Explanation

The Cauchy Mean Value Theorem states that if functions f(x) and g(x) are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that: (f(b) - f(a))g'(c) = (g(b) - g(a))f'(c)

In this case, f(x) = x², g(x) = x², and the interval is [1, 2]. Plugging in the values, we get: (4 - 1)(2c) = (4 - 1)(2c)

Simplifying, we get: c = 14/9

So, the value of c is 14/9.

Explanation

The degree of a function is the highest power of the variable(s) in the function. In this case, the function is: f(x, y) = √x + √y / (x + y)

The highest power of x and y in the function is 1/2, which comes from the terms √x and √y.

Explanation

• A cyclic group is a group that can be generated by a single element, i.e., every element in the group can be expressed as a power of that element.
• Cyclic groups are always Abelian, meaning that the group operation is commutative (i.e., the order of elements does not matter).

Explanation

A Bernoulli equation is a nonlinear differential equation that can be transformed into a linear equation by substituting v = y^(1-n), where n is the coefficient of the y² term (in this case, n = 2).

Explanation

¬p (read as "not p") represents the negation of statement p.

In this case, p is the statement "x + y = 3".

The negation of this statement would be "x + y ≠ 3", which means any situation where x + y is not equal to 3.

Explanation

This is a alternating series with terms of the form x^n / n, which converges for -1 < x ≤ 1.

Explanation

The p-series is a series of the form:

1 + 1/2^p + 1/3^p + 1/4^p + ...

The p-series is convergent if p > 1. If p ≤ 1, the series is divergent.

Explanation

The given differential form can be rewritten as:

xdy - ydx/x^2 = d(y/x) - (y/x^2)dx

Using the quotient rule for derivatives, we can simplify this to:

d(y/x) - (y/x^2)dx = d(ln(y/x))

Explanation

The correct answer is: n!.

If X has n elements, then the number of permutations (i.e., arrangements) that can be taken on X is given by the factorial function:

n! = n × (n-1) × (n-2) × ... × 2 × 1

This counts the number of ways to arrange the n elements in a particular order.

Explanation

The area of the circle with radius 10 cm is A₁ = π(10)² = 100π cm².

The area of the circle with radius 10.1 cm is A₂ = π(10.1)² = 102.01π cm².

The percentage change in area is:

((A₂ - A₁) / A₁) × 100%

= ((102.01π - 100π) / 100π) × 100%

= (2.01 / 100) × 100%

= 4%

So, the percentage change in the area is 4%.