Local Linearity
Includes defining tangents, differentiability, local linearity, and linearization (or tangent line approximation).
Table of Contents
- Tangents
- Tangents, defined poorly
- Tangents, redefined for calculus
- Local Linearity
- Differentiability and Local Linearity
- Linearization (or Tangent Line Approximation)
- Linearization and Tangent Line Approximation
Tangents, defined poorly
You first encountered the notion of a tangent in geometry. I found the following definition of a tangent in a geometry textbook, which is probably the definition you've been carrying around in your head ever since. [br][quote]A [b]tangent [/b]to the circle [math]C[/math] is a line that contains exactly one point of [math]C[/math].[/quote][br]It is natural to try to generalize this definition to suit our purposes in calculus.
Bad Definition #1
A [b]tangent [/b]to the function [math]f[/math] is a line that contains exactly one point of [math]f[/math].[br][br]This is no good. Below, plot:[br][list][*]a function,[/*][*]a point on the function, and [/*][*]the line tangent to the function at that point [/*][/list]such that the tangent contains more than one point of the function.
Bad Definition #2 (which isn't all that bad; it just fails to describe what it means for two points to be "infinitely close")
Calculus was famously independently invented by two mathematicians, Isaac Newton and Gottfried Leibniz, in the 1600s. Leibniz described a tangent line as a line passing between two "infinitely close" points on a curve. But what does that mean?
What mathematical tool do we use to describe two things as "infinitely close"? How does Leibniz's description relate to the definition of the derivative?
Bad Definition #3
The 1828 edition of Webster's Dictionary defines a tangent as a "line which touches a curve, but which when produced, does not cut it".[br][br]This too is no good. If we were to accept this definition, functions would not have tangent lines at their inflection points. Below, plot:[br][list][*]a function,[/*][*]an inflection point of the function, and [/*][*]the line tangent to the function at that inflection point,[/*][/list]and notice that the tangent line "cuts" the function.
Differentiability and Local Linearity
Differentiable functions have a property called [i]local linearity[/i], meaning that if [math]f[/math] is differentiable around [math]x=a[/math] and you zoom in far enough on [math](a,f(a))[/math], [math]f[/math] will appear linear.[br][br]Use the applet below to observe this zooming for several different functions, including some with points where the function is not differentiable.
Were you able to find an example of a function that did not have the property of local linearity at a point? If so, which function (and which point)?
All differentiable functions have the property of local linearity, but so do some non-differentiable functions. Name one (and a corresponding point).
Linearization and Tangent Line Approximation
Here's some vocabulary:[br][quote]The line tangent to a function [math]f[/math] (that's differentiable) at [math]x=a[/math] is also called the [i]linearization [/i]of [math]f[/math] at [math]a[/math]. [/quote]You can use the linearization of a function [math]f[/math] at [math]a[/math] to approximate values of [math]f[/math] near [math]x=a[/math]. This technique is also called [i]tangent line approximation[/i].[br][br]Here's an example. Suppose someone asks you to estimate [math]\sqrt{125}[/math] (without a calculator).
Without using any calculus, you know [math]\sqrt{125}[/math] is a number between which two consecutive integers? Why?
But with calculus, we can do much better than that, even in our heads (or at least with paper and pencil only).[br][br]We're going to use the linearization of [math]f(x)=\sqrt{x}[/math] at [math]121[/math] (because [math]121[/math] is the nearest perfect square to [math]125[/math]). Determine this linearization and input your answer in the applet below as the function [math]l(x)[/math].[br]
How does what you observed in the applet related to Good Definition #2 of a tangent?
In other words, plugging [math]125[/math] into [math]l(x)[/math] (which you can do without a calculator) should yield a result that is extremely close to [math]\sqrt{125}[/math] because [math]l(125)\approx f\left(125\right)[/math].[br][br]Use the scientific calculator below to compute these two quantities.
Use the linearization of [math]f(x)=x^2[/math] at [math]3[/math] to approximate [math]\pi^2[/math].
Use tangent line approximation to estimate the cube root of [math]\sqrt[3]{30}[/math]. (This means you'll have to determine what function and linearization to use.)