Difference between revisions of "Tangent-squared function"
From Calculus
(Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function is defined as the composite of ...") |
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| interval description based on increase/decrease and concave up/down || For each integer <math>n</math>:<br>decreasing and concave up from <math>n\pi - \pi/2</math> to <math>n\pi</math><br>increasing and concave up from <math>n\pi</math> to <math>n\pi + \pi/2</math>. | | interval description based on increase/decrease and concave up/down || For each integer <math>n</math>:<br>decreasing and concave up from <math>n\pi - \pi/2</math> to <math>n\pi</math><br>increasing and concave up from <math>n\pi</math> to <math>n\pi + \pi/2</math>. | ||
|} | |} | ||
+ | |||
+ | ==Differentiation== | ||
+ | |||
+ | ===First derivative=== | ||
+ | |||
+ | The first derivative can be computed by combining the [[chain rule for differentiation]] and knowledge of the derivatives of the [[square function]] and the [[tangent function]]: | ||
+ | |||
+ | <math>\frac{d}{dx}[\tan^2x] = \frac{d}{d(\tan x)}[\tan^2x] \frac{d(\tan x)}{dx} = (2\tan x)(\sec^2 x)</math> | ||
+ | |||
+ | ==Integration== | ||
+ | |||
+ | ===First antiderivative=== | ||
+ | |||
+ | We use the identity: | ||
+ | |||
+ | <math>\tan^2x + 1 = \sec^2x</math> | ||
+ | |||
+ | Using this, we rewrite: | ||
+ | |||
+ | <math>\int \tan^2x \, dx = \int (\sec^2x - 1) \, dx = \int \sec^2 x \, dx - \int 1 \, dx = \tan x - x + C</math> | ||
+ | |||
+ | where we use that the [[tangent function]] is an antiderivative for the [[secant-squared function]] | ||
+ | |||
+ | ===Second antiderivative=== | ||
+ | |||
+ | We can antidifferentiate the function one more time: | ||
+ | |||
+ | <math>\int \int \tan^2x \, dx \, dx = \int (\tan x - x + C)\, dx = -\ln|\cos x| - x^2/2 + Cx + C_0</math> |
Revision as of 22:48, 12 February 2012
This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki
For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle of is measured as .
Contents
Definition
This function is defined as the composite of the square function and the tangent function. Explicitly, it is the function:
is written as for shorthand.
Key data
Item | Value |
---|---|
Default domain | all real numbers except the odd integer multiples of . |
range | , i.e., . All nonnegative reals. no absolute maximum value; absolute minimum value of 0 |
period | , i.e., . |
local maximum values and points of attainment | No local maximum values |
local minimum values and points of attainment | 0 at all integer multiples of . |
points of inflection (both coordinates) | None |
vertical asymptotes | at all odd multiples of , with the function going to from both directions in each case. |
derivative | |
antiderivative | . Note that the value must be constant within each interval between consecutive odd multiples of , but can be different in different intervals. The domain overall is not connected. |
interval description based on increase/decrease and concave up/down | For each integer : decreasing and concave up from to increasing and concave up from to . |
Differentiation
First derivative
The first derivative can be computed by combining the chain rule for differentiation and knowledge of the derivatives of the square function and the tangent function:
Integration
First antiderivative
We use the identity:
Using this, we rewrite:
where we use that the tangent function is an antiderivative for the secant-squared function
Second antiderivative
We can antidifferentiate the function one more time: