Fundamental Theorem Of Calculus

Step 1: Expand the integral near We have: Use the Taylor expansion: So, Step 2: Expand Step 3: Combine expressions inside the limit Step 4: Apply the limit For the limit to be finite, the coefficient of must vanish: Now substitute: Now plug in : Final Answer:

Step 1: Recall that if , then by the chain rule, Using the formula for derivative of inverse function, Thus,
Step 2: We need to find . First find . Given, Set , So,
Step 3: Calculate . Using chain rule, Evaluate at :
Step 4: Calculate . Differentiating, Thus, Evaluate at : Step 5: Finally, compute Since this value is not listed among the options, there might be a typo in the question or options. But mathematically, the value of the derivative is .

Step 1: Expand the integral near We have: Use the Taylor expansion: So, Step 2: Expand Step 3: Combine expressions inside the limit Step 4: Apply the limit For the limit to be finite, the coefficient of must vanish: Now substitute: Now plug in : Final Answer:

Step 1: Given
Step 2: Expression for is:
Step 3: We need to find: Let’s define: Differentiate using L'Hôpital's Rule since the limit is of the form .
Step 4: Apply L'Hôpital’s Rule: Now, differentiate : Let Then Compute derivatives: - - Evaluate all at : Substitute and simplify: Final value: Now substitute , compute and simplify to get: