Calculus

The series is infinite, implying .
Assume a geometric series or misinterpretation; let’s try . Then: The given equation suggests a different function. Assume the problem intends a recursive or specific form. Let’s derive for a general . Differentiate : This is inconsistent, so assume a finite sum or typo. For , check at : Try the given derivative: The series definition seems incorrect; assuming , the result holds at .
Answer: .

Convert . Since radians, radians. Use linear approximation: Answer: .

Use integration by parts: Let , . Then , . For , let , , , so: Answer: .

The curve intersects at . Area between and from to : Since is even: Answer: square units.

A function is strictly increasing if for all .
Since and , for all .
Thus, is strictly increasing.

Use identity: .
Use : Evaluate: Answer: .

The curve (since ). Area between , , X-axis, and curve: Answer: square units.

Use substitution: Let , so , and: Alternatively, the result can be written as:

Rewrite: .
Substitute , : Rewrite: .
Adjust constants for options: Compute derivative of (a): Correct integral: . Thus: Check options: Option (a) seems to have a typo; correct form should yield .

Differentiate implicitly: .
Left: . Right: .
At , find y: .
Substitute , :