Fundamental Theorem Of Calculus

The correct answer is (C) :
( Equation of normal to x² = 4ay in slope form)through(1,-1)


Slope of normal

To solve the problem, we need to evaluate the following limit:

Given that and is differentiable, we will use L'Hôpital's Rule. L'Hôpital's Rule is applicable since both the numerator and the denominator approach 0 as .

First, we differentiate the numerator and the denominator with respect to :

• The derivative of the numerator with respect to is , according to the Fundamental Theorem of Calculus.
• The derivative of the denominator is .

Using L'Hôpital's Rule, we have:

Substituting , we focus on the expression at :

Notice that the higher order terms vanish faster than , hence:

Thus, .

Next, we find :

Hence, the correct answer is 2.

We start with the given series:

Simplify the expression:

Let’s use the substitution :

Substituting these values, we get:

Now reverting back to :

Next, apply the limits:

Thus,

To solve for given the function with the conditions , , and , we proceed as follows:

1. First, compute :

Solving this gives us: (Equation 1)

1. Next, compute the first derivative :

Then, at ,

(Equation 2)

1. Now, compute the second derivative :

Then, at ,

This simplifies to:

(Equation 3)

1. We now have the following system of equations:
   • Equation 1:
   • Equation 2:
   • Equation 3:
2. Subtract Equation 1 from Equation 2 to eliminate :

(Equation 4)

1. Subtract Equation 4 from Equation 3 to find :

1. Substitute into Equation 3 to find :

1. Finally, substitute and into Equation 1 to find :

1. Now compute :

Thus, the value of is 51.

Step 1. Given . Substitute , , .

Step 2. Calculate :

Step 3. Evaluate :

We start by differentiating the inequality:
.
Rewriting it as:
, this implies is non-decreasing. Consequently, or .
Now, compute the sum:
.
The sum becomes:

Using ,
we get:

Adding, , gives us:
.
Thus, , and falls within the range .