# Please do not post homework
# Matthew Esaia, 2/2/2025, Homework 2

# 4.1 Sequences

print(`Section 4.1`);
x := 1,2,3:
y := 4,5,6:
s := x,y:
s[4];

f:='f':		seq(f(i), i=1..6):
seq(i^2, i=1..5):

x := 'x':
x$4: 	# $ is the repetition operator

# 4.2 Sets
# Order is not important

print(`Section 4.2`);

y:= 'y':
{x,y,z,y};

a:={1,2,3,4}: b:={2,4,6,8}:
a union b;
a intersect b;
a minus b;

# 4.3 Lists
# Order is important

print(`Section 4.3`);

s := seq( i/(i + 1), i=1..6):
M := [s];

# 4.4 Tables

print(`Section 4.4`);
a := 'a': b := 'b':
T := table([a,b]):
T[2];

S := table([(1) = A, (3) = B + C, (5) = A * B * C]);
S[3];
S;
op(S);

# 4.5 Arrays

print(`Section 4.5`);

A := array(1..2, 1..3, []);
B := array(1..2, 1..2, 1..2);

C := array(1..2, 1..2):
C[1, 1] := 1: C[1, 2] := 2: C[2, 1] := 3: C[2, 2] := 7:
op(C);

# 4.6 Data Conversions

# type() checks the type of an object
# convert() converts an object to the specified type

# 5.1 Defining functions

print(`Section 5.1`);
f := x -> x^2 - 3*x + 5;
f(2);

# 5.2 Composition of functions

print(`Section 5.2`);
(sin @ cos)(x);

f := x -> x^2:
g := x -> sqrt(1 - x):
(f@g)(x);
(g@f)(x);

f@@(-1)(x):

# 5.3 Summation and Product

print(`Section 5.3`);
f:='f':
Sum(f(i), i = 1..n):

Sum(i^2,i=1..10);
value(%);
sum(i^2,i=1..10);

# Product works similarly

# 5.4 Limits

print(`Section 5.4`);
Limit(f(x), x=a, right);

# 5.5 Differentiation

print(`Section 5.5`);

f := sqrt(1-x^2):
diff(f, x);

g:= z -> z^2*exp(z) + sin(z):
D(g);

# 5.6 Extrema

print(`Section 5.6`);

maximize(sin(x) + cos(x));
minimize(x^2+y^2, x);

# 5.7 Integration

print(`Section 5.7`);

Int(x^2/sqrt(1-x^3),x);
value(%);

# 5.7.1 Techniques of integration

# changevar -> substitution
# intparts -> integration by parts

# 5.8 Taylor and series expansions

print(`Section 5.8`);
y := 1/sqrt(1-x);
taylor(y, x=0,5);

# 5.9 Solving differential equations

print(`Section 5.9`);
f:='f':
y := f(x):
dy := diff(y,x):
ddy := diff(dy, x):

dsolve(ddy + 5*dy + 6*y = sin(x)*exp(-3*x), y);