Question

Внимание задания на английском языке!!!1.Observe that 18 = 4² + 1² + 1² + 0². How many of the first fifteen positive integers can be written as the sum of the squares of four integers ?2. If c is a constant, what is (2 − 3x²)² dx ?3 If y = xe^x then dy/dx =4 For each real number x, let [x] be the biggest integer which is less than or equal to x. What can you say about the following three equations ?(i) [p + 3] = [p] + 3(ii) [p + q] = [p] + [q](iii) [5p] = 5 [p]

Answer 1

1.Observe, что 18 = 4 ² + 1 ² + 1 ² + 0 ². Сколько из первых пятнадцати натуральных чисел можно записать в виде суммы квадратов четырех целых чисел? 2. Если с-постоянная, то, что (2 - 3x ²) ² дх? 3 Если у = х ^ х, то ду / дх = 4 Для каждого вещественного числа х, пусть [х] самое большое целое число, которое меньше или равно х. Что вы можете сказать о следующих трех уравнений? (Я) [р + 3] = [р] + 3 (II) [р + д] = [р] + [Q] (III) [5р] = 5 [р]

Answer 2

1. The first question asks how many of the first fifteen positive integers can be written as the sum of the squares of four integers. To solve this, we can iterate through the positive integers from 1 to 15 and check if they can be expressed as the sum of four squares.

Here's the code snippet that checks each number from 1 to 15:

count = 0
for num in range(1, 16):
    for a in range(int(num**0.5) + 1):
        for b in range(int(num**0.5) + 1):
            for c in range(int(num**0.5) + 1):
                for d in range(int(num**0.5) + 1):
                    if a2 + b2 + c2 + d2 == num:
                        count += 1
                        break
print(count)

The answer to the first question is 15, as all the first fifteen positive integers can be expressed as the sum of four squares.

2. The second question asks to find the integral of the expression (2 - 3x²)² with respect to x. To solve this, we can expand the expression and integrate it term by term.

Here's the code snippet that calculates the integral:

import sympy as sp

x = sp.symbols('x')
expression = (2 - 3*x2)2
integral = sp.integrate(expression, x)
print(integral)

The answer to the second question is the integral of (2 - 3x²)² with respect to x, which can be calculated as (4x - 3x³ + x⁵/5) + C, where C is the constant of integration.

3. The third question involves the function y = xe^x and asks to find dy/dx. To solve this, we can differentiate the function y with respect to x using the product rule.

Here's the code snippet that calculates the derivative:

import sympy as sp

x = sp.symbols('x')
y = x*sp.exp(x)
derivative = sp.diff(y, x)
print(derivative)

The answer to the third question is the derivative of y = xe^x with respect to x, which can be calculated as (x + 1)e^x.

Regarding the three equations involving the greatest integer function [x]:

(i) [p + 3] = [p] + 3: This equation states that the greatest integer less than or equal to p + 3 is equal to the greatest integer less than or equal to p plus 3. This equation is true for all real numbers p.

(ii) [p + q] = [p] + [q]: This equation states that the greatest integer less than or equal to p + q is equal to the sum of the greatest integers less than or equal to p and q. This equation is true for all real numbers p and q.

(iii) [5p] = 5 + [p]: This equation states that the greatest integer less than or equal to 5p is equal to 5 plus the greatest integer less than or equal to p. This equation is true for all real numbers p.