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

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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