Explanation

Average is equal to sum of all numbers and divided by Total numbers

= 9+18+27+36+45 = 135/5

=27

Explanation

Explanation

To form a differential equation from a given function containing arbitrary constants:

• If the function has 'n' arbitrary constants, differentiate the equation 'n' times with respect to the independent variable.
• This process removes the arbitrary constants, leaving a differential equation.

Explanation

Explanation

To find the value of c that satisfies the Mean Value Theorem, we need to find the derivative of the function f(x) = x^2 + 2x + 1.

f'(x) = 2x + 2

The Mean Value Theorem states that there exists a point c in the interval [1, 2] such that:

f'(c) = (f(2) - f(1)) / (2 - 1)

First, let's calculate f(2) and f(1):

f(2) = 2^2 + 2(2) + 1 = 9

f(1) = 1^2 + 2(1) + 1 = 4

Now, we can calculate the slope:

(f(2) - f(1)) / (2 - 1) = (9 - 4) / 1 = 5

Now, we set f'(c) equal to 5:

2c + 2 = 5

Solving for c, we get:

2c = 3

c = 3/2

Since c = 3/2 is between 1 and 2, it satisfies the Mean Value Theorem.

Explanation

Explanation

Explanation

Explanation

Explanation

• 1. The functions f(x) and g(x) are continuous on [a, b].
• 2. The functions f(x) and g(x) are differentiable on (a, b).
• 3. g'(x) ≠ 0 for any x in (a, b).