You've uploaded an assignment image containing calculus-related problems. I’ll help break down the questions and work through the solutions. Here is a brief overview of the types of problems listed in the assignment:

Section A: Short Answer Type Questions

1. nth derivative of $a^x \cos(x)$
2. Find $y_n$ if $y = \tan^{-1}\left(\frac{1}{x}\right)$
3. Find $y_n$ if $y = x^3 \sin(x)$ at $x = 0$
4. Find $y_n$ if $y = e^{-ax} \sin(2x)$
5. Examine if Rolle’s theorem is applicable for $f(x) = \sec(x)$ on $[0, 2\pi]$
6. Expand $\ln(x)$ in powers of $(x - 1)$ up to the third degree term
7. Examine if Lagrange's mean value theorem is applicable for $f(x) = \frac{1}{x^2 - 1}$ on $[1, 2]$

Section B: Long Answer Type Questions

1. Expand $4x^2 + 7x + 5$ in powers of $(x - 3)$
2. If $y = \ln\left(x + \sqrt{1 + x^2}\right)$, prove a given recursive relation for $y_n$
3. If $y = x^n \ln(x)$, show a relation for $y_n$
4. Verify Rolle’s theorem for specific functions:
   • $x^2 - 6x + 8$ on $[2, 4]$
   • $e^x \sin(x)$ on $[0, \pi]$

Would you like me to start with any specific question? I can work through solutions one at a time or provide assistance with a particular concept.

Let me know how you'd like to proceed! Here are 5 questions based on this material:

1. What is the general method to find the nth derivative of a product like $a^x \cos(x)$?
2. How does Rolle’s theorem apply to periodic functions like $\sec(x)$?
3. How can Taylor series expansion be used to approximate $\ln(x)$?
4. What is the significance of the recursive relation for $y_n$ in logarithmic functions?
5. How is Lagrange’s Mean Value Theorem verified for rational functions?

Tip:

For the nth derivative problems, applying Leibniz's rule for differentiating products might simplify your work when dealing with functions like $a^x \cos(x)$.