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Boundless Calculus
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Chapter 1

Building Blocks of Calculus

Book Version 1
By Boundless
Boundless Calculus
Calculus
by Boundless
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Section 1
Precalculus Review
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Real Numbers, Functions, and Graphs

Functions relate a set of inputs to a set of outputs such that each input is related to exactly one output. Graphs can be used to represent these relationships pictorially.

Linear and Quadratic Functions

Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.

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Trigonometric Functions

Trigonometric functions are functions of an angle, relating the angles of a triangle to the lengths of the sides of a triangle.

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Inverse Functions

An inverse function is a function that undoes another function: For a function $f(x)=y$ the inverse function, if it exists, is given as $g(y)= x$.

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Exponential and Logarithmic Functions

Both exponential and logarithmic functions are widely used in scientific and engineering applications.

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Using Calculators and Computers

For numerical calculations and graphing, scientific calculators and personal computers are commonly used in classes and laboratories.

Section 2
Functions and Models
Four Ways to Represent a Function

Functions can be expressed in four different ways.

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Essential Functions for Mathematical Modeling

Mathematical models are used to explain systems, study effects of components, and make predictions about behavior.

Section 3
Limits
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Tangent and Velocity Problems

Iinstantaneous velocity can be obtained from a position-time curve of a moving object.

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Limit of a Function

The limit of a function is a fundamental concept in calculus and analysis concerning the behavior of a function near a particular input.

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Calculating Limits Using the Limit Laws

Limits of functions can often be determined using simple laws, such as L'Hôpital's rule and squeeze theorem.

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Precise Definition of a Limit

The $(\varepsilon,\delta)$-definition of limit (the "epsilon-delta definition") is a formalization of the notion of limit.

Continuity

A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.

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Finding Limits Algebraically

For a real-valued function expressed in terms of other functions, limit values may be computed via algebraic operations.

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Trigonometric Limits

There are several limits of special interest involving trigonometric functions.

Intermediate Value Theorem

For a real-valued continuous function $f$ on the interval $[a,b]$ and a number $u$ between $f(a)$ and $f(b)$, there is a $c \in [a,b]$ such that $f(c)=u$.

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Infinite Limits

Limits involving infinity can be formally defined using a slight variation of the $(\varepsilon, \delta)$-definition.

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Boundless Calculus by Boundless
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Chapter 1
Building Blocks of Calculus
  • Precalculus Review
  • Functions and Models
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Chapter 2
Derivatives and Integrals
  • Derivatives
  • Applications of Differentiation
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