Calculus PYQ Theorems & Theory

math-1109 MATH-1109 Differential and Integral Calculus

Part 1: Differential Calculus

Definitions and Concepts

• Limits:
  • Define limit of a function.
  • Discuss the distinction between $\lim f(x)$ and $f(a)$. Describe the nature of the function.
  • What are necessary and sufficient conditions for existence of limit of a function?.
• Continuity & Differentiability:
  • Define limit and continuity of a function.
  • Define continuity and differentiability of a function.
  • What is meant by the continuity of a function?.
  • What do you mean by continuity of function at a particular point and at an interval? Also, define the differentiability of a function at that cases.
• Geometric Definitions:
  • Define tangent and normal.
  • Define sub-tangent and subnormal.
  • Define radius of curvature.
  • What is meant by curvature?.
  • Define asymptote, radius of curvature, subtangent and subnormal.
  • Define asymptotes.
• Multi-variable & Other Concepts:
  • Define homogeneous function.
  • Give an example of homogeneous function.
  • Define Jacobian of two variables.
  • Define maximum of a function.
  • What is the physical meaning of $d^{2}y/d{x}^2$?.

Statements of Theorems

• Leibnitz’s Theorem:
  • State Leibnitz’s theorem.
  • Write the statement of Leibnitz’s theorem.
• Rolle’s & Mean Value Theorems:
  • State Rolle’s theorem.
  • State Leibnitz’s theorem and Rolle’s theorem.
  • State mean value theorem.
  • State Lagrange’s mean value theorem.
• Euler’s & Taylor’s Theorems:
  • State Euler’s theorem.
  • Write the statement of Euler’s theorem on homogeneous function.
  • State Euler’s theorem for a $n$-th degree homogeneous function of three variables.
  • Define tangent and normal state Taylor’s theorem.

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Part 2: Integral Calculus

Definitions and Concepts

• Integration Basics:
  • What is meant by integration?.
  • What is integration? Write the name of four applications of integration.
• Definite Integrals:
  • Write down some properties of definite integrals.
  • Write down some general properties of definite integral.

Formulas and Functions (Statements only)

• Wallis’s Formula:
  • State Wallis formula.
• Gamma and Beta Functions:
  • Define Gamma and Beta function.
  • Define Beta and Gamma functions. Write down the relationship between them.
  • Define Gamma and Beta function. Write down the different form of Gamma function.