Consider the following function of two variables: ?(?,?)=1+?2?2�(�,�)=1+�2�2 Use SymPy for the following: a) Write a function `S` with a parameter ?� to evaluate the function ?(?)�(�) given by the following integral: ?(?)=∫1?2?(?,?)??�(�)=∫�21�(�,�)�� as a symbolic expression in ?�. b) Write a function `D(x,n)` which takes as arguments a symbol `x` and an integer ?� to evaluate the ?�-th derivative of ?(?)�(�) with respect to ?�. c) Finally write a function `T(a)` with one argument a given by a sympy.Rational to evaluate the integeral ?(?)=∫?0?(?)??.�(�)=∫0��(�)��. The functions ?(?)�(�) and ?(?,?)�(�,�) should both return polynomials of the type `sympy.Poly`, variable `x` and domain `sympy.QQ`. The function ?(?)�(�) should return a sympy.Rational If ?� is not a not an integer greater than or equal to 0 then a ValueError should be raised in ?(?,?)�(�,�) Hint: Note that the the output of "sympy.integrate" is not a SymPy polynomial. Note that the function "sympy.diff" does not accept a SymPy polynomial as input. To convert input and output you might find the functions `.as_expr` and `as_poly` useful. For example: Test Result assert_equal(S(x),sympy.Poly(-x**8/3 - 2*x**2/3 + 1,domain=sympy.QQ)) True assert_equal(D(x,1),sympy.Poly(-8/3*x**7 - 4/3*x, x, domain=sympy.QQ)) True assert_equal(D(x,9),sympy.Poly(0,x,domain=sympy.QQ)) True assert_equal(T(sympy.Rational(1,2)),sympy.Rational(6527,13824))

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Consider the following function of two variables:

?(?,?)=1+?2?2�(�,�)=1+�2�2

Use SymPy for the following:

a) Write a function `S` with a parameter ?� to evaluate the function ?(?)�(�) given by the following integral:
?(?)=∫1?2?(?,?)??�(�)=∫�21�(�,�)��
as a symbolic expression in ?�.
b) Write a function `D(x,n)` which takes as arguments a symbol `x` and an integer ?� to evaluate the ?�-th derivative of ?(?)�(�) with respect to ?�.
c) Finally write a function `T(a)` with one argument a given by a sympy.Rational to evaluate the integeral
?(?)=∫?0?(?)??.�(�)=∫0��(�)��.

The functions ?(?)�(�) and ?(?,?)�(�,�) should both return polynomials of the type `sympy.Poly`, variable `x` and domain `sympy.QQ`.
The function ?(?)�(�) should return a sympy.Rational
If ?� is not a not an integer greater than or equal to 0 then a ValueError should be raised in ?(?,?)�(�,�)

Hint:

Note that the the output of "sympy.integrate" is not a SymPy polynomial.
Note that the function "sympy.diff" does not accept a SymPy polynomial as input.
To convert input and output you might find the functions `.as_expr` and `as_poly` useful.

For example:

Test	Result
assert_equal(S(x),sympy.Poly(-x*8/3 - 2x**2/3 + 1,domain=sympy.QQ))	True
assert_equal(D(x,1),sympy.Poly(-8/3x7 - 4/3x, x, domain=sympy.QQ))	True
assert_equal(D(x,9),sympy.Poly(0,x,domain=sympy.QQ))	True
assert_equal(T(sympy.Rational(1,2)),sympy.Rational(6527,13824))	True

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