Measures Of Dispersion

The Mean Value Theorem (MVT) states that if a function is continuous on the closed interval and differentiable on the open interval , then there exists at least one point such that: Here, on the interval . Therefore, we need to compute the derivative of and apply the MVT.
Step 1: Compute and
Step 2: Apply the Mean Value Theorem We apply the MVT to the given interval : Substitute the values of and :
Step 3: Differentiate The derivative of is: So, we have:
Step 4: Solve for Finally, solving for :

The correct option is (A) :

We are asked to find the limit: First, expand the terms in the numerator: Now, substitute these into the original expression: Simplify the numerator: Factor out from the numerator: Cancel in the numerator and denominator: Now, take the limit as :

The correct option is (B) :

We are given the curve and the point of tangency . The goal is to find the equation of the normal to the curve at this point and determine where it intersects the y-axis.

The derivative of is: At , the slope of the tangent is: Thus, the slope of the tangent line at is .

The slope of the normal line is the negative reciprocal of the slope of the tangent. So, the slope of the normal line is:

The equation of the normal line can be written as: Substituting , , and : Simplifying: \textbf{Step 4: Find the y-intercept} To find the point where the normal line intersects the y-axis, set : Thus, the normal line intersects the y-axis at .

The correct option is (B) :

We are given the limit: First, factor the denominator: Now, substitute into the denominator: For the limit to be finite, the numerator must also approach zero at , otherwise, the limit would be infinite. Therefore, substitute into the numerator: For the numerator to approach zero at , we set: Solving for :

The correct option is (D) :

We are tasked with evaluating the following limit:
Step 1: Simplify the expression First, notice that as , both and approach 0, making the limit of the form , which suggests the use of L'Hopital's Rule. However, before using L'Hopital's Rule, let's rewrite the expression to simplify the computation. We know the following standard limit:
Step 2: Apply the standard limit We can split the given expression as follows: Now, evaluate each part separately: 1. (since and we have instead of ), 2. .
Step 3: Combine the results So the overall limit becomes:

The correct option is (D) :

We are tasked with finding the limit:
Step 1: Apply Taylor series expansions for and
The Taylor series expansion of around is: The Taylor series expansion of around is: Thus, we can approximate: Simplifying: Step 2: Simplify the denominator The denominator is , which is already in a simplified form.
Step 3: Substitute the approximations into the original expression Substituting into the expression: For small , the higher-order terms can be neglected. Therefore, the expression simplifies to: Factor out from the denominator: ### Step 4: Take the limit as As , the expression simplifies to:

The correct option is (E) :

The coefficient of variation (C.V.) is defined as the ratio of the standard deviation ( ) to the mean ( ) of the distribution, multiplied by 100: We are given:


Substitute these values into the formula for C.V: Solving for (standard deviation):

The correct option is (D) :