Calculus
Board: JEE Advanced | Exam: JEE Advanced
JEE Advanced study notes for Calculus by Ajay Yadav. In-depth coverage of advanced topics and problem-solving techniques.
Key Concepts
Limits
L’Hopital (multiple applications). Series expansion (Taylor/Maclaurin). Standard limits using expansion. Evaluation using sandwich theorem.
Continuity and Differentiability
Continuity in an interval. Differentiability in an interval. Rolle’s and LMVT applications. Cauchy’s mean value theorem. Taylor’s theorem with Lagrange remainder.
Integration
Reduction formulas. Differentiation under integral sign (Leibniz). Gamma and Beta functions. Area, volume using integration. Length of curves.
Differential Equations
Higher order linear DE with constant coefficients. Homogeneous and particular integrals. Method of variation of parameters. Cauchy-Euler equation.
Important Formulas
| Taylor series | f(x)=Σfⁿ(a)(x-a)ⁿ/n! |
| Leibniz | d/dx∫anf(x,t)dt |
| 2nd order DE | ay''+by'+cy=0: substitute y=eᵉx |
Solved Examples
Example 1: Find limx→0 (sinx-x)/x³ using series.
Solution: sinx=x-x³/6+… (sinx-x)/x³ ≈ -1/6. -1/6.
Practice Questions
- Find limx→0 (ex-1-x)/x².
- Verify Rolle’s for f(x)=sinx on (0,&pi).
- Find ∫exsinxdx using reduction.
- Solve y”+y=0.
- Find area bounded by y²=4ax and x²=4ay.