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 […]

## Derivatives of Inverse Reciprocal Trigonometric Functions

FoldUnfold Table of Contents Derivatives of Inverse Reciprocal Trigonometric Functions Derivatives of Inverse Reciprocal Trigonometric Functions We will derive the derivatives of the inverse reciprocal trigonometric functions in the same manner that we derived the Derivatives of Inverse Trigonometric Functions by Implicit Differentiation. Theorem 1: The following functions have the following derivatives: a) If $f(x) […]

## Derivatives of Reciprocal Trigonometric Functions

FoldUnfold Table of Contents Derivatives of Reciprocal Trigonometric Functions Example 1 Derivatives of Reciprocal Trigonometric Functions We are going to look at even more derivative rules – this time for the reciprocal trigonometric functions. Theorem 1: The following functions have the following derivatives: a) If $f(x) = \sec x$, then $\frac{d}{dx} \sec x = \sec […]

## Derivatives of Inverse Trigonometric Functions

FoldUnfold Table of Contents Derivatives of Inverse Trigonometric Functions Example 1 Derivatives of Inverse Trigonometric Functions We will now begin to derive the derivatives of inverse trigonometric functions with basic trigonometry and Implicit Differentiation. Theorem 1: The following functions have the following derivatives: a) If $f(x) = \sin^{-1} x$, then $\frac{d}{dx} f(x) = \frac{1}{\sqrt{1 – […]

## Implicit Differentiation

FoldUnfold Table of Contents Implicit Differentiation Example 1 Example 2 Implicit Differentiation So far we have done calculus with explicit functions. That is we have a function $f$ explicitly in terms of x. For example, $f(x) = \sin x + 3x$ is a function explicitly in terms of $x$. However, sometimes we have a curve […]

## Higher Order Differentiation

FoldUnfold Table of Contents Higher Order Differentiation Example 1 Patterns in Higher Order Derivatives Higher Order Differentiation In some applications (soon to come up) and in the physical sciences such as physics, it may be necessary to find what are known as higher order derivatives. Higher order derivatives are simply derivatives of derivatives. A common […]

## Derivatives of Even and Odd Functions

FoldUnfold Table of Contents Derivatives of Even and Odd Functions Derivatives of Even and Odd Functions Recall that a function $f$ is said to be even on its domain if for every $x$ in the domain of $f$ we have that: (1) \begin{align} \quad f(x) = f(-x) \end{align} Similarly, a function $f$ is said to […]

- « Previous Page
- 1
- …
- 276
- 277
- 278
- 279
- 280
- …
- 295
- Next Page »