# Equation of the best quadratic approximation?

• Nov 15th 2009, 09:01 PM
Rumor
Equation of the best quadratic approximation?
I'm not sure what to do for this. Here's the problem:

"(a) Find the equation of the best quadratic approximation to $y=ln(x)$ at x=1. The best quadratic approximation has the same first and second derivatives as $y=ln(x)$ at x=1.

(b) Use a computer or calculator to graph the approximation and $y=ln(x)$ on the same set of axes. What do you notice?

(c) Use your quadratic approximation to calculate approximate values for ln(1.1) and ln(2)"

Parts b and c, I don't doubt that they'd be easy after getting part a.
So could someone help me with part a?
• Nov 16th 2009, 02:08 AM
CaptainBlack
Quote:

Originally Posted by Rumor
I'm not sure what to do for this. Here's the problem:

"(a) Find the equation of the best quadratic approximation to $y=ln(x)$ at x=1. The best quadratic approximation has the same first and second derivatives as $y=ln(x)$ at x=1.

(b) Use a computer or calculator to graph the approximation and $y=ln(x)$ on the same set of axes. What do you notice?

(c) Use your quadratic approximation to calculate approximate values for ln(1.1) and ln(2)"

Parts b and c, I don't doubt that they'd be easy after getting part a.
So could someone help me with part a?

Taylor polynomial:

$
f(x)\approx P_2(x)=f(x_0)+(x-x_0)f'(x_0)+\frac{(x-x_0)^2}{2}f''(x_0)
$

CB