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Overview of Calculus: Key Concepts, Rules, and Example Problems, Study Guides, Projects, Research of Mathematics

A comprehensive introduction to calculus, a fundamental branch of mathematics. It covers the main concepts of limits, functions, derivatives, integrals, and infinite series. The document delves into the two main branches: differential calculus and integral calculus, and presents key definitions, notations, and basic rules for derivatives and integrals. It also includes example problems for finding derivatives and evaluating integrals. This resource is ideal for students studying informatics engineering at gunadarma university in 2024.

Typology: Study Guides, Projects, Research

2023/2024

Available from 05/26/2024

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Calculus
Basic Mathematics
Informatics Engineering
Gunadarma University
2024
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Calculus

Basic Mathematics

Informatics Engineering

Gunadarma University

Overview of Calculus Calculus is a branch of mathematics focused on the concepts of limits, functions, derivatives, integrals, and infinite series. It is divided mainly into two branches:

  1. Differential Calculus : Concerns the concept of a derivative, representing rates of change and slopes of curves.
  2. Integral Calculus : Concerns the concept of an integral, representing areas under curves and accumulation of quantities. **Key Concepts
  3. Limits**
  • Definition : The value that a function approaches as the input approaches some value.
  • Notation : limx→af(x)=L 2. Derivatives
  • Definition : A measure of how a function changes as its input changes. It represents the slope of the function at any point.
  • Notation : f′(x) or 𝑑 𝑑𝑥 f(x)
  • Basic Rules : o Power Rule: 𝑑 𝑑𝑥

xn^ = nxn-^1

o Sum Rule: 𝑑 𝑑𝑥 [f(x) + g(x)] = f′(x) + g′(x) o Product Rule: 𝑑 𝑑𝑥 [f(x) g(x)] = f′(x)g(x) + f(x)g′(x)

2. Evaluating an Integral Problem : Evaluate the integral ∫ (4x^3 − 2 x + 6) dx Solution : 1. Integrate each term separately. 2. ∫ 4x^3 dx − ∫ 2x dx + ∫ 6 dx 3. Using the power rule for integrals: o ∫ 4x^3 dx = 4 ⋅ 𝑥^3 +^1 3 + 1 = x^4 o ∫ 2x dx=2 ⋅ 𝑥^1 +^1 1 + 1 = x^2 o ∫ 6 dx = 6x 4. Combine the results and add the constant of integration C: o x^4 – x^2 + 6x + C