Explanation

∫01​(2x+1)dx

Step 1: Integrate the function

We integrate each term separately:

$$int (2x + 1), dx = int 2x, dx + int 1, dx = x^2 + x + C$$

Step 2: Apply the limits from 0 to 1

$$left[ x^2 + x right]_0^1 = (1^2 + 1) - (0^2 + 0) = 1 + 1 = 2$$

Explanation

This is a product of two functions, so we apply the product rule:

d​[u⋅v]/dx=u′⋅v+u⋅v′

Here,

• u=x^2, sou′=2x

• $v = sin(x)$, so $v' = cos(x)$

Now apply the rule:

$$f'(x) = 2x cdot sin(x) + x^2 cdot cos(x)$$

Explanation

To integrate $x^2$, use the power rule of integration:

$$int x^n dx = frac{x^{n+1}}{n+1} + C$$

Here, $n = 2$, so:

$$int x^2 dx = frac{x^{2+1}}{2+1} + C = frac{x^3}{3} + C$$2+1/2+1​+C

=x^3​/3+C

Explanation

 The correct command in MATLAB to solve the linear system Ax = b is B) Ab. 

The backslash operator `` in MATLAB performs matrix division, which efficiently solves systems of linear equations in the form Ax = b. 

Explanation

The command to obtain a numerical solution in Mathematica is NSolve. 

NSolve is specifically designed for finding numerical solutions to equations or systems of equations.

Explanation

An uncountable set (like real numbers) has more elements than can be listed.

The union of a countable and uncountable set is always uncountable

A ∪ B is uncountable 

Explanation

A metric space is a set with a distance function (metric) that meets three properties: 

1. Non-negativity: Distances are always non-negative.  

2. Symmetry: Distance from A to B equals distance from B to A.  

3.  Triangle Inequality: For any points A, B, and C, the distance from A to C is less than or equal to the sum of the distances from A to B and B to C.  

Explanation

The rank of a tensor is the number of indices; R(i,j,k) has three indices

The rank of the tensor is equal to the number of indices.

Therefore, the rank of R(i,j,k) is 3.

Explanation

  f(x) = |x| has a cusp at x = 0, making it non-differentiable there.

A function is differentiable at a point if its derivative exists at that point. 

The absolute value function, (f(x)=|x|), has a sharp corner at (x=0). 

Functions with sharp corners are not differentiable at those points.

Explanation

The only countable set is the set of all rational numbers

A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers.

Rational numbers can be expressed as a fraction frac{p}{q}, where p and q are integers and q is not zero.