This article was downloaded by: [North Carolina State University] On: 18 October 2012, At: 15:20 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Mathematical Education in Science and Technology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmes20 Some curious properties and loci problems associated with cubics and other polynomials Amal de Alwis a a Department of Mathematics, Southeastern Louisiana University, SLU, Hammond 70402, LA, USA Version of record first published: 29 Mar 2012. To cite this article: Amal de Alwis (2012): Some curious properties and loci problems associated with cubics and other polynomials, International Journal of Mathematical Education in Science and Technology, 43:7, 897-910 To link to this article: http://dx.doi.org/10.1080/0020739X.2012.662288 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and- conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
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This article was downloaded by: [North Carolina State University]On: 18 October 2012, At: 15:20Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
International Journal of MathematicalEducation in Science and TechnologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmes20
Some curious properties and lociproblems associated with cubics andother polynomialsAmal de Alwis aa Department of Mathematics, Southeastern Louisiana University,SLU, Hammond 70402, LA, USA
Version of record first published: 29 Mar 2012.
To cite this article: Amal de Alwis (2012): Some curious properties and loci problems associatedwith cubics and other polynomials, International Journal of Mathematical Education in Science andTechnology, 43:7, 897-910
To link to this article: http://dx.doi.org/10.1080/0020739X.2012.662288
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
International Journal of Mathematical Education inScience and Technology, Vol. 43, No. 7, 15 October 2012, 897–910
Some curious properties and loci problems associated with cubics
and other polynomials
Amal de Alwis*
Department of Mathematics, Southeastern Louisiana University, SLU,Hammond 70402, LA, USA
(Received 19 June 2011)
The article begins with a well-known property regarding tangent lines to acubic polynomial that has distinct, real zeros. We were then able togeneralize this property to any polynomial with distinct, real zeros. We alsoconsidered a certain family of cubics with two fixed zeros and one variablezero, and explored the loci of centroids of triangles associated with thefamily. Some fascinating connections were observed between the originalfamily of the cubics and the loci of the centroids of these triangles. Forexample, we were able to prove that the locus of the centroid of certaintriangles associated with the family of cubics is another cubic whose zerosare in arithmetic progression. Motivated by this, in the last section of thearticle, we considered families of cubic polynomials whose zeros are inarithmetic progression, along with the loci of the special points of certaintriangles arising from such families. Special points include the centroid,circumcentre, orthocentre, and nine-point centre of the triangles.Throughout the article, we used the computer algebra system,Mathematica�, to form conjectures and facilitate calculations.Mathematica� was also used to create various animations to explore andillustrate many of the results.
1. A remarkable property of a tangent line to a cubic and a generalization
We will begin with the following well-known property regarding a third-degreepolynomial with distinct real zeros. When tangent lines are drawn to the graph ofsuch a polynomial, a surprising property can be observed: Namely, the tangent lineat a point on the graph that is halfway between any two zeros will pass through theother zero (Figure 1).
We will state this well-known fact in the following theorem, along with its proof.
Theorem 1.1: Consider the cubic polynomial given by f ðxÞ ¼ kðx� aÞðx� bÞðx� cÞwhere k is any nonzero real number and a, b, c are three distinct, real numbers. Let Mbe the point on the graph of f whose x-coordinate is the average of any two zeros of f.Then, the tangent line to the graph of f at M passes through the third zero.
Proof: Without loss of generality, consider the two zeros x ¼ a and x ¼ b of f.
Then, M has the coordinates M ¼ ððaþ bÞ=2, f ððaþ bÞ=2ÞÞ: Using the product rule,
one can calculate the derivative of f ðxÞ to obtain the following [1,2]:
f 0 xð Þ ¼ k ðx� aÞðx� bÞ þ ðx� bÞðx� cÞ þ ðx� cÞðx� aÞ½ � ð1Þ
Thus, the slope of the tangent line to the graph of f at M is given by
m ¼ f 0 ðaþ bÞ=2ð Þ ¼ �kða� bÞ2=4.By point-slope formula, we can obtain the equation of the tangent line atM to be
the following [3]:
y ¼ �k a� bð Þ
2
4xþ
k a� bð Þ2c
4ð2Þ
when x ¼ c, the right-hand side of Equation (2) vanishes, meaning the tangent line at
M passes through the third zero ðc, 0Þ. œ
Next, we wanted to see if a similar idea would hold for any degree polynomial
with real, distinct zeros. As an example, for the fourth degree polynomial,
gðxÞ ¼ kðx� aÞðx� bÞðx� cÞðx� d Þ, we were curious to find out whether the
tangent line to graph of g at x ¼ ðaþ bþ cÞ=3 would pass the fourth zero, x ¼ d.
Unfortunately, this idea failed as some specific calculations suggested. However,
going back to Theorem 1.1, rather than thinking of ðaþ bÞ=2 as an average, another
way of thinking of it is as a critical point of the function F xð Þ ¼ k x� að Þðx� bÞ that
is obtained by dropping the last factor x� cð Þ from f xð Þ ¼ k x� að Þ x� bð Þðx� cÞ (see
[1,2]). In the same vein, we conjectured that the tangent line to the graph of the above
fourth-degree polynomial gðxÞ at any critical point of G xð Þ ¼ k x� að Þ x� bð Þðx� cÞ
would pass the fourth zero x ¼ d of g. To our surprise, the conjecture was true, and
thus we were able to discover the following theorem.
Theorem 1.2: Consider the nth degree polynomial given by f xð Þ ¼ kðx� a1Þ
x� a2ð Þ x� a3ð Þ . . . ðx� anÞ where k is a nonzero real constant, and ai 2R,
i ¼ 1, 2, . . . , n are all distinct, and n 2N, with n � 3. Let FðxÞ be the polynomial
obtained by omitting any one factor of f ðxÞ, such as ðx� anÞ and x ¼ t be any critical
point of FðxÞ. Then, the tangent line to the graph of f at x ¼ t will pass through the zero
x ¼ an of f:
Figure 1. The tangent line at a point halfway between any two zeros passes through thethird zero.
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Proof: Observe that f xð Þ ¼ F xð Þðx� anÞ. By using the product rule of calculus, onecan obtain [1,2]:
f 0 xð Þ ¼ F xð Þ þ F 0 xð Þ x� anð Þ ð3Þ
Therefore, the slope of the tangent line to the graph of f at x ¼ t is given bym ¼ f 0 tð Þ ¼ F tð Þ þ F 0 tð Þ t� anð Þ. However, since x ¼ t is a critical point of FðxÞ,F 0 tð Þ ¼ 0, which implies that m ¼ FðtÞ. Thus, using the point-slope formula, theequation of the tangent line to the graph of f at x ¼ t is given as follows:
y ¼ F tð Þ x� tð Þ þ f ðtÞ ð4Þ
when x ¼ an, the right-hand side of Equation (4) becomes F tð Þ an � tð Þ þ FðtÞðt� anÞ,which is zero. This means that our tangent line will pass through the zero x ¼ anof f. œ
In the next section, we will consider some interesting locus problems associatedwith a certain family of cubic polynomials.
2. Locus problems associated with a family of cubics with two fixed zeros and one
variable zero
Consider the family of cubic polynomials given by f xð Þ ¼ k x� að Þ x� bð Þðx� tÞwhere k, a, and b, are fixed real numbers, k 6¼ 0, a 6¼ b, and t is a variable realparameter such that t 6¼ a, t 6¼ b. These polynomials have two fixed zeros at x ¼ a,x ¼ b, and one variable zero at x ¼ t. Let Aða, 0Þ, Bðb, 0Þ, and Cðt, 0Þ be the pointsthat correspond to the three zeros of f . Using calculus, it is clear that f has exactlytwo distinct local extrema, say corresponding to the points P and Q on thegraph of f .
The current situation motivates us to consider some geometric properties oftriangles associated with such families of cubic polynomials. For example, letG1ð�1,�1Þ be the centroid of 4APQ as shown in Figure 2 [4–6]. Note that thecentroid, or the centre of gravity, is the balancing point of any triangle. It is also thepoint where all three medians of the triangle meet. As the parameter t changes,4APQ changes along with its centroid G1. Now, let us investigate the locus, or thepath of centroid G1, for changing values of t.
Figure 2. Centroid G1 of 4APQ.
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For any triangle with the vertices at the points Piðxi, yiÞ, where i ¼ 1, 2, 3, its
centroid Gð �x, �yÞ is given by the following formula [6]:
�x ¼ ðx1 þ x2 þ x3Þ=3 �y ¼ ð y1 þ y2 þ y3Þ=3 ð5Þ
Let Aða, 0Þ be ðx1, y1Þ, P be ðx2, y2Þ, and Q be ðx3, y3Þ. To find the x-coordinates of
the vertices P and Q of 4APQ, set f 0ðxÞ ¼ 0 and solve for x to obtain the
Equations (8) and (9) define the parametric equations of the locus of the centroid
G1 of 4APQ. In order to find the Cartesian equation of the locus, one can eliminate
the variable t between the Equations (8) and (9), and replace �1 and �1 by x and y
respectively. To do so, one can use hand-calculations or use the ‘Eliminate’
command of Mathematica�, and the result is as follows [7,8]:
y ¼ �9k
2x3 þ
9k
22aþ bð Þx2 þ
k
2�11a2 � 14ab� 2b2� �
xþka
22aþ bð Þðaþ 2bÞ ð10Þ
To our greatest surprise, the right-hand side of Equation (10) factored completely.
Using the ‘Factor’ command of Mathematica�, one can obtain the following
simplified equation for the locus of the centroid G1 of 4APQ:
y ¼k
22aþ b� 3xð Þ aþ 2b� 3xð Þða� xÞ ð11Þ
Therefore, as suggested by Equation (11), the locus of G1 is also a cubic
polynomial passing through the point ða, 0Þ, just like the original cubic
f xð Þ ¼ k x� að Þ x� bð Þðx� tÞ. Clearly, the three zeros of polynomial in Equation
(11) are z1 ¼ a, z2 ¼ ð2aþ bÞ=3, and z3 ¼ ðaþ 2bÞ=3. We can easily see that z2 is the
average of z1 and z3. In other words, the cubic given by Equation (11) has a
remarkable property, being that all its zeros are equally spaced. In other words, the
zeros of the locus of the centroid of 4APQ are in arithmetic progression!The earlier discussion and similar findings can be summarized in the following,
which is a major theorem of this article.
Theorem 2.1: Consider the family of cubic polynomials given by f xð Þ ¼ k x� að Þ
x� bð Þðx� tÞ where k, a, and b, are fixed real numbers, k 6¼ 0, a 6¼ b, and t is a
variable real parameter such that t 6¼ a, t 6¼ b. Let Aða, 0Þ, Bðb, 0Þ, and Cðt, 0Þ be the
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points corresponding to the three zeros of f , and let P and Q be the points on the graph
of f corresponding to its local extrema.
(a) The locus of the centroid of 4APQ is another cubic polynomial given by
y ¼ k 2aþ b� 3xð Þ aþ 2b� 3xð Þða� xÞ=2, whose zeros are in arithmetic
progression.(b) The locus of the centroid of 4BPQ is also a cubic polynomial given by
y ¼ k 2aþ b� 3xð Þ aþ 2b� 3xð Þðb� xÞ=2, whose zeros are in arithmetic
progression.(c) The locus of the centroid of 4CPQ is also another cubic polynomial given by
y ¼ �2k 4a� b� 3xð Þ aþ b� 2xð Þða� 4bþ 3xÞ=125, whose zeros are in
arithmetic progression.
Proof: Part (a) was proved in the discussion preceding the theorem. Using similar
methods, one can also obtain the coordinates of centroid G2ð�2,�2Þ of 4BPQ as
By eliminating the t variable between Equations (12) and (13), we can obtain the
following locus of the centroid of 4BPQ:
y ¼ k 2aþ b� 3xð Þ aþ 2b� 3xð Þðb� xÞ=2 ð14Þ
Note that the zeros of Equation (14) are z1 ¼ b, z2 ¼ ðaþ 2bÞ=3, and z3 ¼ ð2aþ bÞ=3.Clearly, z2 ¼ ðz1 þ z3Þ=2, so the zeros of (14) are in arithmetic progression, which
proves (b) of the theorem.Similarly, the coordinates of centroid G3ð�3,�3Þ of 4CPQ are given by
Note that the zeros of Equation (17) are z1 ¼ ð4a� bÞ=3, z2 ¼ ðaþ bÞ=2, andz3 ¼ ð�aþ 4bÞ=3. Clearly, z2 ¼ ðz1 þ z3Þ=2, so the zeros of (17) are in arithmetic
progression, which proves (c) of the theorem. œ
Let us illustrate part (a) of the above Theorem 2.1, using Mathematica�.
In addition to being a computer algebra system, Mathematica� has its own
programming language, which can be used to write the following Program 2.1
(see [7,8]). This program creates an animation of the centroid G1 of 4APQ for
changing values of t.
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Observe that each of the above graphs exhibit a certain type of symmetry around
their middle zeros. Namely, f 2þ tð Þ ¼ �f ð2� tÞ, and g 1þ tð Þ ¼ �gð1� tÞ, where t is
any real number. This is the same type of symmetry occurring in any odd function,
such as y ¼ x3 or y ¼ sinðxÞ around x ¼ 0. These observations led us to the following
theorems, whose proofs we had to omit.
Theorem 3.1: Consider the odd degree polynomial whose zeros are in arithmetic
progression given by f xð Þ ¼ k x� a½ � x� aþ dð Þ½ � . . . ½x� ðaþ n� 1ð Þd � where k, a, and
d are real numbers with k and d being nonzero, and n a positive odd integer. Then, the
graph of f is symmetric about its middle zero x ¼ aþ n� 1ð Þd=2, which means that
f ½aþ ðn� 1Þd=2� t� ¼ �f ½aþ ðn� 1Þd=2þ t� where t is any real number.
Figure 4. Graphs of odd degree polynomials f ðxÞ and gðxÞ, respectively.
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Theorem 3.2: Consider the polynomial f defined in the previous theorem. Lat Ai be the
area of the region bounded by the graph of f, the x-axis, and between the consecutive
zeros x ¼ aþ i� 1ð Þd and x ¼ aþ id for i ¼ 1, 2, . . . , n� 1. Then Ai ¼ An�i for
i ¼ 1, 2, . . . , n� 1.
Note that similar results can be obtained for even degree polynomials whose
zeros are in arithmetic progression. However, those results are not included here.
(b) As the last item of the article, we will make a series of interesting observations
regarding a certain family of cubic polynomials whose zeros are in arithmetic
progression. Consider the family of cubic polynomials given by
f xð Þ ¼ k x� a½ � x� aþ tð Þ½ �½x� aþ 2tð Þ�, where k, a are real constants, k 6¼ 0, and t
is a nonzero real parameter. Note that all the polynomials f have one fixed zero at
x ¼ a and two variable zeros at x ¼ aþ t and x ¼ aþ 2t. Let Aða, 0Þ, Bðaþ t, 0Þ, and
Cðaþ 2t, 0Þ be the points corresponding to the zeros of f. Let P be the point on the
graph that corresponds to the local extremum of f between x ¼ a and x ¼ aþ t. As t
changes, 4APC changes (Figure 5).
We would now like to investigate several geometric properties of 4APC, for
changing values of t. Some of these geometric properties include the centroid G,
circumcentre O, orthocentre H, and centre N of the Nine-Point circle [4,9].
By definition, the circumcentre of a triangle is the point where the perpendicular
bisectors of each side intersect. Similarly, the orthocentre of a triangle is the point
where the three altitudes meet. Lastly, according to the famous Feuerbach theorem,
the nine-point circle of a triangle is the circle passing through the midpoints of the
sides, feet of the altitudes, and the midpoints of line segments joining the vertices to
the orthocentre [9]. See Figure 6.In order to find the centroid Gð�1,�1) of 4APC, we will first calculate the
coordinates of Pðx1, y1Þ. By setting the derivative of f xð Þ equal to zero and solving for
x, we can obtain the following expressions for x1 and y1:
x1 ¼ ð3aþ 3t� tffiffiffi3pÞ=3 ð18Þ
y1 ¼ f ðð3aþ 3t� tffiffiffi3pÞ=3Þ ð19Þ
Figure 5. The graph of f xð Þ ¼ k x� a½ � x� aþ tð Þ½ �½x� aþ 2tð Þ� and the associated triangle4APC.
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Since we now know the coordinates of each vertex of 4APC, as done in Section 2
of the article, we can average the x and y-coordinates separately to obtain the
following coordinates of centroid Gð�1,�1).
�1 ¼ aþ t�t
3ffiffiffi3p ð20Þ
�1 ¼2kt3
9ffiffiffi3p ð21Þ
In order to find the Cartesian equation for the locus of centroid G, eliminate the t
variable between Equations (20) and (21) to obtain the following equation:
y ¼9k x� að Þ
3
45ffiffiffi3p� 41
ð22Þ
Now, we can use Mathematica� program similar to Program 2.1 to create an
animation of G along with its locus (Figure 7).
Figure 7. A frame of the animation of centroid G of 4APC with corresponding locus.
Figure 6. The (a) circumcentre, (b) orthocentre, and (c) nine-point centre of 4APC.
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Now, let us find the equation of the locus of circumcentre Oð�2,�2Þ of 4APC.For any triangle with the vertices Piðxi, yiÞ, where i ¼ 1, 2, 3, its circumcentre
Equations (23) and (24) can be obtained either by hand or Mathematica�
calculations, by using the definition of the circumcentre of the triangle. Since we
already know the coordinates of 4APC, the above Equations (23) and (24) can be
reduced to obtain the following coordinates for circumcentre C:
�2 ¼ aþ t ð25Þ
�2 ¼�9þ 2k2t4
6ffiffiffi3p
ktð26Þ
In order to find the Cartesian equation for the locus of circumcentre O, eliminate
the t variable between Equations (25) and (26) to obtain the following:
y ¼1
6ffiffiffi3p
k2k2 x� að Þ
3�
9
x� a
� �ð27Þ
Here is a frame of the animation of circumcentre O and its locus (Figure 8).Next, we will find the equation of the locus of orthocentre Hð�3,�3Þ of 4APC.
For any triangle with the vertices Piðxi, yiÞ, where i ¼ 1, 2, 3, its orthocentreHð�3,�3Þis given by the following formulas:
Equations (28) and (29) can be obtained either by hand or Mathematica�
calculations, by using the definition of the orthocentre of the triangle. Since wealready know the coordinates of 4APC, the above Equations (28) and (29) can bereduced to obtain the following coordinates for orthocentre H:
�3 ¼ aþ t�tffiffiffi3p ð30Þ
�3 ¼
ffiffiffi3p
ktð31Þ
In order to find the Cartesian equation for the locus of orthocentre H, eliminatethe t variable between Equations (30) and (31) to obtain the following:
y ¼
ffiffiffi3p� 1
kðx� aÞð32Þ
Here is a frame of the animation of orthocentre H and its locus (Figure 9).Lastly, we will discuss the equation of the locus of the Nine-point centre
Nð�4,�4Þ. A well-known fact in geometry is that, for any triangle, the nine-pointcentre is the midpoint of the line segment joining the circumcentre to the orthocentre[9]. Therefore, since we know the coordinates of circumcentre Cð�2,�2Þ andorthocentre Hð�3,�3Þ from Equations (25), (26), (30) and (31), we can now find thecoordinates of Nð�4,�4Þ by using �4 ¼ ð�2 þ �3Þ=2 and �4 ¼ ð�2 þ �3Þ=2. Uponsimplification, we will obtain the following expressions for the coordinates of N:
�4 ¼ aþ t�t
2ffiffiffi3p ð33Þ
Figure 9. A frame of the animation of orthocentre H of 4APC with corresponding locus.
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�4 ¼9þ 2k2t4
12ffiffiffi3p
ktð34Þ
As usual, eliminate the t variable between Equations (33) and (34) to acquire the
equation of the locus of nine-point centre N of 4APC:
y ¼1
8 30ffiffiffi3p� 37
� �k
32k2 x� að Þ3þ217� 104
ffiffiffi3p
x� a
� �ð35Þ
Here is a frame of the animation of nine-point centre N and its locus (Figure 10).Just to summarize, Equations (22), (27), (32), and (35) represent the loci of the
centroid, circumcentre, orthocentre, and nine-point centre respectively, of 4APC.
What is interesting is to observe that each of these loci is a combination of the basic
cubic function y ¼ ðx� aÞ3 and the hyperbola y ¼ 1=ðx� aÞ with zero or nonzero
constant coefficients. We will now collect the results of the preceding discussions in
the following theorem, which is the last, major result of the article.
Theorem 3.3: For changing t values, the equations of the loci of the centroid,
circumcentre, orthocentre, and Nine-Point centre of 4APC are given by the following,
respectively:
centroid ðxÞ ¼ 9kðx� aÞ3=ð45ffiffiffi3p� 41Þ
circumcenterðxÞ ¼ ½1=ð6ffiffiffi3p
kÞ�½2k2ðx� aÞ3 � 9=ðx� aÞ�
orthocenterðxÞ ¼ ðffiffiffi3p� 1Þ=½kðx� aÞ�
Figure 10. A frame of the animation of Nine-Point centre N of 4APC with correspondinglocus.
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Nine-point centre ðxÞ ¼1
8 30ffiffiffi3p� 37
� �k
32k2ðx� aÞ3 þ217� 104
ffiffiffi3p
x� a
� �
Each of these loci equations are of the form �ðx� aÞ3 þ � 1ðx�aÞ for some nonzero or
zero constants � and �.
Now, we can use a Mathematica� program similar to Program 2.1. to create ananimation which simultaneously displays all four special points, centroid G,circumcentre O, orthocentre H, and nine-point centre N of 4APC, for changingvalues of t. It also displays the graphs of the loci of these four points as dotted lines.A frame of the animation is given in Figure 11.
The most remarkable feature of this animation is that the four special points arealways collinear at any instant. This supports a well-known theorem in geometry,which states that for any triangle, its centroid, circumcentre, orthocentre, and Nine-Point centre are always collinear [4,5,10]. A modern day computer algebra systemsuch as Mathematica� enables us to discover new results, and at the same time, it canalso reinforce old results.
4. Conclusion
In conclusion, we were able to combine well-known concepts of calculus andgeometry to observe some curious phenomena. While making these discoveries, wewere able to use modern day technology such as Mathematica� to form conjectures,perform calculations, and create animations to display the results. Using methodssimilar to the ones employed in this article, the reader can also investigate a variety ofother geometric situations. For example, referring to Figure 2, it would be interestingto explore the locus of the midpoint of line segment PQ as well as the envelope oflines PQ. For the same figure, the reader could also consider the loci of the fourspecial points for 4’s ABR,ACR, and BCR, where R is the point of inflection of the
Figure 11. A frame of the animation of the four special points of 4APC simultaneously.
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cubic polynomial f xð Þ ¼ k x� að Þ x� bð Þ x� tð Þ (see [1,2]). Furthermore, we would
encourage the reader to supply the proofs of those results that were omitted from this
article due to space limitations. Many of the concepts relating to this article are
considered to be ‘classical mathematics’. However, with the aid of modern computer
algebra systems, one can shine new light on this traditional material to obtain some
unexpected and exciting results.
References
[1] R. Larson, Calculus of a Single Variable, Cengage, Mason, OH, 2010.[2] R. Finney and G. Thomas, Calculus, 2nd ed., Addison-Wesley, Reading, MA, 1994.
[3] M. Dugopolski, College Algebra, Addison-Wesley/Pearson, Boston, MA, 2003.[4] D. Kay, College Geometry, HarperCollins College Publishers, New York, 1994.[5] R. Larson, L. Boswell, and L. Stiff, Geometry, McDougal Littell, Evanston, IL, 2004.[6] S.L. Loney, Coordinate Geometry, Macmillan, London, 1962.
[7] R. Maeder, Programming in Mathematica, Addison-Wesley, Redwood City, CA, 1991.[8] S. Wolfram, Mathematica� Book, 5th ed., Cambridge University Press, Cambridge, 2003.[9] S.L. Loney, Plane Trigonometry, Part-I, A.L.T.B.S. Publishers, New Delhi, India, 1996.
[10] E. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, Boca
