FoldUnfold Table of Contents Rolle’s Theorem Rolle’s Theorem Theorem 1 (Rolle’s Theorem): If $f$ is a function that satisfies the following conditions: a) $f$ is a continuous function over the closed interval $[a, b]$. b) $f$ is differentiable on the open interval $(a, b)$. c) $f(a) = f(b)$. Then there exists a value $c \in […]

## The Extreme Value Theorem

FoldUnfold Table of Contents The Extreme Value Theorem Example 1 The Extreme Value Theorem Theorem 1 (The Extreme Value Theorem): If $f$ is a continuous function on the closed interval $[a, b]$, then $f$ contains both an absolute maximum and absolute minimum on $[a, b]$. Example 1 Find the absolute maximum values of $f(x) = […]

## The Existence of Roots Theorem

FoldUnfold Table of Contents The Existence of Roots Theorem Example 1 The Existence of Roots Theorem We now show an important application of The Intermediate Value Theorem often called the Existence of Roots Theorem. Under certain conditions we can guarantee the existence of a root. Theorem 1 (The Existence of Roots Theorem): If $f$ is […]

## The Intermediate Value Theorem

FoldUnfold Table of Contents The Intermediate Value Theorem Uses of The Intermediate Value Theorem Example 1 The Intermediate Value Theorem Theorem 1 (The Intermediate Value Theorem): Suppose that $f$ is a continuous function on the closed interval $[a, b]$ where $a . Then for all $p$ such that $f(a) or $f(b) , there exists a […]

## Linear Approximation of Single Variable Functions

FoldUnfold Table of Contents Linear Approximation of Single Variable Functions Example 1 Linear Approximation of Single Variable Functions Suppose that we have a function $f$, and we take a look at the tangent line at point $(a, f(a))$ on $f$. As we zoom in closer and closer towards $(a, f(a))$, we notice that $f$ begins […]

## The Second Derivative Test

FoldUnfold Table of Contents The Second Derivative Test Example 1 Example 2 The Second Derivative Test We have already looked at one way to determine whether a critical number $c$ is a local maximum, minimum, or neither by the The First Derivative Test. Sometimes it may be difficult to compute values to the left and […]

## The First Derivative Test

FoldUnfold Table of Contents The First Derivative Test Example 1 Example 2 The First Derivative Test The first derivative test is a strategy that allows us to determine whether a critical number $c$ is either an extrema (maximum or minimum), or neither. It is important to be familiar with Concavity and Inflection Points as well […]

## Concavity of a Function

FoldUnfold Table of Contents Concavity of a Function Inflection Points of Functions Concavity of a Function Before we describe what an inflection point is, it is first important to describe what it means for a curve to be “concave up” or “concave down“. Definition: A function $f$ is described to be Concave Up on the […]

## Fermat’s Theorem for Extrema

FoldUnfold Table of Contents Fermat’s Theorem for Extrema Fermat’s Theorem for Extrema When looking at Local Maxima and Minima, and Absolute Maxima and Minima, we assumed that if the point $(a, f(a))$ is an extreme (that is, the point is either a local maximum or local minimum), then $f'(a) = 0$. We will now prove […]

## Local Maxima and Minima

FoldUnfold Table of Contents Local Maxima and Minima Example 1 Absolute Maxima and Minima Example 2 Local Maxima and Minima Definition: Suppose that $f$ is a function and $D(f)$ is the domain of $f$. Let $a \in D(f)$. We say that $f(a)$ is a Local (Relative) Maximum Value on $f$ or a Local (Relative) Maxima […]

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