Explanation

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

To find the smallest Bragg scattering angle, we can use Bragg's law:

2d sin(θ) = nλ

where d is the distance between the atomic planes, θ is the scattering angle, n is an integer (which is 1 for the smallest angle), and λ is the wavelength of the X-ray.

Given:

d = 0.3 nm

λ = 0.03 nm

Rearranging Bragg's law to solve for θ, we get:

sin(θ) = λ / (2d)

= 0.03 nm / (2 × 0.3 nm)

= 0.05

θ = arcsin(0.05)

≈ 2.9°

Explanation

Explanation

Explanation

Explanation