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I see that you have uploaded an assignment containing several mathematical problems. Let's go through each of them step by step. From your file, it appears you would like to solve the following problems:

1. Evaluate the truth of several statements regarding functions and derivatives.
2. Differentiation and integration of specified functions.
3. Analysis and tracing of a quadratic curve.
4. Calculation involving curve lengths and angles between curves.
5. Integral evaluation and approximation using Simpson's method.
6. Derivative calculation of a logarithmic function related to an inverse trigonometric function.
7. Area calculation under a specific curve.

Let's start with the first question:

**1. State whether the following statements are True or False? Justify your answers with the help of a short proof or a counter example:**

   a) The function \( f(x) = \cos x + \sin x \) is an odd function.

For a function to be considered odd, it must satisfy the condition \( f(-x) = -f(x) \) for all \( x \). We compute:
\[ f(-x) = \cos(-x) + \sin(-x) = \cos x - \sin x \]
\[ -f(x) = -(\cos x + \sin x) = -\cos x - \sin x \]

Comparing these, we see:
\[ f(-x) \neq -f(x) \]
Therefore, \( f(x) = \cos x + \sin x \) is **not** an odd function. This statement is **False**.

If you'd like me to continue with the other parts of question 1 or move to another question, let me know!

Would you like a detailed analysis on the next part or another question answered? Here are some related questions we can explore next:

1. What defines a function as odd or even?
2. Can you show how to verify if a function is differentiable in a given interval?
3. What are points of inflection and how can you identify them in a function?
4. How do you determine whether a function is increasing or decreasing in a specified interval?
5. Could you explain the fundamentals of integral calculus?

Tip: Always check the symmetry properties of trigonometric functions when analyzing if a combination of them results in an even or odd function.

Solve the given mathematical problems: State whether the following statements are True or False? Justify your answers with a short proof or counterexample.

Explore key calculus problems, including determining whether functions are odd, evaluating integrals, differentiating functions, and analyzing curve properties. This set of problems tests knowledge in differentiation, integrals, and curve tracing, suitable for first-year undergraduate students.

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