Explanation

In a sentence, the main verb and the direct object usually come together without any formal separator.

Example: "She reads books." — "reads" (main verb) is directly followed by "books" (direct object).

Explanation

The rank of a matrix is the number of non-zero rows in its row echelon form.

It represents the maximum number of linearly independent rows or columns.

Explanation

The pattern of the sequence appears to be formed by adding consecutive increments:

1 + 4 = 5

5 + 9 = 14

14 + 16 = 30

30 + 25 = 55

55 + 36 = 91

The next term in the sequence would be 91.

Explanation

The formula ∂f/∂x = lim(h → 0) [f(x + h, y) - f(x, y)]/h represents the partial derivative of f with respect to x, and it's evaluated at the point (x, y).

The answer is x at (x, y).

Explanation

The Maclaurin series expansion of cos(x) is derived from its Taylor series expansion around x = 0:

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

This series alternates signs and involves even powers of x, with factorials in the denominators.

The answer is cosx = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Explanation

To find ∫ln(x)dx, we use integration by parts:

Let u = ln(x) and dv = dx

Then du = 1/x dx and v = x

∫ln(x)dx = uv - ∫v du

= x_ln(x) - ∫x * 1/x dx

= x_ln(x) - ∫1 dx

= x*ln(x) - x + C

The answer is xlnx – x + c.

Explanation

Lagrange's Mean Value Theorem is a special case of Cauchy's Mean Value Theorem when one function is f(x) and the other is g(x) = x.

Cauchy’s theorem is more general and applies to two functions.

Explanation

Given f(x) = x³ - (1/x³):

f(1/x) = (1/x)³ - (1/(1/x)³) = (1/x³) - x³

Now, let's calculate f(x) + f(1/x):

f(x) + f(1/x) = (x³ - 1/x³) + (1/x³ - x³)

= x³ - x³ - 1/x³ + 1/x³

= 0

Explanation

Given the function f(x) = 3x - 2

This is a linear function with a slope of 3. Since it's linear and defined for all real numbers x, its range is all real numbers.

As x varies from -∞ to ∞, f(x) also varies from -∞ to ∞.

Explanation

Most questions in English use an auxiliary verb like do, does, did, is, are, will, etc.

Example: "Do you like tea?" — "Do" is the auxiliary verb used to form the question.