MTH 102 Test Compilation 4

Study Summary: MAT 122 Calculus Examination (2023/2024 Session)

This document outlines the key topics and concepts tested in the MAT 122 Calculus examination from the University of Lagos. The exam covers a broad spectrum of differential and integral calculus, along with limits, series, and their applications.

I. Differential Calculus

• Definition of the Derivative: Understanding the limit definition of the derivative (e.g., `dy/dx = lim h→0 [g(x+h) - g(x)]/h`).
• Basic Differentiation Rules:
  • Product Rule: `d/dx [f(x)g(x)]` (e.g., `d/dx [7x^3 tan 3x]`).
  • Quotient Rule: `d/dx [f(x)/g(x)]` (e.g., `d/dx [(x^2+3)/(2x-7)]`, and recalling the formula itself).
  • Chain Rule: For composite functions (e.g., `d/dx [cot(5x^2+3)]`, `d/dx [√(7-2x^3)]`, `d/dx [e^(f(x))]`).
• Differentiation of Common Functions:
  • Polynomial functions.
  • Logarithmic functions (e.g., `d/dx [ln(f(x))]`).
  • Exponential functions (e.g., `d/dx [e^(ax)]`).
  • Trigonometric functions (e.g., `d/dx [cot(u)]`, `d/dx [sin^2(u)]`, `d/dx [tan(u)]`, `d/dx [cosec(u)]`).
  • Functions involving radicals (e.g., `d/dx [√(f(x))]`).
• Implicit Differentiation: Finding `dy/dx` for equations where `y` is not explicitly defined in terms of `x` (e.g., `cos(x^2+2y) + xe^y = 1`, `y = cos(5x-3y)`, `e^(xy) = x-y`, `tan(x^2y^4) = 3x+y^2`).
• Partial Differentiation:
  • First-order partial derivatives (e.g., `∂f/∂x` for `e^(-x)sin(x+y)`).
  • Higher-order partial derivatives (e.g., `∂^3u/∂x∂y∂z` for `u(x,y,z) = e^(xyz)`, `∂^2u/∂y∂x` for `u(x,y) = x^y`).
• Applications of Differentiation:
  • Finding extrema (minimum/maximum values) of functions (e.g., for `y = 2x^3 - 21x^2 + 36x - 20`, `f(x) = x^3 + 2x^2 - 4x + 6`, `f(x) = p sin x + sin 3x / 3`).

II. Integral Calculus

• Indefinite Integrals:
  • Basic forms (e.g., `∫(4x^3+3x-1)dx`, `∫(e^(3x)+8x)dx`).
  • Logarithmic form (e.g., `∫ (2x)/(2-3x) dx`, `∫ 1/(x-1) dx`).
  • Trigonometric forms (e.g., `∫ sin^8 x cos x dx`, `∫ sec^2 x tan x dx`).
  • Exponential forms (e.g., `∫ x e^(x^2-4) dx`).
• Integration Techniques:
  • Substitution Method: (e.g., `∫ (ln x)/x dx`, `∫ (3x)/(4+9x^2) dx`, `∫ x e^(x^2-4) dx`).
  • Integration by Parts: (e.g., `∫ ln x dx`).
  • Integration of Rational Functions: Using long division (e.g., `∫ (x^2-1)/(x+3) dx`) and partial fractions (e.g., `∫ 1/(x^2-x) dx`).
  • Power Reduction Formulas: For trigonometric integrals (e.g., `∫ sin^2 x dx`, `∫ cos^4 x dx`).
• Reduction Formulas: Specifically for `∫ sec^n x dx` (as identified in Q32) and implicitly for `∫ cos^n x dx` (Q43) and `∫ sin^n x dx`.
• Definite Integrals:
  • Wallis Formula for `∫[0,π/2] cos^n x dx` and `∫[0,π/2] sin^n x dx`.

III. Limits

• Evaluating Algebraic Limits: Direct substitution after simplification (e.g., `lim x→2 (8x^3-4)/(2x-5)`, `lim x→3 (4x-2x^2)/(7x^3-8)`).
• Limits Involving Indeterminate Forms: Implies knowledge of techniques like L'Hopital's Rule (e.g., `lim x→0 (x^5)/(1-cos 3x)` which is `0/0`).

IV. Series Expansions

• Maclaurin Series: Identifying the nth term for standard functions (e.g., `cos x`, `sin 2x`, `e^(-3x)`).

V. Function Analysis and Applications

• Function Evaluation: Evaluating functions and composite functions (e.g., `f(f(3))`, `f(5)` given `f(u-2)`).
• Inverse Functions: Finding the inverse `f^(-1)(x)` for a given function.
• Composite Functions: Evaluating `g(f(x))`.
• Continuity: Determining parameters for a piecewise function to be continuous.

VI. Conceptual Understanding and Formula Recall

• Identifying correct and incorrect mathematical statements or formulas related to differentiation and integration (e.g., Quotient Rule, product rule for differentiation).