The document you provided is a comprehensive question bank for a first-year B.Tech course on Mathematics I (Course Code: NYBS103) from Sandip University, covering several key topics:

Key Topics Covered:

1. Differential Calculus and Its Applications (Unit 1)

   • Finding nth-order derivatives of various functions (trigonometric, logarithmic, etc.)
   • Using Leibniz’s theorem for nth-order derivatives.
   • Expansion of functions using Maclaurin's and Taylor's theorems.
   • Limits and evaluations of complex expressions.

2. Multivariate Calculus (Unit 2)

   • Partial differentiation and higher-order derivatives.
   • Euler’s theorem for homogeneous functions.
   • Application of multivariable functions to real-world problems.
   • Verifying functional dependency of variables.

3. Applications of Multivariate Calculus (Unit 3)

   • Stationary points and error analysis in engineering problems.
   • Extreme values using Lagrange's multipliers.
   • Real-life applications like calculating errors in measurements, power, and volume.

Would you like detailed solutions or explanations for specific questions from this question bank? Let me know which area you'd like to dive deeper into!

Follow-up Questions:

1. Would you like help with nth-order derivatives or expanding functions?
2. Do you need assistance with finding extreme values using calculus techniques?
3. Should I explain partial differentiation or Euler’s theorem for homogeneous functions?
4. Would you like a step-by-step solution for a specific limit or expansion problem?
5. Do you want to explore Lagrange’s method for optimization problems?

Tip:

In multivariable calculus, remember that partial derivatives treat all other variables as constants, focusing only on one variable at a time!