CoCalc -- 2023-05-21-file-2.ipynb

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License: GPL3
ubuntu2204

Kernel: SageMath 9.6

$\hspace{20pt}$ Experiment Number 1

AIM: To Use SageMath as Advanced Calculator

Name: Snehal G Rakas

Roll no: 61

Section: E

Date: 8/05/2023

Basic numerical Computations

In [0]:

In [5]:

454672-258252

Out[5]:

196420

In [4]:

56455+678978

Out[4]:

735433

In [6]:

565*676

Out[6]:

381940

759**11 Returns 11th power of 759

In [7]:

759**11

Out[7]:

48157012271803961449924277680359

In [8]:

81^(1/4)

Out[8]:

3

In [9]:

(79/12)

Out[9]:

79/12

In [10]:

(79/12.0)

Out[10]:

6.58333333333333

In [11]:

(79/12).n()

Out[11]:

6.58333333333333

In [12]:

79//12

Out[12]:

6

In [13]:

(79%12)

Out[13]:

7

In [18]:

(sin(81)).n(digits=20)

Out[18]:

-0.62988799427445387857

In [15]:

(sin(81.0))

Out[15]:

-0.629887994274454

In [16]:

cos(-5.3)

Out[16]:

0.554374336179161

In [17]:

tan(1/2.0)

Out[17]:

0.546302489843790

In [19]:

asin(-0.2)

Out[19]:

-0.201357920790331

In [20]:

exp(1.0)

Out[20]:

2.71828182845905

In [21]:

pi.n(digits=50)

Out[21]:

3.1415926535897932384626433832795028841971693993751

In [22]:

F=factorial(576)

In [23]:

print(F)

Out[23]:

4256894463620625471399338897976462846890230941821769298896221080707434902055990415666782999834652467105561645005135247510594550578570361904546802042930914680310390642433197638242918512304047044258861660945550782976855100391711791398500713833285909054634688620911207051946259440915349012504839365184529533518755428177370879723111743622910445124706269593256104219501887578284982300380039220456095383116526730085743873032838237773072593976177499478159793530051818413365412709223554005276531798500164384916741118247588065706396358679530916434991681240368497544405247600864215548296506680601289148053255684716608821820352021050028244591277208996891498042521087458841329716703518091675689367527272995883290329676158091022259824243125389761208343649293452081505395610821671083260346432722100756628487076760917971350441734334132117633025186198636599163163040922447934465322909545018661306558666998323928906394462759431335113344451854455546674490813919856400793256248896167909254939021724785145027873918136392996266973257470068635273673116279231080146181208818573897317247122622658823827561947836713192608335310969980444617575725662453866484897648923060190912757314122770275982764251413066524055645779981762560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

In [38]:

F.ndigits()

Out[38]:

1342

In [41]:

w=F.digits()

In [42]:

w[0]

Out[42]:

0

In [43]:

w[1]

Out[43]:

0

Defining Variables

In [44]:

a=2345
b=4567
print(a,b)

Out[44]:

2345 4567

first_number = 4565748
second_number = 45366673
p=first_number*second_number

In [46]:

b.is_prime()

Out[46]:

True

In [48]:

b.factor()

Out[48]:

4567

In [49]:

gcd(a,b)

Out[49]:

1

In [51]:

lcm(a,b)

Out[51]:

10709615

In [52]:

list(primes(50,100))

Out[52]:

[53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

In [50]:

nth_prime(100)

Out[50]:

541

Defining Functions

In [54]:

f(x)=sin(x^2)*exp(-x)+3*x+1

In [55]:

f(2.1)

Out[55]:

7.18309969277139

In [56]:

f.diff()

Out[56]:

x |--> 2*x*cos(x^2)*e^(-x) - e^(-x)*sin(x^2) + 3

In [57]:

f.diff(3)

Out[57]:

x |--> -8*x^3*cos(x^2)*e^(-x) + 12*x^2*e^(-x)*sin(x^2) + 6*x*cos(x^2)*e^(-x) - 12*x*e^(-x)*sin(x^2) - 6*cos(x^2)*e^(-x) - e^(-x)*sin(x^2)

In [58]:

f.integral(x)

Out[58]:

x |--> 3/2*x^2 - 1/16*sqrt(pi)*((-(I + 1)*sqrt(2)*cos(1/4) + (I - 1)*sqrt(2)*sin(1/4))*erf(-1/2*(-1)^(3/4)*(2*I*x + 1)) + (-(I + 1)*sqrt(2)*cos(1/4) + (I - 1)*sqrt(2)*sin(1/4))*erf(-(1/4*I - 1/4)*sqrt(2)*(2*I*x + 1)) + ((I - 1)*sqrt(2)*cos(1/4) - (I + 1)*sqrt(2)*sin(1/4))*erf(-(1/4*I + 1/4)*sqrt(2)*(2*I*x - 1)) + (-(I - 1)*sqrt(2)*cos(1/4) + (I + 1)*sqrt(2)*sin(1/4))*erf(1/2*(2*I*x - 1)/sqrt(-I))) + x

In [61]:

var('x,y')
h(x,y)=sin(x^2-y^2)
h(1.0,2.0)

Out[61]:

-0.141120008059867

In [65]:

var('a,b,c')
show((solve(a*x^2+b*x+c==0,x)))

Out[65]:

$\displaystyle \left[x = -\frac{b + \sqrt{b^{2} - 4 \, a c}}{2 \, a}, x = -\frac{b - \sqrt{b^{2} - 4 \, a c}}{2 \, a}\right]$

In [1]:

var('x1,x2')
solve([3*x1-2*x2==7,2*x1+3*x2==11],x1,x2)

Out[1]:

[[x1 == (43/13), x2 == (19/13)]]

In [3]:

def quad(a,b,c):
    x1=(-b+sqrt(b**2-4*a*c))/(2*a)
    x2=(-b-sqrt(b**2-4*a*c))/(2*a)
    return(x1,x2)

In [4]:

a,b,c=2,3,-2
x1,x2=quad(a,b,c)
show(f'The roots of the quadratic are {x1} and {x2}')

Out[4]:

In [6]:

show((x1.n())),(x2).n()

Out[6]:

$\displaystyle 0.500000000000000$

(None, -2.00000000000000)

In [7]:

def Heron(a,b,c):
    s=(a+b+c)/2.0
    A=sqrt(s*(s-a)*(s-b)*(s-c))
    return A

In [8]:

a,b,c=3,7,9
Area=Heron(a,b,c)
print(f'The area of Triangle is {Area.n()}')

Out[8]:

The area of Triangle is 8.78564169540279

In [9]:

#Find the sum of first 50 numbers
k=0
for i in range (1,51):
    k=k+i
k

Out[9]:

1275

In [13]:

#1.Find the roots of:
var('x')
solve(x^3-2*x^2-5*x+6==0,x)

Out[13]:

[x == 3, x == -2, x == 1]

In [17]:


#2.Solve the system of non-linear equations
var('x,y')
solve([(x^2-y^2==4),(y==(x^2)-2)],x,y)

Out[17]:

[[x == -sqrt(1/2*I*sqrt(7) + 5/2), y == 1/2*I*sqrt(7) + 1/2], [x == sqrt(1/2*I*sqrt(7) + 5/2), y == 1/2*I*sqrt(7) + 1/2], [x == -sqrt(-1/2*I*sqrt(7) + 5/2), y == -1/2*I*sqrt(7) + 1/2], [x == sqrt(-1/2*I*sqrt(7) + 5/2), y == -1/2*I*sqrt(7) + 1/2]]

In [19]:

var('x,y')
eq1 = x^2 - y^2 == 4
eq2 = y == x^2 - 2
sol = solve([eq1, eq2], x, y)
sol

Out[19]:

[[x == -sqrt(1/2*I*sqrt(7) + 5/2), y == 1/2*I*sqrt(7) + 1/2], [x == sqrt(1/2*I*sqrt(7) + 5/2), y == 1/2*I*sqrt(7) + 1/2], [x == -sqrt(-1/2*I*sqrt(7) + 5/2), y == -1/2*I*sqrt(7) + 1/2], [x == sqrt(-1/2*I*sqrt(7) + 5/2), y == -1/2*I*sqrt(7) + 1/2]]

In [25]:

F=factorial(1275)

Out[25]:

$\displaystyle 1410188200745141400252996925676992827609427883860681102196565498836336759274700571301062342565763667730574384945641799898255183843797277503390234956935070394479008695945171075361751490266255689101148400638952696890761606725201668092011058223584197411362725808136521522042355331881040677201517524336650583177256411366707450357076114921552137231355444540586815705134573123811035652300829258848866779288445345880939714666466690931600352950947986039996432744118754275201288974846584827623137897792122234828866268708808373651276334848174069438274783771825876627128631153710955178645802336438821598373242525172486367974756812461614902930111917189057714749908028994429977492262359339566774968421933826661845573051906637774350159842777201017432451096856729583848901554363821690871963837317325736243872364148002548730698225622458872512196920911448958125730334065147981692621576246557948729307409911196659268977449824737887320133166571859768610902901473947123457650502571110106359050311523447786812407139365812881583625481044458877349531018685432954716465488394115659122958734526255536933591914359738150594232473711291251674238772555271930793815825985192385373959108468804902794091333308997906654387218990210896432316359969692946647542629459647405723220544167040523889230769503152154177529658591214286091920818334885793018545403374623359640495796839709709384014013012687979103444399633192318216277501509029976296018451103357572826768431048432485483904156377949766394422704783101532471054714207760435511464546800957250453788222377134546592645406967452675959535949763038875966331731971129199721214052529127807792669916640919732183579343359239597918703189536631735584104688709458017493330567389003035699210216919159335873580170482127718603055097898778534952746567262220574405875417584645973964359858336242322940751239914457356958486503721361054938154425602356741007273505642819375318812777259929805604278204113178492040885487253497253599744730743581176164725418840085566636342828519091247494824276253539056065078460524088183488674664479782491935751641451986001983862705359408369452209318497499455387591386320304376709778016052197826525984538725046076767350582105555907369126126599357472133535283786257752742361071712371083162246333227880188819486706970859272715412889313443305257503562357007473653496252819471178088373765780168140628517776493336902711979149445247736765162103192806316969933073702060508523674642586433795221440274867588588796999943822914833791154288328704000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$

In [6]:

#4.Find the fators of sum of digits of 1275!
n = factorial(1275)
t_sum = sum(int(d) for d in str(n))
t_sum
factors = []
for i in range(1, t_sum+1):
    if t_sum % i == 0:
        factors.append(i)
factors

Out[6]:

In [4]:

#3.Find the number digits in 1050! and how many zeros are there in 1050!
F = factorial(1050)
digits = F.ndigits()
zeros = str(F).count("0")
print("Number of digits:", digits)
print("Number of zeros:", zeros)

Out[4]:

Number of digits: 2719
Number of zeros: 472

In [11]:


#5.Evaluate the \int\frac{\cos(x)}{\sqrt{\sin(x)+1}}dx
a=cos(x)
b=sqrt(sin(x)+Integer(1))
f(x)=a/b
f.integral(x)

Out[11]:

x |--> 2*sqrt(sin(x) + 1)

In [15]:

#6.Find the fifth order derivative
f=ln(x)+3*x^3*cos(2*x)
f.diff(5)

Out[15]:

-96*x^3*sin(2*x) + 720*x^2*cos(2*x) + 1440*x*sin(2*x) + 24/x^5 - 720*cos(2*x)

In [23]:

def invest():
    amt=int(input("Enter the amount invested:"))
    yr=int(input("Enter the years of investment"))
    interest_amt=(amt*yr*4)/100
    return(interest_amt)

In [24]:

invest()

Out[24]:

Enter the amount invested:

Enter the years of investment

1000

In [0]:

$\hspace{20pt}$ Experiment Number 1

AIM: To Use SageMath as Advanced Calculator

Name: Snehal G Rakas

Roll no: 61

Section: E

Date: 8/05/2023

Basic numerical Computations

Defining Variables

Defining Functions

What You Learn:

In this practical I learnt to use the basic mathematical functions of the sage math and utilize the advanced calculator for various functions.

Product

Resources

Company

\hspace{20pt}Experiment Number 1

AIM: To Use SageMath as Advanced Calculator

Name: Snehal G Rakas

Roll no: 61

Section: E

Date: 8/05/2023

Basic numerical Computations

Defining Variables

Defining Functions

What You Learn:

In this practical I learnt to use the basic mathematical functions of the sage math and utilize the advanced calculator for various functions.

$\hspace{20pt}$ Experiment Number 1