Instructions PART II: Function Tracer and Graphing Calculator
Content PART II:
- Non-differentiable functions
min Minimum of several values, e.g. min(1#x#x^(1/3)) as minimum of 1, x and third root of x.
max Maximum of several values, e.g. max(abs(x)#x*x) as maximum of the absolute value of x and x2.
% Modulo division, whole-numbered remainder, e.g. 10%x. Example 1: enter 10%3 in the field Function of the Scientific Calculator. You will get 1 as a result because the integer division of 10 by 3 is 3 with a remainder of 1. Example 2: plot the graph of 10 modulo x by entering 10%x in the field First Graph of the Graph Plotter. You will get a stair graph. Be carefull, if you enter decimal numbers in this formula, those will be rounded down.
fmod Modulo division, floating point remainder, e.g. fmod(x#1) displays only the position after the decimal point of the input value. Example 1: if you enter fmod(3.6#1.4) in the field Function of the Scientific Calculator, you will get 0.8 as a result because the division of 3.6 by 1.4 is 2 with a remainder of 0.8. Example 2: enter fmod(x#2.4) in the field First Graph of the Graph Plotter to plot the graph of the remainder of the division of the variable x by 2.4.
R Round, e.g. R(x#2) rounds two decimal places, R(x) rounds to an integer. Example 1: R(3,5681#2) rounds to the second decimal and the result is 3.57.
R0 Floor (rounding down), e.g. R0(x)
R1 Ceil (rounding up), e.g. R1(x)
dist Distance function, e.g. dist(x) gives the distance to the nearest integer.
prime Prime number function. A prime number is a number that can be divided only by 1 and by itself. e.g. prime(x) This returns the next lower prime number (or x itself, if prime) for all x≥2 and x≤100000. Non-integers are rounded.
prime1 Prime number detecting function, e.g. prime1(x) Displays a number only if prime, else 0.
prime2 Distinct prime factor counting function, e.g. prime2(x) returns the amount of different prime factors for an integer. Example 1: prime2(15) returns 2 as a result because 15 can be divided only by two prime numbers, 3 and 5.
prime3 Prime factor counting function, e.g. prime3(x) returns the amount of prime factors for an integer, including multiples. E.g. prime2(4) = 1 because 4 can be divided by 2, whereas prime3(4) = 2 because 4 can be divided by 2 and by 4 (4=2x2). If prime3(x) = 1, then x is prime.
div Divisor function, e.g. div(x) returns the number of divisors of an integer. Non-integers are rounded.
dig Digit sum, e.g. dig(x) returns the digital sum of an integer. Non-integers are rounded, - is ignored. Example 1: dig(289) returns 19 as a result because 2+8+9=19.
dig2 Iterated (one-digit) digit sum, e.g. dig2(x) returns the iterated digital sum of an integer. Example 1: dig2(59) returns 5 as a result because 5+9=14 and 1+4=5.
H Heaviside step function, e.g. H(x) returns 0 if x≤0, else 1.
Hm Multivariate Heaviside step function, e.g. We have two function of x, x2-1 and sin(x), then Hm(x*x-1#sin(x)) will return 0 if at least one value ≤0, else 1. We advise you to trace the following three function on the same picture to understand better: Hm(x*x-1#sin(x)) on First Graph, x2-1 on Second Graph and sin(x) on the Third Graph
sig Signum function (sign function), this function identifies the sign of a number. Example 1: sig(-58) returns -1 as a result because de sign of -58 is negative. Example 2: the graph of sig(x) is a step function whose value is -1 for x≤0 and +1 if x is positive.
gcf Greatest common factor (or greatest common divisor, gcd), e.g. gcf(8#x) returns the greatest common factor between two integers. Non-integers are rounded. Example 2: gcf(18#27) returns 9 as a result.
lcm Least common multiple, e.g. lcm(8#x) returns the least common multiple between two integers. Non-integers are rounded. Example 2: lcm(3#2) returns 6 as a result.
toti Euler's totient function, e.g. toti(x) counts all positive integers less than x that are comprime to x. Non-integers are rounded. Example 1: toti(15) returns 7 because there are 7 prime numbers smaller than 15 (1, 2, 3, 5, 7, 11 et 13).
odd Find odd numbers, e.g. odd(x) returns numbers only when odd. Non-integers are rounded. Trace the graph of odd(x) and you will see that only odd numbers are displayed.
even Find even numbers, e.g. even(x) returns numbers only when even. Non-integers are rounded. Plot the graph of even(x) and you will see that only even numbers are displayed.
bin Binomial coefficient, Example 1: bin(5#3) those two values are n = 5 et k = 3. The result is 10. The formula used by bin is n!/[k!(n-k)!]. Example 2: trace the graph of bin(5#x). Non-integers are rounded.
tri Triangle curve, e.g. tri(2#3#x) The first value is the period, the second is the amplitude. This formula used with the graph plotter will trace triangles with a base of 2 (width) and a height of 3.
rect Rectangle curve, e.g. rect(1#-1#2#x) The first value is the upper limit (on y-axis), the second is the lower (x-axis) and the third is the period.
saw Sawtooth wave, e.g. saw(2#1#x) The first value is the period, the second is the amplitude.
saw2 Reverse sawtooth wave, e.g. saw2(2#1#x) The first value is the period, the second is the amplitude.
ramp Ramp function, e.g. ramp(1#2#1#x) The first value is the start x-value, the second is the end x-value and the third is the height of the ramp.
ramp2 Reverse ramp function, e.g. ramp2(1#2#1#x) The first value is the start x-value, the second is the end x-value and the third is the height.
trap Trapezium (trapezoid) function, e.g. trap(-4#-1#3#2#3#x) The first value is the start x-value of the climb, the second is the end x-value of the climb, the third is the trapezoid's height, the fourth is the start x-value of the descent and the fifth is the end x-value of the descent.
poly Polygon or chart line or Chart-curve, this function is useful, for instance, to trace the graph of the measured temperature by a meteorological station as a function of the time, with connections between experimental dots. Example 1: poly(-4#2#-3#4#-2#1#-1#0#0#3#1#2#2#-1#3#3#4#1#x) gives a chart, respectively a half polygon. Here, (-4,2) is connected to (-3,4), this to (-2,1) and so on. The first value of each pair is the x-value, the second one is the y-value. The x-values must increase with each step. To get a full polygon, enter a second term (in the field Second Graph) with the same start and end points, like poly(-4#2#0.5#-4#4#1#x)
rand Integer random number between two integers, e.g. rand(0#3) returns 0, 1, 2 or 3 in the Scientific Calculator (Mersenne twister is used for generating - Developed by Makoto Matsumoto and Takuji Nishimura in 1997).
rand2 Random number between two numbers with decimal places (maximal 9), e.g. Enter rand2(0#10#3) in the Scientific Graphing Calculator and this will return a number with three decimal places between 0 and 10 (Mersenne twister is used for generating - Developed by Makoto Matsumoto and Takuji Nishimura in 1997).
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- Probability functions and statistics
norm Normal or Gaussian distribution (bell curve), e.g. norm(0#1#x) for the uniform distribution. The first value is the expected value, the second is the standard deviation.
phi Φ, Cumulative Gaussian distribution function, e.g. phi(0#1#x) This is an approximation based on the displayed interval. It delivers reasonable values, if the normal distribution in the chosen interval starts at very low values near 0. A common display of both functions is advisable.
chi2 Chi-square distribution, e.g. chi2(3#x) The first value is the number of the degrees of freedom.
ichi2 Inverse-chi-square distribution, e.g. ichi2(3#x) The first value is the number of the degrees of freedom.
sichi2 Scale-inverse-chi-square distribution, e.g. sichi2(3#1#x) The first value is the number of the degrees of freedom, the second is the scale parameter, both must be >0.
chi Chi distribution, e.g. chi(3#x) The first value is the number of the degrees of freedom.
stud Student's t-distribution, e.g. stud(2#x) The first value is the number of the degrees of freedom.
F F-distribution (Fisher-Snedecor), e.g. F(5#2#x) The first two values are the numbers of the degrees of freedom.
Fz Fisher's z-distribution, e.g. Fz(5#2#x) The first two values are the numbers of the degrees of freedom.
lnorm Log-normal distribution, e.g. lnorm(0#1#x) The first value is the mean, the second is the standard deviation.
cau Cauchy distribution or Lorentz distribution, e.g. cau(0#1#x) for the standard Cauchy distribution. The first value is the location parameter, the second is the scale parameter.
lapc Laplace distribution, e.g. lapc(0#1#x) The first value is the location parameter, the second is the scale parameter. The second parameter must be >0.
logd Logistic distribution, e.g. logd(1#2#x) The first value is the location parameter, the second is the scale parameter.
hlogd Half-logistic distribution, e.g. hlogd(x)
rlng Erlang distribution (developed by Agner Krarup Erlang), e.g. rlng(5#1#x) The first value is the shape parameter, the second is the rate parameter. The first parameter must be a natural number (1, 2, 3, 4, 5,...).
pon Exponential distribution, e.g. pon(1#x) The first value is the rate parameter.
cosd Raised cosine distribution, e.g. cosd(0#1#x) The first value is the location parameter, the second is the scale parameter. cosd is defined in the interval [location-scale;location+scale].
scahd(x) Hyperbolic secant distribution, e.g. scahd(x)
kum Kumaraswamy distribution, e.g. kum(2#3#x) The first two values are the shape parameters a and b.
levy Lévy distribution, e.g. levy(1#x) The first value is the scale parameter.
rlgh Rayleigh distribution, e.g. rlgh(1#x) The first value is the scale parameter.
wb Weibull distribution, e.g. wb(2#1#x) The first value is the shape parameter, the second is the scale parameter.
wig Wigner semicircle distribution, e.g. wig(1#x) The first value gives the radius.
gammad Gamma distribution, e.g. gammad(2#3#x) The first value is the shape parameter, the second is the scale parameter.
igammad Inverse-gamma distribution, e.g. igammad(2#1#x) The first value is the shape parameter, the second is the scale parameter.
igauss Inverse Gaussian distribution, e.g. igauss(1#0.25#x) The first value is the shape parameter, the second is the scale parameter.
betad Beta distribution, e.g. betad(2#3#x) The first two values are the shape parameters, these must be ≥0. betad is defined for x in [0;1].
betap Beta prime distribution, e.g. betap(2#3#x) The first two values are the shape parameters, these must be >0.
par Pareto distribution or Pareto's law or 80 20 rule (Vilfredo Pareto, Italian economist: 1848-1923), e.g. par(2#1#x) The first value is the location parameter, the second is the shape parameter.
pear Pearson distribution - type III (Karl Pearson), e.g. pear(1#1#2#x) The first value is the location parameter, the second is the scale parameter and the third is the shape parameter.
nak Nakagami distribution, e.g. nak(4#1#x) The first value is the shape parameter, the second is the spread parameter.
shg Shifted Gompertz distribution, e.g. shg(0.5#1#x) The first value is the scale parameter, the second is the shape parameter, both must be >0.
brw Relativistic Breit-Wigner distribution, e.g. brw(1#2#x) The first value is the mass of the resonance, the second is the resonance's width and the third is the energy.
gen Generalized extreme value distribution, e.g. gen(0#1#0.2#x) The first value is the location parameter, the second is the scale parameter and the third is the shape parameter.
Ft Fisher-Tippett distribution, e.g. Ft(1#2#x) The first value is the location parameter, the second is the scale parameter. The second parameter must be >0.
rossi Rossi distribution, or mixed extreme value distribution, e.g. rossi(0#3#1#4#x) The first four values are c1, c2, d1 and d2.
gum1 Gumbel distribution type 1, e.g. gum1(2#1#x) The first two values are the parameters a and b.
gum2 Gumbel distribution type 2, e.g. gum2(2#1#x) The first two values are the parameters a and b.
trid Triangular distribution, e.g. trid(1#2#4#x) The first value is the lower limit, the second is the most probable and the third is the upper limit.
- Discrete distributions
bind Binomial distribution, e.g. bind(5#0.4#x) The first value is the number of trials, the second is the success probability.
nbin Negative binomial distribution, e.g. nbin(3#0.4#x) The first value is a paremater >0, the second is a probability.
poi Poisson distribution, e.g. poi(3#x) The first value is λ, the second is the expected value.
skel Skellam distribution, e.g. skel(1#2#x) The first two values are the means of two different Poisson distributions.
gk Gauss-Kuzmin distribution, e.g. gk(x)
geo Geometric distribution (variant A), e.g. geo(0.8#x) The first value is a probability.
hgeo Hypergeometric distribution, e.g. hgeo(8#3#2#x) The first value is the total number of objects, the second is the total number of defective objects, the third is is the number of sample objects and the fourth the number of defective objects in the sample.
yule Yule-Simon distribution, e.g. yule(2#x) The first value is the shape parameter.
logs Logarithmic series distribution, e.g. logs(0.1#x) The first value is a probability.
zipf Zipf or zeta distribution, e.g. zipf(3#x) The first value is a parameter >0.
zm Zipf-Mandelbrot law or Pareto-Zipf law, e.g. zm(100#1#2#x) The first three values are N, q and s. Maximum for N is 100.
uni Uniform distribution, e.g. uni(1#2#x) The first value is the lower limit, the second is the upper limit.
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- Special functions
traj Trajectory parabola, path of a thrown object, e.g. traj(45#20#9.81#x) The first value is the angle, the second is the speed (e.g. in meters per second). The third value is the gravitational acceleration (e.g. in m/s²), the normal value on earth for this is g = 9.81 m/s². The value of g on the moon is g = 1.63 m/s² (which is one sixth of the value of g on earth). The axes scale in this example is meters. Air resistance is ignored.
scir Semicircle curve, e.g. scir(x#1) for a semicircle with the radius 1. The formula is sqr(r*r-x*x), r gives the radius.
ell Semielliptic curve, e.g. ell(2#1#x) for a semiellipse with the horizontal radius 2 and the vertical radius 1. The formula is sqr((1-x*x/(a*a))*b*b).
ell2 Semi-superellipse or semi-hyperellipse, e.g. ell2(2#3#4#x) for a semiellipse with the horizontal radius 2, the vertical radius 3 and n=4.
lmn Lemniscate of Bernoulli (Jacques Bernoulli, Swiss mathematician, 1654-1705. This is Daniel Bernoulli's uncle. Daniel Bernoulli is famous for his work in the basic properties of fluid flow, pressure, density and velocity, and gave the Bernoulli principle), e.g. enter lmn(1#x) in the field First Graph, this returns a half lemniscate. For the other half, use -lmn(1#x) in the field Second Graph of the Graphing Calculator.
lmn2 Lemniscate of Gerono, e.g. lmn2(x) This returns a half lemniscate. For the other half, use -lmn2(x)
lmn3 Lemniscate of Booth, e.g. lmn3(1#x) This returns a half lemniscate. For the other half, use -lmn3(1#x)
pyth Pythagorean theorem (USA) or Pythagoras' theorem (UK), e.g. pyth(4#3) computes de value of the hypotenuse c of a right triangle with its sides a=4 and b=3, try it with the Scientific Calculator. Example 1: enter pyth(4#x) in the Graphing Calculator to trace de curve that gives the value of the hypotenuse of a right triangle with its sides a=4 and b=x. The formula is c=sqr(a*a+b*b).
thr Rule of three. Example 1: if you have to pay 0,85 euros to buy 5 apples, how much will you have to pay to buy 3 apples ? Look at the proportionality we are observing here: 5 apples/0,85 eur = 3 apples/x eur. We can say that x = price of 3 apples = (3 apples * 0,85eur) / (5 apples). The formula thr(5#3#0,85) works out this calculation: 3*0,85/5, which gives as a result the price of 3 apples knowing that 5 apples cost 0,85 euros. The answer is 0,51 euros. Example 2: thr(5#x#0,85) enables you to trace the graph that gives the price y of x apples knowing that 5 apples cost 0,85 euros.
fib Fibonacci numbers (the Fibonacci mathematical series was discovered in 1209 by Leonardo of Pisa, known as Fibonacci). The Fibonacci numbers describe the growth of a rabbit population as follow: "Initially we have only one pair of rabbits, then how many pairs will we have in 12 months if each pair produces a new baby pair every months ? We assume that each baby rabbit starts producing offsprings when he is 2 months old.". The rabbits never die, thus we have here a strictly increasing sequence. Initially, at month 0, we have no rabbits. At month 1 there is one pair of rabbits. At month 2 we still have one pair of rabbits. At month 3, there are 2 pairs because the first pair just produced a new pair. And so on. To know the number of pairs at month 9, enter fib(9) in the Scientific Calculator. You will get 233 as a result. At month 9, there are 233 pairs of rabbits.
If you enter fib(x) in the Function Tracer, you will draw a curve with the month number on the x-axis and the rabbit number on the y-axis.
dc Exponential Decay. Example: dc(5#1#x) The first value is the initial value, for instance the initial number of Uranium 238 nucleus in a sample. The second value is the decay constant. We are considering the radioactive decay of the isotope Uranium 238. The formula used is N=N0e-λx where x is the time and λ is the decay constant. N is the number of nucleus of Uranium 238 that are still intact (those who have not decayed yet), and N0 is the initial number of Uranium 238. At the moment x=0, thus at the initial instant, N=N0. Then in dc(5#1#x) N0 = 5 and λ = 1.
erf Gaussian error function, e.g. erf(x) For the computation its Taylor series is used.
HY4 Hyper 4 or Hyper-4 or Hyper4, also known as tetration or super-exponentiation, e.g. HY4(x#3) for x to the power of (x to the power of x) xxx. Here the maximum value can be excessed very quickly!
lambda Lambda function, e.g. lambda(x#3) for x to the power of (x to the power of (3-1)) xx(3-1).
sgm Sigmoid function, e.g. sgm(x) for 1/(1+e^(-x)).
gom Gompertz curve, e.g. gom(2#-5#-3#x) The first value is the upper asymptote, the second is the parameter b and the third is the growth rate. Second and third value must be negative.
zeta Riemann zeta function for values >1, e.g. zeta(x)
eta Dirichlet eta function, e.g. eta(x)
stir Stirling's approximation for large factorials, e.g. stir(x) The formula is (2*pi*x)^(1/2)*(x/e)^x.
gamma Gamma function (Euler and Weierstrass definition, approximation), e.g. gamma(x) as extension of the factorial function and for many statistical distributions. It gives a good approximation for the factorial of a large number. For numbers greater than 10, we use the Stirling function, stir(x-1), which saves computing time.
beta Euler beta function, e.g. beta(2#x)
digamma Digamma function, e.g. digamma(x) for D(gamma(x))/gamma(x). D() is a function that computes the derivative.
omega Lambert-W function or Omega function or product log (approximation), e.g. omega(x)
theta Ramanujan theta function, e.g. theta(x#0.3) The two values are a and b. abs(a*b) must be <1.
bump Bump function psi, ψ, e.g. bump(x) for exp(-1/(1-x*x)) between -1 and 1, else 0.
srp Serpentine curve, e.g. srp(2#1#x) The formula is a*a*x/(x*x+a*b). The first two values are a and b.
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- Programmable functions
bool characteristic boolean function, e.g. bool(1/x) Returns nothing, if the input value is not defined, 0, if 0, else 1.
bool0 defined boolean function, e.g. bool0(x) Returns 0, if the input value is 0 or not defined, else 1.
bool1 undefined boolean function, e.g. bool1(prime1(x)) Returns nothing, if the input value is 0 or not defined, else 1.
con Condition function, e.g. con(0#sin(x)#1) The first value is the lower limit, the third is the upper limit. If the second value is between these two, the result is 1, else 0.
rcon Reverse condition function, e.g. rcon(0#sin(x)#1) The first value is the lower limit, the third is the upper limit. If the second value is between these two, the result is 0, else 1.
wcon Weighted condition function, e.g. wcon(0#sin(x)#1) Only returns the second value, if this lies between the first and the third value.
rwcon Reverse weighted condition function, e.g. rwcon(0#sin(x)#1) Only returns the second value, if this doesn't lie between the first and the third value.
&& (AND) can be simulated with the minimum function, e.g. min{ con[0#sin(x)#1] # con[0#cos(x)#1] }, if sin(x) AND cos(x) (AND means both at the same time) are within the interval [0;1] then the result displayed will be 1. Else 0.
|| (OR) can be simulated with the maximum function, e.g. max{ con[0#sin(x)#1] # con[0#cos(x)#1] }, if sin(x) OR cos(x) (OR means one of them or both at the same time) are within the interval [0;1] then the result displayed will be 1. Else 0
⊕ (XOR) can be simulated with the maximum minus the minimum function, e.g.
max{ con[0#sin(x)#1] # con[0#cos(x)#1] } - min{ con[0#sin(x)#1] # con[0#cos(x)#1] }, the result will be 1 if the sin(x) OR cos(x) BUT NOT both at the same time belong to the interval [0;1]. Else the result is 0.
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- Iterations (iterative functions)
y Previous function value, e.g. for y(0)+0.01 is 0 the initial value for y, the next value is the last result of the input value x and so on.
y2 Pre-previous function value, e.g. y2(1)+0.001
step Number of the iteration steps done (always 500), divided by the parameter value, e.g. step(100) counts up to five (500/100=5).
mean Arithmetic mean, e.g. mean(sin(x)) gives the arithmetic mean of the y-values returned to the so far reached x-values.
man Mandelbrot function, e.g. man(0#-1.9) for y(0)*y(0)-1.9.
Attention: derivative and integral with the iteration don't lead to very reasonable results. As well a logarithmic scale won't work here.
- Fractals
rsf Random singular function (a kind of devil's staircase), e.g. rsf(0#2) for y(a)+0.008*rand(0#1)*rand(0#1)*(b-a), from a (first value) to b (second value). The first value is the start point on the y-axis, the second is the average end value.
wf Weierstrass function, e.g. wf(x#0.5#17#10) The second value is a parameter between 0 and 1, the third value is a positive, odd integer. The second multiplied with the third must be larger than 1+3/2*pi. The fourth value is the number of steps done. In theory this is infinite, but here the maximum is 100.
blanc Blancmange curve, e.g. blanc(x#10) The second value is the number of steps done, maximum is 1000.
tak Takagi-Landsberg curve, e.g. tak(x#0.7#10) The second value is a parameter, which should be between 0 and 1. The third is the number of steps done, maximum is 1000.
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Differential and integral equations
The Function Tracer and its Scientific Calculator enable you to compute and to trace the derivative and the integral of a function. The first derivative and the integral of a function must be written as follow:
D Derivative. Example: D(x*x) or D(x^2). The following expression is not allowed: D(D(...))
S Integral. Example: S(x*x) or S(x^2). The following expression is not allowed: S(S(...))
The second derivative can be drawed with the Free Graphing Calculator by checking the box Derivative and using at the same time the function D().
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