides for a much

more geometrical approach to these ideas. Fig. 7 provides an illustration based on the

following question used in a Calculus Test with the TI-92 class:

Fig. 7. A geometrical computation.

Test question

(i) Determine, by hand calculation, the point(s) of intersection of the curve

y = 4 + 5x x2 and the line y = 4 x2

(ii) Check your result in (i) using the TI-92.

(iii) Sketch (i) and (ii) using the TI-92.

(iv) Determine, by hand calculation, the equations of the tangent lines to the curve

in (i) at the computed intersection points.

(v) Superimpose the tangent lines in (iv) on the sketch in (iii). In this way check your

equations.(vi) Determine the area of the triangle formed by the tangents in (v) and the x-axis.


Here hand computations are asked for since both the tangent lines themselves and their

equations (together with many other facilities) are available from the Graph Math menu.

This is the F5 : Math toolbar available when in a graphical application. In addition we may


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Fig.

8.

SomeTI-92

toolbars.


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recall (Example 3) that graphs can be labelled using the Text option on the F7 toolbar, and

this can be a great aid in subsequent computations. See Fig. 7.

5. CONCLUSIONS

We have only considered a few aspects of the types of ideas that can be, and indeed have

been, investigated with the TI-92 at an elementary level. The three most important points

we wish to make in conclusion are the following:

(1) The five examples we have considered involve the following topic areas: Arithmetic,

Algebra, Graphs, Geometry, Calculus. A single hand-held calculatorsuffices for all

these topics, and we have not even discussed the (very extensive) programming

features, nor the Data/Matrix editor.

(2) Although one cannot expect TI-92, and its inevitable successors, to provide auniversal panacea for the effective teaching of mathematics, one can, with proper

use, expect the TI-92 to aid substantially in this goal. Relying too much on the

technology is an ever-present danger but, once appreciated, such pitfalls can be over-

come. The gains to be made from using the technology vastly outweigh the dangers.

(3) The portability of the TI-92 encourages its constant use and this in turn lets the

students develop a more investigative view of their mathematics. This mustbe benefi-

cial, not only for students, but also for teachers and the general health and well-being

of the subject.

Student reactions to the course were, after some initial apprehension, very positive andencouraging [3, 9]. Certainly none of the students wished to sell their TI-92s back to us at the

end of the year! It remains to be seen in later years of the course whether this translates into

enhanced skills, or at least more confidence in their mathematical ability, and increased

mathematical maturity. However, for both students and teacher alike, be warned:

Once you start using the TI-92 you will not want to look at any other calculator,

nor will you want to teach without it. I certainly dont.

For anybody thinking about employing the technology, in either a teaching capacity, or

merely for personal use, we give a brief indication of some of the possible options by listing

a few of the toolbars available. See Fig. 8.Some applications of the TI-92, suitable with first-year engineering classes, will be

considered in reference [7].

REFERENCES

[1] Perez, R. A., and Blake, J., Using symbolic algebra to enhance the learning environment ofdynamic systems, International Journal of Mechanical Engineering Education, 24(3), 207225(1996).

[2] Waits, B. K., and Demana, F., TI-92, the hand-held revolution in computer enhanced maths

teaching and learning, CTI Centre for Mathematics and Statistics Newsletter, 6(2), 210 (1995).[3] Short, L., The TI-92 Pilot Engineering Mathematics Project at Napier, Scottish MathematicalCouncil Journal, 27, 3944 (1998).

[4] Short, L., Some teaching implications of the TI-92. Part I: The Napier Pilot Project, InternationalJournal for Mathematical Education in Science and Technology(unpublished).

[5] Short, L., Teaching and learning with the TI-92, Scottish Mathematical Council Journal, 27, 4552 (1998).


26 L. Short


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Engineering mathematics teaching via the TI-92 27

[6] Lawson, D. A., How well prepared are students for undergraduate engineering mathematics?,International Journal of Mechanical Engineering Education, 23(4), 352362 (1995).

[7] Short, L., Engineering mathematics teaching via the TI-92. Part II: First year applications,International Journal of Mechanical Engineering Education (to be submitted).

[8] Townend, M. S., Unified reciprocals, Teaching Mathematics and its Applications, 14, 6770(1995).

APPENDIX

The questions are arranged according to topic, and not according to difficulty. The students

are told not to waste time over those they cannot do, and to answer [E] rather than guessing.

Arithmetic

1. The value of 11 (6) 5 2 is

[A] 0 [B] 24 [C] 5 [D] 7 [E] I dont know

2. The value of 1223+ is

[A] 35 [B]76 [C]

36 [D]

86 [E] I dont know

Algebra

3. The simplified form of a a2 3 is

[A] a

23

[B] a

6

[C] a

1

[D] a

5

[E] I dont know

4. A pen costs P and a refill for such a pen is p pence. If you buy 5 pens, each with a refill,

you pay, in pence,

[A] 5 5100P p+ [B]5

100 5P p+ [C] +5 500P p

[D] 500P + 5p [E] I dont know

5. Transforming the formula1 1 1

f u v= + to make v the subject of the equation gives

[A] f u [B]fu

u f[C]

1

uf+

[D]u f

uf

[E] I dont known


6. The expression

[A] is always negative [B] is always positive [C] is sometimes

zero

[D] can be positive or negative, depending on the value of x [E] I dont know

Graphs

7. The straight line joining the points (3, 11) and (2, 4) has equation

[A] y = 2 3x [B] y = 10 7x [C] y = 7x 10

[D] y = 3x 2 [E] I dont know


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28 L. Short

8 The graph of y = 4 sin (2x) has

[A] amplitude 2 and period 2 [B] amplitude 2 and period 1 [C] amplitude 4

and period 2 [D] amplitude 4 and period 1 [E] I dont know

9. The graphs of y = x2 and y = x3 intersect in

[A] one point [B] two points [C] three points [D] no points

[E] I dont know

Trigonometry

10. In radian measure 45 degrees is

[A]

2 [B]

3 [C]

4 [D]

6 [E] I dont know

11. The Sine and Cosine Rules apply

[A] to any triangle [B] to any triangle with no angle greater than a right-angle

[C] only to right-angled triangles [D] only to isosceles triangles

[E] I dont know

Geometry

12. In any triangle the sum of the (internal) angles, in degrees, is

[A] 90 [B] 180 [C] 270 [D] 360 [E] I dont know

13. The angle in a semicircle, in degrees, is

[A] 90 [B] 180 [C] 270 [D] 360 [E] I dont know

Calculus

14. The derivative of x3 is

[A] x2 [B] x4 [C] 3 x2 [D] 14 x 4 [E] I dont know

15. The integral of velocity is

[A] distance [B] acceleration [C] work done [D] energy

[E] I dont know

Engineering Mathematics TI-92

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