Questions
1–2 questions per paper
Difficulty
Medium
Importance
Moderate yield for HPCL/NTPC/BHEL
Overview
Calculus for PSU exams focuses on the practical application of derivatives and integrals to solve engineering problems. It is a foundational pillar that tests your proficiency in optimization, area calculation, and multivariable functions. Mastering these concepts ensures you can solve analytical problems appearing in HPCL, NTPC, and BHEL examinations quickly.
Mean Value Theorems
Mean Value Theorems provide a bridge between local derivative behavior and global function behavior over an interval. These are frequently tested via conceptual MCQs regarding the existence of points where slopes match the average rate of change.
- Rolle's Theorem: f(a) = f(b) implies f'(c) = 0 for some c in (a, b)
- Lagrange's Mean Value Theorem: f'(c) = [f(b) - f(a)] / (b - a)
- Cauchy's Mean Value Theorem: f'(c)/g'(c) = [f(b) - f(a)] / [g(b) - g(a)]
Definite & Improper Integrals
Definite integrals are essential for finding area under curves and physics-based quantities like work or center of gravity. Improper integrals, specifically those involving infinite limits or singular points, appear in signal processing contexts in electrical engineering papers.
- Leibniz Rule for Differentiation under Integral Sign
- Gamma Function: Integral from 0 to infinity of x^(n-1)e^(-x)dx = Γ(n)
- Beta Function: Integral from 0 to 1 of x^(m-1)(1-x)^(n-1)dx
- Wallis's Formula for trigonometric integrals
Partial Derivatives, Maxima & Minima
Partial derivatives are used for multivariable functions, essential for field theory and thermodynamics. Maxima and Minima questions often involve finding critical points of surfaces using the Hessian matrix.
- Euler's Theorem on Homogeneous Functions
- Total Derivative: dz = (∂z/∂x)dx + (∂z/∂y)dy
- Maxima/Minima condition: f_xx * f_yy - (f_xy)^2 > 0 for local extrema
- Lagrange Multipliers for constrained optimization
Multiple Integrals
Double and triple integrals are used to compute volume and mass distribution. Changing variables to Polar, Cylindrical, or Spherical coordinates is the standard technique to simplify these calculations during exams.
- Change of order of integration for double integrals
- Polar coordinates conversion: dx dy = r dr dθ
- Dirichlet's Integral theorem for evaluating triple integrals
- Jacobian transformation: J = ∂(x,y)/∂(u,v)
Formula Sheet
Rolle's Theorem: f'(c) = 0
Lagrange MVT: f'(c) = (f(b)-f(a))/(b-a)
Leibniz Rule: d/dx [integral from g(x) to h(x) of f(t,x)dt] = f(h(x),x)h'(x) - f(g(x),x)g'(x) + integral from g(x) to h(x) of ∂f/∂x dt
Euler's Theorem: x(∂u/∂x) + y(∂u/∂y) = nu for a homogeneous function of degree n
Jacobian: J = det | ∂x/∂u ∂x/∂v | / | ∂y/∂u ∂y/∂v |
Gamma Function: Γ(n+1) = nΓ(n) = n!
Beta-Gamma Relation: B(m,n) = Γ(m)Γ(n) / Γ(m+n)
Exam Tip
When solving multiple integrals, always sketch the region of integration to easily identify the limits and determine if swapping the order of integration simplifies the integrand.
Common Mistakes
- Neglecting to check the conditions of Rolle's theorem (f(a)=f(b) and differentiability) before applying it.
- Forgetting the Jacobian factor (like 'r' in polar or 'r^2 sin(θ)' in spherical coordinates) during multiple integration changes.
- Incorrectly identifying the nature of critical points by miscalculating the second-order partial derivative determinants.
More Revision Notes
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