Finding Tangent Curves
Inhaltsverzeichnis
- Let's Polynomial Long Divide!
- A Quadratic Polynomial
- A Cubic Polynomial
- Let's Taylor Series Expand!
- Taylor Polynomials
- Back to that Quadratic
- Back to that Cubic
- Let's Conjecture
- Quartic Conjectures
A Quadratic Polynomial
1. Consider the polynomial [math]f\left(x\right)=x^2-5x+2[/math]. Let [math]r_1\left(x\right)[/math] be the remainder when we divide [math]f\left(x\right)[/math] by [math]\left(x-1\right)[/math]. That is, [math]r_1\left(x\right)[/math] satisfies [math]\frac{f\left(x\right)}{x-1}=q\left(x\right)+\frac{r_1\left(x\right)}{x-1}[/math] where the degree of [math]r_1\left(x\right)[/math] is less than the degree of the divisor, [math]\left(x-1\right)[/math] . Find and graph [math]r_1\left(x\right)[/math] on the set of axes below.
2. Now, find the remainder [math]r_2\left(x\right)[/math] that results from dividing [math]f\left(x\right)[/math] by [math]\left(x-1\right)\left(x-1\right)[/math]. Graph [math]r_2\left(x\right)[/math] on the set of axes below.
3. What do you notice about each remainder at the point P?
4. What do you think the remainder graph would have looked like if we considered [math]\frac{f\left(x\right)}{x^2-4x+4}[/math]?
Taylor Polynomials
Define the [math]n[/math]th [b]Taylor polynomial [math]T_n[/math] of [math]f[/math] centered at [math]x=a[/math][/b] in the following way:[br][br][math]T_n\left(x\right)=f\left(a\right)+\frac{f'\left(a\right)}{1!}\left(x-a\right)+\frac{f''\left(a\right)}{2!}\left(x-a\right)^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}\left(x-a\right)^n[/math].[br][br]
1. Let [math]f\left(x\right)[/math] be a polynomial of degree [math]n[/math]. What is the degree of [math]T_m\left(x\right)[/math] if [math]0\le m\le n[/math]?
2. Let [math]f\left(x\right)[/math] be a polynomial of degree 2. What do you expect the graphs of [math]T_0[/math], [math]T_1[/math], and [math]T_2[/math] to look like?
3. If [math]f\left(x\right)[/math] is a polynomial of degree 3, what behavior will the graphs of [math]T_0[/math], [math]T_1[/math], and [math]T_2[/math] have near the point [math]\left(a,f\left(a\right)\right)[/math]?
Quartic Conjectures
Consider the function [math]f\left(x\right)=x^4-2x^2[/math].
1. Find the tangent line to [math]f\left(x\right)[/math] at the point [math]x=-1[/math]. Can you do this in multiple ways?
2. Find a quadratic function whose graph is tangent to the graph of [math]f\left(x\right)[/math] at the point [math]x=-1[/math].
3. Find a cubic function whose graph is tangent to the graph of [math]f\left(x\right)[/math] at [math]x=-1[/math].
4. If dividing [math]f\left(x\right)[/math] by [math]\left(x+1\right)^2[/math] gives a remainder whose curve is tangent to [math]f\left(x\right)[/math] at [math]x=-1[/math], what do you think I must divide [math]f\left(x\right)[/math] by in order to find a curve that is tangent to [math]f\left(x\right)[/math] at the points [math]x=-1[/math] AND at [math]x=3[/math]? Find such a polynomial and graph it below.