Calculus

Step 1: Analyze the function.
We are given the function in the interval . First, let’s express in a more convenient form. We know that: Using the angle addition identity: Step 2: Find the maximum and minimum values.
The maximum and minimum values of occur at , so: Step 3: Determine the values within the interval.
In the interval , the function reaches its maximum at and its minimum at . However, since the interval is only , the minimum is (since ). Thus, the maximum value is and the minimum value is . Final Answer:

Step 1: Differentiate the function.
We are given a recursive exponential function. To differentiate it, we use the chain rule. Let , so . Step 2: Apply the chain rule.
Using the chain rule, we differentiate with respect to , and we arrive at the result . Final Answer:

Step 1: Use the residue theorem.
We are asked to evaluate the contour integral , where is the circle . The integrand has singularities at and . The contour only encloses the singularity at . Step 2: Find the residue.
We need to find the residue of at . Factorizing the denominator: The residue at is given by: Since , we evaluate the integral using the residue theorem. Final Answer: