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.
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: limx→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.
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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