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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 \( a^x \cos(x) \).

Assignment 1 contains calculus problems covering nth derivatives, Rolle's theorem, Lagrange's mean value theorem, expansions, and recursive formulas.

Explore solutions to IILM University calculus assignment problems involving nth derivatives of trigonometric and exponential functions, verification of Rolle's and Lagrange's theorems, and logarithmic expansions using Taylor series. Suitable for university-level calculus students.

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