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Comprehensive Lecture Notes on Single-Variable and Multivariable Calculus: Concepts, Techniques, and Applications $15.49
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Comprehensive Lecture Notes on Single-Variable and Multivariable Calculus: Concepts, Techniques, and Applications
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Calculus is not just a collection of techniques; it is a fundamental language for understanding change and motion across various fields. From physics to economics, biology to engineering, calculus provides the tools necessary for modeling complex systems and solving real-world problems.

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  • October 3, 2024
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Lecture Notes on Calculus: Single-Variable and Multivariable Calculus
Introduction to Calculus

Calculus is a branch of mathematics that studies continuous change, primarily through the concepts
of differentiation and integration. It is divided into two main areas: single-variable calculus, which deals
with functions of one variable, and multivariable calculus, which extends these concepts to functions of
multiple variables.

Single-Variable Calculus

Key Concepts

1. Functions and Graphs

 A function f(x)f(x) assigns a unique output for each input xx.

 Understanding the graph of a function is crucial for visualizing its behavior.

2. Limits

 The limit of a function describes its behavior as the input approaches a certain value.

 Notation: lim⁡x→cf(x)=Llimx→cf(x)=L means as xx approaches cc, f(x)f(x) approaches LL.

3. Derivatives

 The derivative represents the rate of change of a function.

 Notation: f′(x)f′(x) or dfdxdxdf.

 Fundamental rules include:

 Power Rule: ddx(xn)=nxn−1dxd(xn)=nxn−1

 Product Rule: (uv)′=u′v+uv′(uv)′=u′v+uv′

 Quotient Rule: (uv)′=u′v−uv′v2(vu)′=v2u′v−uv′

 Chain Rule: (f(g(x)))′=f′(g(x))g′(x)(f(g(x)))′=f′(g(x))g′(x)

4. Applications of Derivatives

 Finding local maxima and minima using critical points.

 Analyzing the concavity and inflection points through the second derivative test.

5. Integrals

 The integral represents the accumulation of quantities, such as area under a curve.

 Notation for definite integrals:
∫abf(x)dx∫abf(x)dx

 Fundamental Theorem of Calculus connects differentiation and integration:
∫abf′(x)dx=f(b)−f(a)∫abf′(x)dx=f(b)−f(a)

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