Section 21: Errors in linear approximation
Problem types
- First Taylor Polynomials
- Error Bounds
Taylor Polynomials:
For the following functions, find the first Taylor polynomial $T_1(x)$ for the given $f(x)$ and $b$:
- $\frac{1}{1-x}$ at $b = 0$
- $\tan x$ at $b = 0$
- $\cos (x - \pi/6)$ at $b = \pi/6$
- $\frac{1}{3-2x}$ at $b = 2$
Error Bounds
Use the tangent line error bound to bound the error $|f(x) - T_1(x)|$ on the given interval $I$.
- $\frac{1}{1-x}$ on $I = [-0.9,0.9]$
- $\tan x$ on $I = [-0.5,0.5]$
- $\cos (x - \pi/6)$ on $I = [\pi/6-a,\pi/6+a]$
- $\frac{1}{3-2x}$ on $I=[1.5,2.5]$
Error Bounds
Use the tangent line error bound to find an interval on which $|f(x) - T_1(x)|\lt 1/10$ near $b$.
- $\frac{1}{1-x}$ where $b = 0$
- $\tan x$ where $b = 0$
- $\cos (x - \pi/6)$ where $b = \pi/6$
- $\frac{1}{3-2x}$ where $b = 2$