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Calculus I Limits and Derivatives

Sunil Karna

Chapter 2 Introduction to Calculus

Calculus is a study of change. It gives tools to describe and analyze change with precision. For example, how fast something is changing at a specific instant or how much has total change occured over a period of time. Calculus has two parts: Differential calculus and Integral calculus. Differential calculus describes rate of change which can be determined from slope of a curve and integral calculus describes total accumulation over time which can be determined by area under the curve. Calculus studies properties like:
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Limits: Behavior of \(f(x)\) as \(x\) approaches a certain value.
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Derivatives: Rate of change of \(f(x)\) with respect to \(x.\)
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Integrals: Accumulation of quantities represented by \(f(x).\)
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Example 2.0.1.

for
\begin{equation*} f(x) = x^2 \end{equation*}
Domain: all real numbers \(\mathbb R\text{.}\)
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Range: all non-negative real numbers \([0, \infty).\)
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Derivative:
\begin{equation*} f'(x) = 2x, \end{equation*}
this can give an idea of slope or critical points in a function.
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