Explanation

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

Explanation

To evaluate the limit, we can start by rewriting the expression as:

lim (sin(2x) + a*sin(x)) / x^3

As x approaches 0, the sine functions can be approximated by their arguments, so we get:

lim (2x + a*x) / x^3 = lim (2 + a) / x^2

For this limit to exist, the numerator must be 0, so we set 2 + a = 0, which gives a = -2.

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

To solve the inequality x−3/x+7 < 0, we need to find where the expression is negative.

We first determine the critical points by setting the numerator and denominator equal to zero:

x−3 = 0 gives x = 3, and x+7 = 0 gives x = −7.

Now, we create a sign chart with these critical points and test intervals to see where the expression is negative:

Test x = −8 (a number less than -7): −8−3/-8+7 = −11/-1 ​= 11>0 (positive)

Test x = 0 (a number between -7 and 3): 0−3/0+7 = −3/7 <0 (negative)

Test x = 4 (a number greater than 3): 4−3/ 4+7 = 1/11 >0 (positive)

From this, we see the expression is negative when x is between -7 and 3, so the solution is (−7,3).

Explanation

• The derivative of a^u with respect to x is a^u(ln(a))(du/dx).
• In this case, a = 3 and u = 2x.
• So, the derivative is: d(3^2x)/dx = 3^2x(ln(3))(2) = 2(3^2x)ln(3)

Explanation

• The formula 5*n!/(n-2) can be expanded as:
• 5 * (n * (n-1) * (n-2) * ... ) / (n-2)
• The (n-2) terms cancel out, leaving: 5 * n * (n-1)
  

Explanation

The given equation is a permutation of the numbers, where [E] = {4, 1, 2} represents the set of values.

The correct permutation is E = (2, 1, 4), which means:

• E1 = 2
• E2 = 1
• E3 = 4

Explanation

The permutation of 1, 3, 5, and 6 is 1356.

Explanation

The degree of the polynomial x^2yz^2+xyz+2 is 3.

Explanation

The expression √8-√2 simplifies to √2.