Mathematics Archives

January 12, 2025 Calculus, Mathematics FC

BASIC to INTERMEDIATE problems for Limits, Derivatives, and Integrals.

1. Limit Exercises Exo 1: Basic Limit Calculation Problem: \[ \lim_{x \to 0} \frac{\sin(x)}{x} \] Solution: This is a standard limit which is part of the limit formulas: \[ \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \] Exo 2: Using L’Hopital’s Rule Problem: \[ \lim_{x \to 0} \frac{1 – \cos(x)}{x} \] Solution: This is a \(\frac{0}{0}\) […]

January 12, 2025 Calculus, Mathematics FC

Most useful calculus formulas from BASIC to ADVANCED

Basic Calculus Formulas: Limits: \(\lim_{x \to a} f(x) = L\), where \(f(x)\) approaches \(L\) as \(x\) approaches \(a\). \(\lim_{x \to a} \frac{f(x) – f(a)}{x – a} = f'(a)\) (Definition of the derivative). \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\). \(\lim_{x \to 0} \frac{1 – \cos(x)}{x} = 0\). Derivative Formulas: Power Rule: \(\frac{d}{dx}[x^n] = n x^{n-1}\). Constant […]

January 2, 2025 Algebra, Mathematics FC

Simplifying algebraic expressions

Exo 1 Simplify the following expressions with positive exponents. \((5x^{-7}y^3)^{-5}\) Solution: Step 1: Apply the exponent to each term inside the parentheses Use the property \( (ab)^n = a^n \cdot b^n \) to distribute the exponent \(-5\) to each factor: \[ (5x^{-7}y^3)^{-5} = 5^{-5} \cdot (x^{-7})^{-5} \cdot (y^3)^{-5} \] Step 2: Simplify each term \( […]

December 27, 2024 Algebra, Mathematics FC

Most important and commonly used algebraic formulas

1. Basic Arithmetic Formulas Addition and Subtraction: \( a + b = b + a \) (Commutative property) Multiplication and Division: \( a \times b = b \times a \) (Commutative property) \( \frac{a}{b} \) (Quotient) 2. Exponents and Powers Power of a product: \[ (ab)^n = a^n \times b^n \] Power of a quotient: […]

December 27, 2024 Algebra, Mathematics FC

Word Problem : Book

Find the total number of pages in the book: Marie has finished \( \frac{3}{4} \) of a book. She found that she has read 106 more pages than the number of pages left to read. What is the total number of pages in the book? Solution: Let \(x\) be the total number of pages in […]

December 27, 2024 Algebra, Mathematics FC

Word Problem: Salesman

Word problem 1: A salesman sold twice as many pears in the afternoon as he did in the morning. If he sold 360 kilograms of pears that day, how many kilograms did he sell in the morning and how many in the afternoon? Solution Let’s break the problem down step by step: Let \( x […]

Category: Mathematics

BASIC to INTERMEDIATE problems for Limits, Derivatives, and Integrals.

Most useful calculus formulas from BASIC to ADVANCED

Simplifying algebraic expressions

Most important and commonly used algebraic formulas

Word Problem : Book

Word Problem: Salesman