Grade 12
Table of Contents
- Definite integral
- Upper and lower Riemann Sums
- Definite Integral - Upper and Lower Sum
- Cont in uity
- Inverse Trigonometric functions
- Range of Inverse Trigonometric Functions
- Trigonometric Functions and its Inverses
- Inverse trig functions
- The Cosecant Inverse Function
- The Secant Inverse Function
- The Arccotangent Function
- The Inverse Sine Function
Upper and lower Riemann Sums
This applet shows how upper and lower Riemann sums can approximate an integral [math]\int^b_a f(x) \, \textrm{d}x[/math][br]Further, they show that as the number of strips [math]n[/math] increases, the Riemann sums converge to true value of the definite integral.[br]Input your own function into the textbox and set the limits to different values.
Range of Inverse Trigonometric Functions
Description
In this applet you adjust the output range of inverse trigonometric relations so that only one [math]y[/math] value exist for each [math]x[/math] value. This makes them inverse trigonometric functions.
Explorations
Select each trigonometric function on the left drop down. On the right side adjust the minimum and maximum allowed output values so there is only one output for each input. This range is shown with the brown curve. Note the corresponding domain of the function on the left is highlighted in brown to match.[br][br]Some ranges are better than other ranges, especially tangent and cotangent functions. Mathematicians have selected ranges for the inverse trigonometric functions that are very standard. These ranges are either [math]\left[-\frac{\pi}{2},\frac{\pi}{2}\right][/math] or [math]\left[0,\pi\right][/math]. Which functions work best with each range and why?[br][br]The inverse functions with standard output ranges are prefixed with "arc". For example, the inverse function of [math]\sin\left(x\right)[/math] may be written as [math]\sin^{-1}\left(x\right)\text{ , }\arcsin\left(x\right)\text{ or asin}\left(x\right)[/math]. Note that [math]\sin^{-1}\left(x\right)\ne\frac{1}{sin\left(x\right)}[/math] which may be easily misinterpreted since [math]\sin^2\left(x\right)=\left(\sin\left(x\right)\right)^2[/math].