SymPy Basic Flashcards
1
Q
A
2
Q
How do you import Sympy and create a symbolic variable (x)?
A
```python
import sympy
x = sympy.Symbol(‘x’)
~~~
3
Q
How do you define an expression (x^2 + 2x + 1) in Sympy using the symbol (x)?
A
```python
expr = x**2 + 2*x + 1
~~~
4
Q
How do you simplify the expression ((x^2 + 2x + 1) / (x + 1)) using sympy.simplify?
A
```python
import sympy
x = sympy.Symbol(‘x’)
expr = (x**2 + 2*x + 1) / (x + 1)
simpl_expr = sympy.simplify(expr)
# Output: x + 1
~~~
5
Q
How do you factor the polynomial (x^3 - 6x^2 + 11x - 6)?
A
```python
p = x3 - 6x**2 + 11x - 6
factor_p = sympy.factor(p)
# (x - 1)(x - 2)(x - 3)
~~~
6
Q
How do you expand ((x + 1)^3) into a polynomial form using sympy.expand?
A
```python
expr = (x + 1)3
expanded = sympy.expand(expr)
# x^3 + 3x^2 + 3x + 1
~~~
7
Q
How do you collect like terms of (x) in the expression (x^3 + 2x*x^2 + x)?
A
```python
expr = x3 + 2xx2 + x
collected = sympy.collect(sympy.expand(expr), x)
# x^3 + 2x^3 + x -> 3x^3 + x
~~~
8
Q
How do you substitute (x = 2) into the expression (x^2 + x + 1)?
A
```python
expr = x**2 + x + 1
sub_expr = expr.subs(x, 2)
# 2^2 + 2 + 1 = 7
~~~
9
Q
How do you solve the equation (x^2 - 4 = 0) for (x) using sympy.solve?
A
```python
solution = sympy.solve(sympy.Eq(x**2, 4), x)
# [ -2, 2 ]
~~~
10
Q
How do you solve the system of linear equations: 2x + y = 5, x - 3y = -4 for (x, y)?
A
```python
x, y = sympy.symbols(‘x y’)
sol = sympy.solve([2x + y - 5, x - 3y + 4], [x, y])
# {x: 2, y: 1}
~~~
11
Q
How do you compute the derivative of (x^3 + 2x^2) with respect to (x)?
A
```python
expr = x3 + 2x**2
deriv_expr = sympy.diff(expr, x)
# 3x^2 + 4*x
~~~
12
Q
How do you compute the partial derivative of (f(x,y) = x^2 y^3) with respect to (y)?
A
```python
x, y = sympy.symbols(‘x y’)
f = x2 * y3
partial_y = sympy.diff(f, y)
# 3x^2y^2
~~~
13
Q
How do you compute the indefinite integral of (x^2) with respect to (x)?
A
```python
integrand = x**2
indef_int = sympy.integrate(integrand, (x))
# x^3/3
~~~
14
Q
How do you compute the definite integral of (x^2) from (x=0) to (x=2)?
A
```python
def_int = sympy.integrate(x**2, (x, 0, 2))
# 8/3
~~~
15
Q
How do you calculate the limit of (rac{sin(x)}{x}) as (x) approaches 0?
A
```python
limit_expr = sympy.limit(sympy.sin(x)/x, x, 0)
# 1
~~~
16
Q
How do you find the series expansion of (e^x) around (x=0) up to (x^4)?
A
```python
series_expr = sympy.series(sympy.exp(x), (x, 0, 5))
# 1 + x + x^2/2 + x^3/6 + x^4/24 + O(x^5)
~~~
17
Q
How do you compute the first 4 terms of the series expansion of (sin(x)) around (x = 0)?
A
```python
series_sin = sympy.series(sympy.sin(x), (x, 0, 5))
# x - x^3/6 + x^5/120 - …
~~~
18
Q
How do you rewrite (sin(x)) in exponential form using sympy.rewrite(sympy.exp)?
A
```python
rewritten = sympy.sin(x).rewrite(sympy.exp)
# (exp(Ix) - exp(-Ix))/(2*I)
~~~
19
Q
How do you rewrite (log(x, 2)) as a fraction of natural logs using rewrite(sympy.log)?
A
```python
expr = sympy.log(x, 2)
rewritten_log = expr.rewrite(sympy.log)
# log(x)/log(2)
~~~
20
Q
How do you evaluate the expression (sqrt{2}) as a floating-point number using evalf()?
A
```python
val = sympy.sqrt(2).evalf()
# 1.414213562…
~~~
21
Q
How do you approximate (pi) to 10 decimal places using sympy.nsimplify or .evalf(10)?
A
```python
approx_pi = sympy.pi.evalf(10)
# 3.141592654
~~~
22
Q
How do you create a 2x2 symbolic matrix (M = egin{pmatrix} x & 1 \ 2 & y end{pmatrix})?
A
```python
M = sympy.Matrix([[x, 1],
[2, sympy.Symbol(‘y’)]])
~~~
23
Q
How do you multiply two symbolic matrices A and B of compatible shapes using A*B?
A
```python
A = sympy.Matrix([[1, 2], [3, 4]])
B = sympy.Matrix([[x, 0], [0, y]])
product = A * B
~~~
24
Q
How do you compute the determinant of the 2x2 matrix (egin{pmatrix} x & 2 \ 3 & y end{pmatrix})?
A
```python
mat = sympy.Matrix([[x, 2],
[3, y]])
det_mat = mat.det()
# x*y - 6
~~~
25
How do you compute the inverse of the matrix (egin{pmatrix} x & 2 \ 3 & y end{pmatrix}), assuming it’s invertible?
```python
mat_inv = mat.inv()
```
26
How do you find the eigenvalues of the 2x2 matrix (egin{pmatrix} x & 1 \ 0 & y end{pmatrix})?
```python
mat = sympy.Matrix([[x, 1],
[0, y]])
eigs = mat.eigenvalues()
# [x, y]
```
27
How do you symbolically sum the series ( sum_{k=0}^n k ) using `sympy.summation`?
```python
k, n = sympy.symbols('k n', positive=True)
sum_expr = sympy.summation(k, (k, 0, n))
# n*(n+1)/2
```
28
How do you symbolically compute the product (prod_{k=1}^n (k)) using `sympy.product`?
```python
prod_expr = sympy.product(k, (k, 1, n))
# factorial(n)
```
29
How do you define a piecewise function (f(x) = egin{cases} x^2, & x<0 \ x+1, & x ge 0 end{cases}) using `sympy.Piecewise`?
```python
f = sympy.Piecewise(
(x**2, x < 0),
(x + 1, True)
)
```
30
How do you solve the inequality (x^2 - 4 < 0) using `sympy.solve` and `sympy.StrictInequality` or `sympy.solve_inequality`?
```python
ineq = x**2 - 4 < 0
solution_ineq = sympy.solve(ineq, x)
# -2 < x < 2
```
31
How do you simplify a complicated expression using `sympy.simplify(expr)` in general?
```python
expr = (x**3 - x**2 + 2*x) / x
simpl_expr = sympy.simplify(expr)
# x^2 - x + 2
```
32
How do you factor the expression ((x^2 - 1)(x^2 + 1) + (x - 1)(x + 1)) then simplify?
```python
expr = (x**2 - 1)*(x**2 + 1) + (x - 1)*(x + 1)
fact_expr = sympy.factor(sympy.simplify(expr))
```
33
How do you perform partial fraction decomposition on ( rac{x+1}{x(x-1)})?
```python
fraction = (x + 1) / (x*(x - 1))
part_frac = sympy.apart(fraction)
# 1/x + 2/(x-1)
```
34
How do you rewrite (ln(x^2)) as (2 ln(x)) using `sympy.logcombine` or direct rewriting?
```python
expr = sympy.log(x**2)
rewritten = sympy.logcombine(expr)
# 2*log(x)
```
35
How do you compute an indefinite integral of (sin^2(x)) using `sympy.integrate`?
```python
int_sin2 = sympy.integrate(sympy.sin(x)**2, x)
# x/2 - sin(2*x)/4 + C
```
36
How do you solve a polynomial equation (x^3 - 3x + 1 = 0) symbolically?
```python
poly_eq = sympy.Eq(x**3 - 3*x + 1, 0)
solutions = sympy.solve(poly_eq, x)
```
37
How do you solve a transcendental equation (sin(x) = x) for a real solution, approximately, using `sympy.nsolve`?
```python
guess = 0
solution_approx = sympy.nsolve(sympy.sin(x) - x, guess)
```
38
How do you declare a symbol (x) with the assumption that (x) is positive?
```python
x = sympy.Symbol('x', positive=True)
```
39
How do you create a symbolic function (f(x) = x^2) using `sympy.Function`?
```python
f = sympy.Function('f')(x)
# Then define expression: f_expr = x**2
# Or simpler for direct usage, just write x**2 as expression.
```
40
How do you compute the derivative of a defined function (f(x) = x^2 + 2x) symbolically?
```python
f_expr = x**2 + 2*x
df = sympy.diff(f_expr, x)
# 2*x + 2
```
41
How do you compute the indefinite integral of a function (f(x) = x^2 + 2x)?
```python
sympy.integrate(f_expr, x)
# x^3/3 + x^2 + C
```
42
How do you evaluate a symbolic expression `expr` at multiple points, say x=1 and x=2, quickly in a Python loop?
```python
for val in [1, 2]:
print(expr.subs(x, val))
```
43
How do you define two symbolic variables `a` and `b` with the assumption they are integers?
```python
a = sympy.Symbol('a', integer=True)
b = sympy.Symbol('b', integer=True)
```
44
How do you use `sympy.solve` to solve for `x` in the equation `a*x + b = 0`, where `a` and `b` are also symbols?
```python
a, b, x = sympy.symbols('a b x')
solution = sympy.solve(sympy.Eq(a*x + b, 0), x)
# [-b/a]
```
45
How do you use `sympy.nsimplify` to simplify an expression numerically, for example \(\sqrt{2}/2\)?
```python
expr = sympy.sqrt(2)/2
simpl_num = sympy.nsimplify(expr, [sympy.sqrt(2)])
# 2**(1/2)/2, might remain the same or become sqrt(2)/2
```
46
How do you evaluate the Gamma function \(\Gamma(x)\) at \(x = 1/2\) symbolically?
```python
from sympy import gamma
val = gamma(sympy.Rational(1,2))
# sqrt(pi)
```
47
How do you expand the binomial \((x+y)^3\) in Sympy?
```python
y = sympy.Symbol('y')
expr = (x + y)**3
expanded = sympy.expand(expr)
# x^3 + 3*x^2*y + 3*x*y^2 + y^3
```
48
How do you do a piecewise integral of the piecewise function \(f(x) = x\) for x>=0, and 0 for x<0, from -1 to 2?
```python
f = sympy.Piecewise((0, x < 0), (x, True))
# Integrate from x=-1 to x=2
value = sympy.integrate(f, (x, -1, 2))
# = (1/2)*2^2 = 2
```
49
How do you evaluate the expression \(\sin(x)\cos(x)\) at \(x = \frac{\pi}{4}\)?
```python
value = (sympy.sin(x)*sympy.cos(x)).subs(x, sympy.pi/4)
# sin(pi/4)*cos(pi/4) = sqrt(2)/2 * sqrt(2)/2 = 1/2
```
50
How do you compute the Taylor series expansion of \(\ln(1+x)\) around \(x=0\) up to \(x^4\)?
```python
series_ln = sympy.series(sympy.log(1+x), (x, 0, 5))
# x - x^2/2 + x^3/3 - x^4/4 + ...
```
51
How do you perform numeric approximation of a symbolic expression `sympy.sqrt(3)` to 5 decimal places?
```python
approx_val = sympy.sqrt(3).evalf(5)
# 1.7321...
```
52
How do you declare multiple symbolic variables at once, for example `x0, x1, x2`, using `sympy.symbols`?
```python
import sympy
x0, x1, x2 = sympy.symbols('x0 x1 x2')
```
53
How do you sympify a string "cos(x) + sin(x)**2" into a Sympy expression?
```python
expr_str = "cos(x) + sin(x)**2"
expr = sympy.sympify(expr_str)
```
54
How do you create a rational number \(\frac{3}{4}\) in Sympy using `sympy.Rational`?
```python
frac = sympy.Rational(3, 4)
# 3/4
```
55
How do you compute the second derivative of \(x^3 + 5x - 1\) with respect to \(x\)?
```python
x = sympy.Symbol('x')
expr = x**3 + 5*x - 1
second_derivative = sympy.diff(expr, (x, 2))
# 6*x
```
56
How do you compute the definite integral of \(\sin^2(x)\) from \(0\) to \(\pi\)?
```python
integrand = sympy.sin(x)**2
res = sympy.integrate(integrand, (x, 0, sympy.pi))
# pi/2
```
57
How do you find the limit of \(\frac{1}{x}\) as \(x \to \infty\)?
```python
limit_val = sympy.limit(1/x, x, sympy.oo)
# 0
```
58
How do you find the limit of \(\frac{1}{x}\) as \(x \to -\infty\)?
```python
limit_val = sympy.limit(1/x, x, -sympy.oo)
# 0
```
59
How do you solve a linear system given by a matrix? For example, solve:
\[
\begin{pmatrix}
1 & 2 & 1 \\
3 & 1 & 4
\end{pmatrix}
\]
```python
from sympy.matrices import Matrix
x, y = sympy.symbols('x y')
# Coefficients in augmented matrix form:
mat = Matrix([
[1, 2, 1],
[3, 1, 4]
])
sol = sympy.solve_linear_system(mat, x, y)
# {x: ..., y: ...}
```
60
How do you create a 3x3 symbolic matrix and compute its transpose?
```python
a, b, c = sympy.symbols('a b c')
M = sympy.Matrix([[a, b, c],
[1, 2, 3],
[x, x+1, x+2]])
M_T = M.T
```
61
How do you evaluate the matrix exponential `exp(A)` for a 2x2 symbolic matrix `A` using `A.exp()`?
```python
A = sympy.Matrix([[0, 1],
[-1, 0]])
A_exp = A.exp()
```
62
How do you rewrite the trigonometric expression \(\sin^2(x)\) using the identity \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\) via `sympy.trigsimp` or `sympy.expand_trig`?
```python
expr = sympy.sin(x)**2
trig_simpl = sympy.trigsimp(expr)
# 1 - cos(2*x)
# or (1 - cos(2*x))/2 depending on the simplification
```
63
How do you find the series expansion of \(\cos(x)\) around \(x=0\) up to \(x^6\) and then remove the big-O term?
```python
series_cos = sympy.series(sympy.cos(x), (x, 0, 7))
# cos(x) expansion
series_no_O = series_cos.removeO()
```
64
How do you rewrite \(\tan(x)\) in terms of \(\sin\) and \(\cos\) using `rewrite(sympy.sin)`?
```python
expr = sympy.tan(x)
rewritten = expr.rewrite(sympy.sin)
# sin(x)/cos(x)
```
65
How do you evaluate a symbolic derivative at a specific value? For example, \(\frac{d}{dx} x^2\) at \(x=3\)?
```python
f = x**2
df = sympy.diff(f, x) # 2*x
val_at_3 = df.subs(x, 3) # 6
```
66
How do you define a lambda function in Python from a Sympy expression \(\sin(x) + x\) using `sympy.lambdify` so you can evaluate it numerically?
```python
f_expr = sympy.sin(x) + x
f_lam = sympy.lambdify(x, f_expr, 'math')
print(f_lam(3.14))
```
67
How do you compute the greatest common divisor (gcd) of two polynomials, e.g. \(x^2 - 4\) and \(x^3 - 8x\)?
```python
p1 = x**2 - 4
p2 = x**3 - 8*x
g = sympy.gcd(p1, p2)
# x - 2 or x+2, depending on factoring
```
68
How do you compute the least common multiple (lcm) of two polynomials, e.g. \(x - 1\) and \((x-1)(x+1)\)?
```python
p1 = x - 1
p2 = (x - 1)*(x + 1)
l = sympy.lcm(p1, p2)
# (x - 1)*(x + 1)
```
69
How do you convert an expression `expr = x**3 + x**2 - x + 1` to a polynomial object using `sympy.poly(expr, x)`?
```python
p = sympy.poly(expr, x)
degree_p = p.degree()
```
70
How do you evaluate the indefinite integral of \(\exp(-x^2)\) symbolically with Sympy (knowing it doesn't have an elementary antiderivative)?
```python
int_expr = sympy.integrate(sympy.exp(-x**2), (x))
# returns sqrt(pi)*erf(x)/2
```
71
How do you compute the error function \(\mathrm{erf}(x)\) at a specific point, say \(x=1\), in a symbolic exact form or approximate form?
```python
val_exact = sympy.erf(1)
val_approx = val_exact.evalf()
```
72
How do you find the roots of a polynomial \(x^3 - 2x + 1\) using `sympy.roots` or `sympy.nroots`?
```python
p = x**3 - 2*x + 1
# For exact roots:
r_exact = sympy.roots(p, x)
# For numeric approximations:
r_approx = sympy.nroots(p)
```
73
How do you get all solutions (including complex) of \(x^2 + 1 = 0\) using `sympy.solve`?
```python
sol = sympy.solve(sympy.Eq(x**2 + 1, 0), x, dict=True)
# [{x: I}, {x: -I}]
```
74
How do you check the assumptions on a symbol, for example if `x = sympy.Symbol('x', real=True, positive=True)`, how to see `x.is_real` or `x.is_positive`?
```python
x = sympy.Symbol('x', real=True, positive=True)
print(x.is_real) # True
print(x.is_positive) # True
```
75
How do you define a function of two variables, \(f(x, y) = x^2 + y^2\), and compute its gradient (partial derivatives w.r.t. x and y)?
```python
x, y = sympy.symbols('x y', real=True)
f = x**2 + y**2
grad_f = (sympy.diff(f, x), sympy.diff(f, y))
# (2*x, 2*y)
```
76
How do you compute the Hessian matrix of \(f(x, y) = x^2 + 3xy + y^2\)?
```python
f = x**2 + 3*x*y + y**2
H = sympy.hessian(f, (x, y))
# Matrix([[2, 3],
# [3, 2]])
```
77
How do you evaluate the indefinite integral \(\int \ln(x), dx\)?
```python
int_ln = sympy.integrate(sympy.log(x), (x))
# x*log(x) - x
```
78
How do you expand a trigonometric expression like \(\sin(x + y)\) using `sympy.expand_trig`?
```python
expr = sympy.sin(x + y)
expanded_trig = sympy.expand_trig(expr)
# sin(x)*cos(y) + cos(x)*sin(y)
```
79
How do you simplify a logical expression like `(~A & B) | (~A & ~B)` using `sympy.simplify_logic`?
```python
A, B = sympy.symbols('A B')
expr_logical = (~A & B) | (~A & ~B)
simpl_logical = sympy.simplify_logic(expr_logical)
# ~A
```
80
How do you evaluate whether \(\sqrt{2}\) is rational or an integer using Sympy’s `.is_rational` or `.is_integer`?
```python
root2 = sympy.sqrt(2)
print(root2.is_rational) # None or False
print(root2.is_integer) # False
```
81
How do you perform partial fraction decomposition on \(\frac{x}{(x+1)(x+2)}\) using `sympy.apart`?
```python
expr = x / ((x+1)*(x+2))
apart_expr = sympy.apart(expr)
# (x + 2) - 2 / ((x+1)(x+2)) => 1/(x+1) - 1/(x+2), etc.
```
82
How do you solve the rational inequality \(\frac{x-1}{x+2} > 0\) using Sympy’s `solve_rational_inequalities`?
```python
inequalities = [((x-1, x+2), '>')]
sol = sympy.solve_rational_inequalities([inequalities])
# (-∞, -2) U (1, ∞)
```
83
How do you force Sympy to treat a numeric constant as exact rational if possible, for example `sympy.nsimplify(1.5)`?
```python
expr = sympy.nsimplify(1.5)
# 3/2
```
84
How do you convert a Sympy expression to a lambda function that uses NumPy for fast numeric evaluation (e.g., `(x + 1)**2`)?
```python
f_expr = (x + 1)**2
f_np = sympy.lambdify(x, f_expr, 'numpy')
# f_np can now take NumPy arrays
```
85
How do you compute the third derivative of \(\sin(x)\) with respect to \(x\)?
```python
third_deriv = sympy.diff(sympy.sin(x), (x, 3))
# -cos(x)
```
86
How do you solve a polynomial equation \(x^4 - 1 = 0\) and get the four complex roots in a list?
```python
eq = sympy.Eq(x**4 - 1, 0)
solutions = sympy.solve(eq, x, dict=False)
# [-1, 1, -I, I]
```
87
How do you evaluate the numeric approximation of \(\ln(2)\) with 8 decimal places?
```python
val_ln2 = sympy.log(2).evalf(8)
# 0.69314718
```
88
How do you expand a binomial coefficient symbolically, for instance `sympy.binomial(n, 2)`?
```python
n = sympy.Symbol('n', positive=True)
binom_expr = sympy.binomial(n, 2)
expanded_binom = sympy.expand(binom_expr)
# n*(n - 1)/2
```
89
How do you use the `sympy.solve_poly_inequality` to solve \( (x-2)(x+1) \ge 0\)?
```python
ine_expr = (x - 2)*(x + 1)
solution_poly_ineq = sympy.solve_poly_inequality(sympy.Poly(ine_expr, x) >= 0, x)
# (-∞, -1] U [2, ∞)
```
90
How do you compute the numeric approximation of a definite integral, say \(\int_0^1 e^{-x^2}, dx\), using `.evalf()` after symbolic integration?
```python
int_expr = sympy.integrate(sympy.exp(-x**2), (x, 0, 1))
approx_val = int_expr.evalf()
```
91
How do you create a symbolic expression for a sum like \(\sum_{k=1}^n k^2\) using `sympy.summation`?
```python
k, n = sympy.symbols('k n', positive=True)
sum_expr = sympy.summation(k**2, (k, 1, n))
# n*(n+1)*(2*n+1)/6
```
92
How do you declare a symbolic constant `C1` or `C2` often used for integration constants, from the `sympy.symbols` function?
```python
C1, C2 = sympy.symbols('C1 C2', real=True)
```
93
How do you convert a Sympy expression to a string containing LaTeX code using `sympy.latex`?
```python
expr = x**2 + sympy.sin(x)
latex_code = sympy.latex(expr)
# "\\sin{\\left(x \\right)} + x^{2}"
```
94
How do you define a Wild symbol, say `A = sympy.Wild('A')`, and use it to match a pattern in an expression?
```python
A = sympy.Wild('A')
pattern_expr = x**2 + 2*x + 1
match_result = pattern_expr.match(x**2 + A*x + 1)
# {A: 2} if it matches
```
95
How do you factor the expression \(\cos(x)^2 - \sin(x)^2\) using `sympy.trigsimp`?
```python
expr = sympy.cos(x)**2 - sympy.sin(x)**2
fact_trig = sympy.trigsimp(expr)
# cos(2*x)
```
96
How do you check if a symbolic expression `expr = x*(x + 1)` is a polynomial in `x` by using `sympy.poly(expr, x)` inside a try-except?
```python
try:
p = sympy.poly(expr, x)
print("It is a polynomial of degree", p.degree())
except sympy.PolynomialError:
print("Not a polynomial in x")
```
97
How do you solve for y in the equation \(y + \frac{1}{y} = x\) in terms of x, using `sympy.solve`?
```python
y = sympy.Symbol('y', complex=True)
eq = sympy.Eq(y + 1/y, x)
solutions_y = sympy.solve(eq, y)
# [x/2 + sqrt(x^2 - 4)/2, x/2 - sqrt(x^2 - 4)/2]
```
98
How do you define a piecewise function \(f(x) = \begin{cases} x & x>2 \\ 2 & x\le2 \end{cases}\) using `sympy.Piecewise` and then evaluate at x=3?
```python
f = sympy.Piecewise(
(x, x > 2),
(2, True)
)
val_at_3 = f.subs(x, 3)
# 3
```
99
How do you convert a piecewise function to a “standard” expression if possible, using `sympy.piecewise_fold`?
```python
folded_expr = sympy.piecewise_fold(f)
# Might combine piecewise parts if there's a known simplification
```
100
How do you compute \(\frac{\partial^2}{\partial x \partial y} [x^2 + xy + y^2]\) (the mixed partial derivative) in Sympy?
```python
expr = x**2 + x*y + y**2
dxy = sympy.diff(expr, x, 1, y, 1)
# 1
```
101
How do you simplify a piecewise rational expression, say \(\frac{x^2 - 1}{x - 1}\), for \(x \neq 1\), using `sympy.cancel` or `sympy.simplify`?
```python
expr = (x**2 - 1)/(x - 1)
canceled_expr = sympy.cancel(expr)
# x + 1
```
102
How do you declare a Sympy symbol (t) with the assumption that it’s real and nonnegative?
```python
import sympy
t = sympy.Symbol('t', real=True, nonnegative=True)
```
103
How do you symbolically compute the derivative ( rac{d}{dx}igl(cos(3x)igr)) using `sympy.diff`?
```python
x = sympy.Symbol('x')
expr = sympy.cos(3*x)
deriv = sympy.diff(expr, x)
# -3*sin(3*x)
```
104
How do you evaluate (int_0^{pi} sin(2x),dx) using `sympy.integrate`?
```python
res = sympy.integrate(sympy.sin(2*x), (x, 0, sympy.pi))
# 0
```
105
How do you expand (log(x*y)) into (log(x) + log(y)) using `sympy.logcombine` or a direct rewrite method?
```python
x, y = sympy.symbols('x y', positive=True)
expr = sympy.log(x*y)
expanded_log = sympy.logcombine(expr) # or expr.rewrite(sympy.log)
# log(x) + log(y)
```
106
How do you create a symbol (n) that’s intended only for integer values (e.g. for summations)?
```python
n = sympy.Symbol('n', integer=True)
```
107
How do you compute the indefinite integral of ( rac{1}{x}) in Sympy?
```python
expr = 1/x
int_expr = sympy.integrate(expr, (x))
# log(x)
```
108
How do you declare a symbolic function (f) of one variable (x) using `sympy.Function` and then refer to (f(x))?
```python
f = sympy.Function('f')
expr = f(x)
# expr represents f(x) symbolically
```
109
How do you solve the equation (e^x = 5) for (x) symbolically?
```python
eq = sympy.Eq(sympy.exp(x), 5)
sol = sympy.solve(eq, x)
# [log(5)]
```
110
How do you solve the equation (log(x) = 3) for (x)?
```python
eq = sympy.Eq(sympy.log(x), 3)
sol = sympy.solve(eq, x)
# [exp(3)]
```
111
How do you check the structure of a Sympy expression, for instance verifying if ((x+1)^2) is a `Pow` node or an `Add` node, using `.func`?
```python
expr = (x+1)**2
expr_func = expr.func
# sympy.core.power.Pow
```
112
How do you symbolically differentiate (sin(x^2)) with respect to (x)?
```python
expr = sympy.sin(x**2)
dexpr = sympy.diff(expr, x)
# 2*x*cos(x^2)
```
113
How do you compute the Laplace transform of (t^2) with respect to (t) using `sympy.laplace_transform`?
```python
t, s = sympy.symbols('t s', positive=True)
laplace_expr = sympy.laplace_transform(t**2, t, s)
# 2/s^3
```
114
How do you compute the inverse Laplace transform of ( rac{1}{s^2 + 1}) using `sympy.inverse_laplace_transform`?
```python
ilap = sympy.inverse_laplace_transform(1/(s**2+1), s, t)
# sin(t)
```
115
How do you compute the integral (int e^{-x^2}, dx) in terms of the error function ((mathrm{erf}))?
```python
expr = sympy.exp(-x**2)
res = sympy.integrate(expr, x)
# sqrt(pi)*erf(x)/2
```
116
How do you create a piecewise expression such that (g(x) = 1) if (x ge 0) and (-1) otherwise?
```python
g = sympy.Piecewise(
(1, x >= 0),
(-1, True)
)
```
117
How do you convert a Sympy expression, say (x^2 + 2x + 1), into a Python function that uses `math` (not NumPy) via `lambdify`?
```python
expr = x**2 + 2*x + 1
f_math = sympy.lambdify(x, expr, 'math')
val = f_math(2) # 9
```
118
How do you compute the Maclaurin series (i.e. expansion around 0) of ( an(x)) up to (x^5)?
```python
series_tan = sympy.series(sympy.tan(x), (x, 0, 6))
# x + x^3/3 + 2*x^5/15 + ...
```
119
How do you define a function (F(x) = int_0^x sin(t^2), dt) symbolically in Sympy?
```python
t = sympy.Symbol('t', real=True)
F_expr = sympy.integrate(sympy.sin(t**2), (t, 0, x))
```
120
How do you expand a polynomial expression with multiple variables, like ((x + y)^3)?
```python
y = sympy.Symbol('y')
expr = (x + y)**3
expanded = sympy.expand(expr)
# x^3 + 3*x^2*y + 3*x*y^2 + y^3
```
121
How do you factor the expression (x^3 + 3x^2y + 3xy^2 + y^3)?
```python
factored = sympy.factor(x**3 + 3*x**2*y + 3*x*y**2 + y**3)
# (x + y)^3
```
122
How do you solve the equation (x^2 + y^2 = 1) for (y) in terms of (x)?
```python
eq = sympy.Eq(x**2 + y**2, 1)
sols = sympy.solve(eq, y)
# [sqrt(1 - x^2), -sqrt(1 - x^2)]
```
123
How do you find the real part of a symbolic expression like (exp(i x)) using `.expand_complex()` or `.as_real_imag()`?
```python
z = sympy.exp(sympy.I*x)
real_part = sympy.expand_complex(z).as_real_imag()[0]
# cos(x)
```
124
How do you find the imaginary part of (exp(i x)) similarly?
```python
imag_part = sympy.expand_complex(z).as_real_imag()[1]
# sin(x)
```
125
How do you check if the expression (x^2 - y^2) is factorable in the real domain, and then factor it?
```python
expr = x**2 - y**2
fac = sympy.factor(expr)
# (x - y)*(x + y)
```
126
How do you do polynomial long division of ((x^3 + 2x + 1)) by ((x - 1)) using `sympy.div`?
```python
num = x**3 + 2*x + 1
den = x - 1
quotient, remainder = sympy.div(num, den)
```
127
How do you symbolically compute (int 2x , e^{x^2}, dx) using `sympy.integrate` (recognizing a substitution pattern)?
```python
expr = 2*x*sympy.exp(x**2)
res = sympy.integrate(expr, (x))
# exp(x^2)
```
128
How do you compute the gradient of ((x - y)^2 + (y - z)^2 + (z - x)^2) with respect to (x, y, z)?
```python
x, y, z = sympy.symbols('x y z', real=True)
f = (x - y)**2 + (y - z)**2 + (z - x)**2
grad_f = (sympy.diff(f, x), sympy.diff(f, y), sympy.diff(f, z))
```
129
How do you define a sum (sum_{k=1}^n 2^k) in Sympy and then simplify it?
```python
k, n = sympy.symbols('k n', positive=True)
sum_expr = sympy.summation(2**k, (k, 1, n))
sympy_simpl = sympy.simplify(sum_expr)
# 2*(2^n - 1)
```
130
How do you compute the binomial coefficient (inom{n}{k}) symbolically in Sympy?
```python
n, k = sympy.symbols('n k', nonnegative=True)
bin_coeff = sympy.binomial(n, k)
```
131
How do you check if ((x^2 + y^2)) depends on (x) by using `expr.has(x)`?
```python
expr = x**2 + y**2
expr.has(x) # True
expr.has(z) # False if z isn't in the expression
```
132
How do you evaluate the real definite integral (int_{-infty}^{infty} e^{-x^2}, dx)?
```python
res = sympy.integrate(sympy.exp(-x**2), (x, -sympy.oo, sympy.oo))
# sqrt(pi)
```
133
How do you create a symbolic constant for (pi) in Sympy (though it’s already built-in as `sympy.pi`)?
```python
# It's built-in: sympy.pi
pi_val = sympy.pi
```
134
How do you compute the indefinite integral of (x cdot ln(x)) with respect to (x)?
```python
expr = x*sympy.log(x)
res = sympy.integrate(expr, (x))
# (x^2*log(x))/2 - x^2/4
```
135
How do you expand the expression ((x + y)^4) using `sympy.expand`?
```python
expr = (x + y)**4
expanded = sympy.expand(expr)
# x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + y^4
```
136
How do you factor the polynomial ((x + y)^4 - x^4 - y^4)?
```python
poly_expr = (x + y)**4 - x**4 - y**4
factored_poly = sympy.factor(poly_expr)
```
137
How do you compute the indefinite integral (int sec^2(x), dx)?
```python
expr = sympy.sec(x)**2
int_expr = sympy.integrate(expr, (x))
# tan(x)
```
138
How do you compute the indefinite integral (int csc^2(x), dx)?
```python
expr = sympy.csc(x)**2
int_expr = sympy.integrate(expr, (x))
# -cot(x)
```
139
How do you compute the indefinite integral (int sec(x), dx)?
```python
expr = sympy.sec(x)
int_expr = sympy.integrate(expr, x)
# log(sec(x) + tan(x)) or something equivalent
```
140
How do you compute the indefinite integral (int csc(x), dx)?
```python
expr = sympy.csc(x)
int_expr = sympy.integrate(expr, x)
# log(csc(x) - cot(x)) or an equivalent form
```
141
How do you express ( an^2(x)) in terms of (sec^2(x)) using trigonometric simplifications?
```python
expr = sympy.tan(x)**2
simpl_expr = sympy.trigsimp(expr)
# sec(x)^2 - 1
```
142
How do you create a symbolic array (non-Matrix) of shape 2x2, e.g. `sympy.Array([[1, x], [y, 2]])`?
```python
A = sympy.Array([[1, x],
[y, 2]])
```
143
How do you compute the sum of absolute values `|x| + |y|` symbolically in Sympy?
```python
abs_expr = sympy.Abs(x) + sympy.Abs(y)
```
144
How do you evaluate `|x|` at `x=-3`?
```python
val = sympy.Abs(x).subs(x, -3)
# 3
```
145
How do you find an expression’s free symbols, for instance in `expr = x*y + z`, using `.free_symbols`?
```python
expr = x*y + sympy.Symbol('z')
print(expr.free_symbols)
# {x, y, z}
```
146
How do you rename a free symbol in an expression, say rename (x) to (u) in (x^2 + y^2)?
```python
expr = x**2 + y**2
new_expr = expr.subs(x, u)
```
147
How do you rename a free symbol in an expression, say rename (x) to (u) in (x^2 + y), using `.xreplace({old: new})`?
```python
expr = x**2 + y
expr_renamed = expr.xreplace({x: sympy.Symbol('u')})
```
148
How do you convert a Sympy `Matrix` to a nested Python list using `.tolist()`?
```python
mat = sympy.Matrix([[1, 2], [3, 4]])
py_list = mat.tolist()
# [[1, 2], [3, 4]]
```
149
How do you compute the cofactor matrix (matrix of cofactors) of a 2x2 or 3x3 matrix using `.cofactor_matrix()`?
```python
M = sympy.Matrix([[a, b], [c, d]])
cof_mat = M.cofactor_matrix()
```
150
How do you specify that (x) is a complex variable using `sympy.Symbol('x', complex=True)`?
```python
x = sympy.Symbol('x', complex=True)
```
151
How do you compute (sum_{k=0}^{infty} x^k) symbolically, recognizing it as a geometric series ((|x|<1)) using `sympy.summation` and `.simplify()`?
```python
k = sympy.Symbol('k', nonnegative=True)
sum_expr = sympy.summation(x**k, (k, 0, sympy.oo))
geom_simpl = sympy.simplify(sum_expr)
# 1/(1 - x), for |x| < 1
```
152
How do you get the sorted list of all symbols in an expression, for instance in `(x + 2)*(y + 3*z)`?
```python
expr = (x + 2)*(y + 3*z)
sym_list = sorted(expr.free_symbols, key=lambda s: s.name)
# [x, y, z], in alphabetical order
```
153
How do you compute the absolute value of a variable (x) when (x = -3) using `sympy.Abs`?
```python
val = sympy.Abs(x).subs(x, -3)
# 3
```
154
How do you create a piecewise function (f(x)) such that (f(x) = x^2) for (x < 2) and (f(x) = 4) for (x ge 2), and then differentiate it?
```python
import sympy
x = sympy.Symbol('x', real=True)
f = sympy.Piecewise(
(x**2, x < 2),
(4, True)
)
df = sympy.diff(f, x)
# Derivative is 2*x for x<2, and 0 for x>=2
```
155
How do you declare two symbolic parameters `a` and `b`, define `expr = a^2 + 2*a*b + b^2`, and factor it?
```python
a, b = sympy.symbols('a b', real=True)
expr = a**2 + 2*a*b + b**2
factored_expr = sympy.factor(expr)
# (a + b)**2
```
156
How do you compute the residue of the function ( rac{1}{(x - 1)(x - 2)}) at the pole (x=1) using `sympy.residue`?
```python
expr = 1 / ((x - 1)*(x - 2))
resid = sympy.residue(expr, x, 1)
# 1/(1 - 2) = -1
```
157
How do you construct a `Lambda` object for the expression (x^2 + 1) so that it can be called like a function in Python?
```python
f = sympy.Lambda(x, x**2 + 1)
value_at_3 = f(3) # 10
```
158
How do you compute the polynomial remainder of (x^3 + 2x + 1) when divided by (x - 1) using `sympy.rem`?
```python
num = x**3 + 2*x + 1
den = x - 1
remainder = sympy.rem(num, den)
# Evaluate or check the symbolic expression for the remainder
```
159
How do you convert a Sympy expression like (x^2 + 1) into a Python lambda function that uses NumPy for fast numeric operations?
```python
expr = x**2 + 1
f_numpy = sympy.lambdify(x, expr, 'numpy')
import numpy as np
arr = np.array([0, 1, 2])
res = f_numpy(arr) # array([1, 2, 5])
```
160
How do you compute the Taylor (Maclaurin) series expansion of (exp(3x)) around (x=0) up to (x^4)?
```python
series_expr = sympy.series(sympy.exp(3*x), (x, 0, 5))
# 1 + 3*x + 9*x^2/2 + 9*x^3/2 + 81*x^4/24 + O(x^5)
```
161
How do you define a symbolic function (g(x,y) = x + 2y), then partially differentiate it with respect to (y)?
```python
x, y = sympy.symbols('x y', real=True)
g_expr = x + 2*y
dg_dy = sympy.diff(g_expr, y)
# 2
```
162
How do you use `sympy.integrate` to compute (int cos(3x), dx)?
```python
expr = sympy.cos(3*x)
int_expr = sympy.integrate(expr, (x))
# sympy.sin(3*x)/3
```
163
How do you solve the equation ( an(x) = 1) for (x) in the principal domain using `sympy.solve`?
```python
eq = sympy.Eq(sympy.tan(x), 1)
solutions = sympy.solve(eq, x, dict=False)
# [pi/4 + n*pi], but solve() typically gives a principal solution plus a symbolic periodic term.
```
164
How do you compute the complex conjugate of (exp(i x)) using `sympy.conjugate`?
```python
z = sympy.exp(sympy.I*x)
z_conjugate = sympy.conjugate(z)
# exp(-I*x)
```
165
How do you substitute (x = 3) and (y = 4) simultaneously into the expression (x^2 + y^2)?
```python
expr = x**2 + y**2
val = expr.subs({x: 3, y: 4})
# 25
```
166
How do you force an expression like (sqrt{4}) to be simplified to `2` using `sympy.sqrt` plus a simplification?
```python
expr = sympy.sqrt(4)
simpl_expr = sympy.simplify(expr)
# 2
```
167
How do you evaluate the indefinite integral of ( rac{1}{1 + x^2}) (which is (arctan(x)))?
```python
expr = 1/(1 + x**2)
int_expr = sympy.integrate(expr, (x))
# atan(x)
```
168
How do you solve the system: \( \begin{cases} x + y = 10 \ x - y = 2 \end{cases} \) for \(x, y\)?
```python
sol = sympy.solve([x + y - 10, x - y - 2], [x, y])
# {x: 6, y: 4}
```
169
How do you compute the Jacobian matrix of \(\vec{F}(x,y) = (x^2 + y, \; x + y^2)\)?
```python
F1 = x**2 + y
F2 = x + y**2
J = sympy.Matrix([F1, F2]).jacobian([x, y])
# [[2*x, 1 ],
# [ 1, 2*y ]]
```
170
How do you compute the inverse Laplace transform of \(\frac{1}{s^2 + 4}\) in terms of `t`?
```python
s, t = sympy.symbols('s t', real=True, positive=True)
expr = 1/(s**2 + 4)
ilap = sympy.inverse_laplace_transform(expr, s, t)
# sin(2*t)/2
```
171
How do you compute the Laplace transform of \(\cos(2t)\) with respect to \(t\)?
```python
laplace_expr = sympy.laplace_transform(sympy.cos(2*t), t, s)
# s/(s^2 + 4)
```
172
How do you do polynomial long division of \(x^4 + x^2 + 1\) by \(x^2 - 1\) using `sympy.div`?
```python
num = x**4 + x**2 + 1
den = x**2 - 1
quotient, remainder = sympy.div(num, den)
# quotient and remainder are symbolic polynomials
```
173
How do you integrate \(\sec^3(x)\) with respect to \(x\) using Sympy?
```python
expr = sympy.sec(x)**3
int_expr = sympy.integrate(expr, x)
# (1/2)*log|sec(x) + tan(x)| + (1/2)*sec(x)*tan(x)
```
174
How do you factor the polynomial \(x^4 - 1\) using `sympy.factor`?
```python
p = x**4 - 1
factored = sympy.factor(p)
# (x - 1)*(x + 1)*(x^2 + 1)
```
175
How do you use `sympy.radsimp` to simplify a radical expression, for example \(\frac{1}{\sqrt{2}} + \frac{\sqrt{18}}{6}\)?
```python
expr = 1/sympy.sqrt(2) + sympy.sqrt(18)/6
rad_simpl = sympy.radsimp(expr)
# Typically simplifies the radicals (e.g. sqrt(2)/2 + sqrt(2)/2 = sqrt(2))
```
176
How do you compute \(\int x \cdot \ln(x), dx\) again, but confirm by differentiating the result?
```python
int_expr = sympy.integrate(x*sympy.log(x), x)
# (x^2*log(x))/2 - x^2/4
diff_back = sympy.diff(int_expr, x)
# x*log(x)
```
177
How do you define a symbolic summation of \(\sum_{k=1}^n \frac{1}{k}\) and see if Sympy can simplify it?
```python
k, n = sympy.symbols('k n', positive=True)
harm_sum = sympy.summation(1/k, (k, 1, n))
# This is the n-th harmonic number H_n, often returned as sympy.HarmonicNumber(n)
```
178
How do you compute the numeric approximation of \(\zeta(3)\) (the Riemann zeta function at 3) to 6 decimal places in Sympy?
```python
val = sympy.zeta(3).evalf(6)
# ~ 1.20206
```
179
How do you find the limit of \(\left(1 + \frac{1}{n}\right)^n\) as \(n \to \infty\)?
```python
n = sympy.Symbol('n', positive=True)
expr = (1 + 1/n)**n
limit_val = sympy.limit(expr, n, sympy.oo)
# E
```
180
How do you rewrite \(\cos^3(x)\) using `sympy.trigsimp` or `sympy.expand_trig`?
```python
expr = sympy.cos(x)**3
expanded_trig = sympy.expand_trig(expr)
# (3*cos(x) + cos(3*x))/4
```
181
How do you check the derivative of \(\arctan(x)\) using `sympy.diff`, verifying it is \(\frac{1}{1+x^2}\)?
```python
d_atan = sympy.diff(sympy.atan(x), x)
# 1/(1 + x^2)
```
182
How do you define a function \(f(x) = \sqrt{x^2 + 1}\) and find its derivative?
```python
f_expr = sympy.sqrt(x**2 + 1)
df = sympy.diff(f_expr, x)
# x/sqrt(x^2 + 1)
```
183
How do you create a diagonal matrix with symbolic entries `[x, y, z]` using `sympy.diag`?
```python
diag_mat = sympy.diag(x, y, z)
# Matrix([[x, 0, 0],
# [0, y, 0],
# [0, 0, z]])
```
184
How do you declare that a symbol `n` is positive and an integer, and then see `n.is_integer`, `n.is_positive`?
```python
n = sympy.Symbol('n', integer=True, positive=True)
print(n.is_integer) # True
print(n.is_positive) # True
```
185
How do you compute the derivative with respect to `x` of `x^2 + 3*x*y` treating `y` as a constant?
```python
expr = x**2 + 3*x*y
d_expr = sympy.diff(expr, x)
# 2*x + 3*y
```
186
How do you define `f(x, y)` as `x^2 + y^3` and compute the directional derivative in the direction `(1,1)` at the point `(1,1)`?
```python
f = x**2 + y**3
grad_f = (sympy.diff(f, x), sympy.diff(f, y)) # (2*x, 3*y^2)
direction = sympy.Matrix([1, 1])
point = {x: 1, y: 1}
grad_val = sympy.Matrix([g.subs(point) for g in grad_f]) # [2, 3]
dir_deriv = grad_val.dot(direction) # 2 + 3 = 5
```
187
How do you solve the polynomial inequality \(x^2 - 4 \ge 0\) using `sympy.solve` or `solve_inequality`?
```python
ineq = x**2 - 4 >= 0
solution_ineq = sympy.solve(ineq, x)
# (-∞, -2] U [2, ∞)
```
188
How do you evaluate the gamma function \(\Gamma(n)\) for a positive integer `n` so that it returns `(n-1)!` in simplified form?
```python
n = sympy.Symbol('n', positive=True, integer=True)
gamma_expr = sympy.gamma(n)
# gamma_expr simplifies to factorial(n-1) or Gamma(n)
sym_simpl = sympy.simplify(gamma_expr)
```
189
How do you find the minimal polynomial of \(\sqrt{2} + \sqrt{3}\) over the rationals using `sympy.minpoly`?
```python
alpha = sympy.sqrt(2) + sympy.sqrt(3)
min_poly = sympy.minpoly(alpha, x)
# x^4 - 10*x^2 + 1, for example
```
190
How do you convert the polynomial `p = x^4 - 10*x^2 + 1` to a `sympy.polys.polytools.poly` object and get its degree?
```python
p = x**4 - 10*x**2 + 1
poly_obj = sympy.poly(p, x)
deg = poly_obj.degree()
# 4
```
191
How do you compute the indefinite integral \(\int \sinh(x), dx\)?
```python
expr = sympy.sinh(x)
int_expr = sympy.integrate(expr, x)
# cosh(x)
```
192
How do you compute the indefinite integral \(\int \cosh(x), dx\)?
```python
expr = sympy.cosh(x)
int_expr = sympy.integrate(expr, x)
# sinh(x)
```
193
How do you rewrite \(\cosh^2(x)\) in terms of \(\cosh(2x)\) using a trigonometric/hyperbolic identity simplification?
```python
expr = sympy.cosh(x)**2
rewritten = sympy.trigsimp(expr)
# (1 + cosh(2*x))/2
```
194
How do you compute the real part of \(\exp(x + i,y)\) (i.e., \(e^x \cos y\)) in Sympy?
```python
z = sympy.exp(x + sympy.I*y)
real_z = sympy.re(z)
# exp(x)*cos(y)
```
195
How do you compute the imaginary part of \(\exp(x + i,y)\) (i.e., \(e^x \sin y\)) similarly?
```python
imag_z = sympy.im(z)
# exp(x)*sin(y)
```
196
How do you find a power series expansion of \(\ln(1 + x)\) at \(x=0\) up to \(x^4\)?
```python
series_log = sympy.series(sympy.log(1+x), (x, 0, 5))
# x - x^2/2 + x^3/3 - x^4/4 + O(x^5)
```
197
How do you define a function \(F(z) = z^2 + 1\) where `z` is a complex symbol, and find its derivative w.r.t. `z` in Sympy?
```python
z = sympy.Symbol('z', complex=True)
F = z**2 + 1
dF = sympy.diff(F, z)
# 2*z
```
