1. First Finite Divided Difference Calculator Using MIT
App Inventor 2
Gomez, Camille Bianca C.
Gokongwei College of Engineering
BS Electronics and Communications Engineering
Biñan, Laguna, Philippines
camille_gomez@dlsu.edu.ph
Abstract—This project is a user-friendly mobile application
of solving the first derivative of a polynomial equation using the
three methods of solving the first derivative of a polynomial
equation: Forward Differentiation, Backward Differentiation,
and Centered Differentiation. The software that was used for
this mobile application is MIT App Inventor 2. The concept of
Algorithm on Discrete Mathematics was applied in this project.
This paper includes the pseudocodes for getting the true value
of a given polynomial equation, true value derivative of a given
polynomial equation and the first finite divided difference.
Keywords—derivative; first finite divided difference; forward
differentiation; backward differentiation; centered
differentiation; MIT App Inventor; mobile application; algorithm;
pseudocode
I. INTRODUCTION
MIT App inventor is a tool created to help aspiring app
developers to build their own application right in the web
browser. It is the beginner’s introduction to programming and
making a mobile app that consists of a drag-and-drop building
blocks [1]. All things needed to understand how the software
works is included in their resources.
The concept of algorithm from discrete mathematics and
numerical differentiation from numerical methods will be
implemented using the app inventor tool.
Algorithm is a finite set of clear instruction or methods for
performing a computation or to solve a problem. It can be
implemented using a programming software (e.g. MATLAB,
DevC) however the coding scheme for a certain programming
software needs a correct command for it to be implemented
and for this reason, a Pseudocode is used [2]. Pseudocode is
considered to be the convention of any programming language
that can be understood easily by humans to aid them on
generating a program or code. The steps or instructions used
to implement an algorithm are well described operations or
statements in a pseudocode for it to be a non-specific
programming language [2].
Finite Divided Difference is the approximation derived
from Taylor series and is used to approximate certain
problems numerically. Finite divided difference is commonly
used in a problem with boundaries [3].
II. THEORETICAL CONSIDERATION
A. Forward, Backward, and Centered Difference
Finite difference is a topic from numerical method and is
used to solve differential equation by using approximation
and represented as:
𝑓′(𝑥𝑖) =
𝑓(𝑥 𝑖+1)−𝑓(𝑥 𝑖)
𝑥 𝑖+1−𝑥 𝑖
+ 𝑂(𝑥𝑖+1 − 𝑥𝑖) (1)
or
𝑓(𝑥𝑖) =
∆𝑓𝑖
ℎ
+ 𝑂(ℎ) (2)
Where ∆𝑓𝑖 is the first forward difference and ℎ is the stepz
size. The term ∆𝑓/ℎ is known as the first finite divided
difference. The forward divided difference is derived from
Taylor series, also backward difference approximation for the
first derivative Eq. (3) and the centered difference
approximation for the first derivative can also be derived Eq.
(4) [3].
𝑓′(𝑥𝑖) ≅
𝑓( 𝑥𝑖)−𝑓( 𝑥𝑖−1)
ℎ
=
∇𝑓
ℎ
(3)
𝑓′(𝑥𝑖) ≅
𝑓( 𝑥𝑖+1)−𝑓( 𝑥𝑖−1)
2ℎ
− 𝑂(ℎ2
) (4)
For a better understanding of the equations for forward,
backward and centered difference, a graphical representation
was shown in Fig. 1.

2. Fig.1 A graphical depiction of forward (a), backward (b) and centered (c)
finite-divided difference approximations of the first derivative [3]
For the forward difference, we compute for the slope
nearby the secant line (Approximation), the points (𝑥𝑖, 𝑓(𝑥𝑖))
and (𝑥𝑖+1, 𝑓(𝑥𝑖+1)). We choose a small number of ℎ which
represents as the small change in 𝑥 and can be either positive
or negative. Thus, it will give us the formula for forward
difference [4]:
𝑓′(𝑥) =
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
(5)
This is the same with backward difference, only differs
with the position of the two points (𝑥𝑖−1, 𝑓(𝑥𝑖−1)) and
(𝑥𝑖, 𝑓(𝑥𝑖)), giving us the formula for backward difference
[4]:
𝑓′(𝑥) =
𝑓(𝑥)−𝑓(𝑥−ℎ)
ℎ
(6)
Another formula to compute the slope nearby the secant
line through the points (𝑥𝑖−1, 𝑓(𝑥𝑖−1)) and (𝑥𝑖+1, 𝑓(𝑥𝑖+1)),
giving us the formula for centered difference [4]:
𝑓′(𝑥) =
𝑓(𝑥+ℎ)−𝑓(𝑥−ℎ)
2ℎ
(7)
It is more accurate to the true derivative value, if the
given ℎ is much closer to 0 because as ℎ approaches to 0,
the slope of the secant line will approaches the tangent line
which is the true derivative line shown in Fig. 1.
B. Approximation and Round-off Errors
Numerical methods are the approximations for analytical
solutions. Solutions that are derived manually will have some
deviation thus producing errors [3].
Round-off error is the most known error when using
numerical method. It happens when only a certain amount of
significant digits are required to be presented. Learning how
to determine or quantify error is a must in using numerical
methods [3].
The following equations governs the quantification of an
error:
𝑇𝑟𝑢𝑒 𝑉𝑎𝑙𝑢𝑒 = 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 + 𝑒𝑟𝑟𝑜𝑟 (8)
𝐸𝑡 = 𝑇𝑟𝑢𝑒 𝑉𝑎𝑙𝑢𝑒 − 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 (9)
Where Et, true error, is the difference between the true
value and the approximated value. Resulting for a relative
true error:
Ɛ 𝑡 =
𝐸 𝑡
𝑇𝑟𝑢𝑒 𝑉𝑎𝑙𝑢𝑒
𝑥 100% (10)
Where Ɛt is the ratio between the true error with respect
to the true value.

3. III. DESIGN CONSIDERATIONS
The design of this project is to calculate for the first
derivative of a given polynomial equation in any order. It will
also determine the percentage error of the forward, backward,
and centered difference approximations of the first derivative
to the true value of the first derivative using the concept of
absolute and relative error.
This mobile app is user friendly in a way that it tells the
user how the calculator works where it is defined what the
calculator is use for. It is show in Fig. 2 what the home screen
looks like.
Fig.2 Home Screen
It has three buttons where it will lead the user to its
preference, whether the details about the app, how to use the
app, and the main objective of the app. On the “Definition”
section shown in Fig. 3, it gives the description about
Numerical Differentiation including the formulas that were
used for the calculator. There is also a detail about the mobile
app itself.
Fig. 3 Definition Screen
The “How to Use” button gives a specific example of a
problem from the book [3] where it will tell the user how to
use the calculator by telling what should be the input on the
textboxes. It is shown in Fig. 4 what the screen looks like.

4. Fig. 4 Tips Screen
The last button “Calculator” will lead the user to the main
objective of the mobile app shown in Fig. 5. It will ask the
user for the polynomial, value of x, and the step size (ℎ). It is
already shown in Fig. 4 how to do that. Then the user will
click the “Solve” button and it will give the output below.
Fig. 5 First Finite Divided Difference Calculator Screen
Fig. 6 First Finite Divided Difference Calculator Process in mobile app
Fig. 6 shows the process on how the calculator works. It’s
only similar to what calculator really do, it only differs that
in just one click it will solve for the true first derivative value,
given the approximations and the relative error of each.
In MIT App Inventor, it is important to know first how the
program will work, and it is by making an algorithm. This
design consists of four algorithms: First, determining how to
get the polynomial evaluation of an equation, second,
determining the numerical value of the first derivative
polynomial, third, is by combining the two and the three
approximations of the first derivative, and lastly, is the
overall algorithm of the calculator.
Alg. 1: Getting the polynomial evaluation
PRE-CONDITION: polynomialeval (m1,m2,m3,...mn:
polynomial coefficients which corresponds to mxn-1
,
mxn-2
, mxn-3
,…,mxn-n
, value of x : integer)
fx = 0
e = (n-1) to 0, decrement 1
for N = 1 to n
y = mN * x(eN)
fx = fx + y
end
POST-CONDITION: fx : Output the value of a
polynomial evaluation
Input the coefficients of a polynomial
(any order)
Input a value of x
Input a value of step size (h)
Click "Solve!"
Output: True Value, Forward-Backward-
Centered Difference Approximations, and
Relative Error

5. Alg. 2: Getting the first derivative of a polynomial
PRE-CONDITION: polynomialderivative
(m1,m2,m3,...,mn: polynomial coefficients which
corresponds to mxn-1
, mxn-2
, mxn-3
,…,mxn-n
, value of x :
integer)
fxd = 0
e = (n-1) to 0, decrement 1
for N = 1 to n
dy = eN*mN*(x(eN)-1
)
fxd = fxd + dy
end
POST CONDITION: fxd : Output the derivative value of
a polynomial
Alg. 3: Getting the approximations of the first derivative
PRE-CONDITION: firstfinitedivideddifference
(m1,m2,m3,...,mn: polynomial coefficients which
corresponds to mxn-1
, mxn-2
, mxn-3
,…,mxn-n
, value of x
and step size (h) : integers)
fx = 0
fxd = 0
e = (n-1) to 0, decrement 1
forward = 0
backward = 0
for N = 1 to n
y = mN * x(eN)
dy = eN * mN * x(eN-1)
fwd = mN * (x-h)(eN)
bwd = mN * (x-h)(eN)
fx = fx + y
fxd = fxd + dy
forward = forward + fwd
backward = backward + bwd
end
forwarddiff = (forward-fx)/h
backwarddiff = (forward-fx)/(-h)
centerdiff = (forward-backward)/(2*h)
POST CONDITION: forwarddiff, backwarddiff
centerdiff : Output the forward, backward, and center
differences
Alg. 4: First Finite Divided Difference Calculator
PRE-CONDITION: firstfinitedivideddifferencecalc
(m1,m2,m3,...,mn: polynomial coefficients which
corresponds to mxn-1
, mxn-2
, mxn-3
,…,mxn-n
, value of x
and step size (h) : integers)
fx = 0
fxd = 0
e = (n-1) to 0, decrement 1
forward = 0
backward = 0
for N = 1 to n
y = mN * x(eN)
dy = eN * mN * x(eN-1)
fwd = mN * (x-h)(eN)
bwd = mN * (x-h)(eN)
fx = fx + y
fxd = fxd + dy
forward = forward + fwd
backward = backward + bwd
end
forwarddiff = (forward-fx)/h
backwarddiff = (forward-fx)/(-h)
centerdiff = (forward-backward)/(2*h)
forwardrelativeerror = absolute((forwarddiff-
fxd)/fxd*100);
backwardrelativeerror = absolute((backwarddiff-
fxd)/fxd*100);
centerrelativeerror = absolute((centerdiff-fxd)/fxd*100);
POST CONDITION: forwarddiff, backwarddiff
centerdiff : Output the forward, backward, and center
differences
IV. DATA AND RESULTS
Algorithm is the process on how to solve a problem with
the use of pseudocodes. In this section, it will give the results
of the four algorithms that were formulated.
It is given a polynomial equation of 2𝑥4
+ 5𝑥3
− 6𝑥2
−
𝑥 + 3 at 𝑥 = 2 and step size ℎ = 0.1. The following tables
demonstrates how the given algorithms work.
TABLE I. ALGORITHM FOR GETTING THE POLYNOMIAL EVALUATION
N mN eN x(eN)
y fx
1 2 4 16 32 32
2 5 3 8 40 72
3 -6 2 4 -24 48
4 -1 1 2 -2 46
5 3 0 1 3 49
TABLE II. ALGORITHM FOR GETTING THE FIRST DERIVATIVE OF A
POLYNOMIAL
N mN eN (x(eN)-1
) dy fxd
1 2 4 8 64 64
2 5 3 4 60 124
3 -6 2 2 -24 100
4 -1 1 1 -1 99
5 3 0 1 0 99

6. TABLE III. ALGORITHM FOR GETTING APPROXIMATIONS OF THE FIRST
DERIVATIVE
N fx fxd forward backward forwarddiff backwarddiff centerdiff
1 32 64 38.8962 26.0642 - - -
2 72 124 85.2012 60.3592 - - -
3 48 100 58.7412 38.6992 - - -
4 46 99 56.6412 36.7992 - - -
5 49 99 59.6412 39.7992 106.412 92.01 99.21
TABLE IV. ALGORITHM FOR FIRST FINITE DIVIDED DIFFERENCE
CALCULATOR
N fxd forward
diff
backward
diff
centerdi
ff
ferror berror cerror
1 64 - - - - - -
2 124 - - - - - -
3 100 - - - - - -
4 99 - - - - - -
5 99 106.4 92.01 99.21 7.48% 7.06% 0.21%
According to the calculations, we have come up to that
answers by simply following the algorithm that was
formulated. Using the Numerical Differentiation mobile app,
it also come up with the same answers (shown in Fig. 7),
verifying that the algorithms are correct.
Fig. 7 Answer to the given example using the First Finite Divided Difference
Calculator
In the given problem, the true first derivative value is 99,
and the approximations value for forward, backward, and
centered differences are 106.412, 92.01, and 99.21,
respectively. Those are only the approximated numerical
value of the first finite divided difference the reason why a
relative errors are included in the computations.
V. ANALYSIS
The results for getting the first finite divided difference
are verified that the algorithms that were formulated are
correct. As it was compared with the Numerical
Differentiation mobile app, they have the same results. With
the help of pseudocode, it became easier to code it by the use
of blocks programming that the MIT App Inventor 2 use.
VI. CONCLUSION
In conclusion, MIT app inventor has been a great tool for
helping app developers such as myself. It helps oneself
cultivate their skills and knowledge to prepare and contribute
for the growing demand for app developers. The use and
application of one’s learning in an academia such as discrete
mathematics (DISMATH) and numerical methods
(NUMMETH) is not only limited to what they have tackled.
It can be applied to many different ways such as this project.
This project, had combined the two by applying a concept
of DISMATH, in which the topic of algorithm, in another
subject course such as NUMMETH. There are a lot of other
things that can be solved or created by integrating inherited
knowledge.

7. APPENDIX A: HOME SCREEN BLOCKS
APPENDIX B: DEFINITION AND TIPS SCREEN BLOCKS

8. APPENDIX C: FIRST FINITE DIVIDED DIFFERENCE CALCULATOR SCREEN

9. ACKNOWLEDGMENT
The accomplishment of this final project in DISMATH
would not be possible without the help of some people so the
student would like to acknowledge with deep appreciation
and gratitude the following persons:
Engr. Melvin Kong Cabatuan, for teaching the concept of
algorithms, making it understandable by giving detailed
examples.
Dr. Argel Bandala, for teaching the concept of Numerical
Differentiation which gives me an idea how it will give
relation to DISMATH.
Paul Arroyo, for guidance and support throughout the
hardships in making this paper.
I also want to thank my family for understanding how
busy I am to make this project be a success.
And finally, I thank the Almighty God for giving me
strength and knowledge, guiding me throughout the days of
finishing this project.
REFERENCES
[1] "MIT App Inventor," 2012. [Online]. Available:
http://appinventor.mit.edu/explore/about-us.html. [Accessed April
2016].
[2] K. H. Rosen, Discrete Mathematics and Its Applications, New York:
McGraw-Hill, 2012.
[3] S. C. Chapra and R. P. Canale, Numerical Methods for Engineering,
New York: McGraw-Hill, 2010.
[4] R. L. Burden and J. D. Faires, "Numerical Analysis," in Numerical
Analysis, Ninth Edition, Boston, Brooks/Cole, Cengage Learning,
2011, pp. 174-182.