Question

Problem Set 3: 2A-12d: Give the quadratic approximation for y=ln(cosx).

I understand how the solution is -(1/2)x^2 (because L(ln(1+u))=u), but when I first attempted the problem, I did the following:

y=ln(cosx)
e^y=e^(ln(cosx))
e^y=cosx

approximating,
1+y+(1/2)y^2=1-(1/2)x^2
y=-(1/2)x^2-(1/2)y^2

plugging in ln(cosx) for y,
Q(y)=-(1/2)x^2-(1/2)(ln(cosx))^2

This solution seems to be a more accurate (although uglier-looking) approximation of y=ln(cosx) when x is near 0. Looking for insight as to if it is incorrect and if so, why?

Answer 1

actually the formula for quadratic approximation is :

f(x) = f(0) + f ' (0) * x + f ''(0) (x^2)/2

For, y = ln (cos x)

f(0) = 0

f ' (0)= 0

and, f '' (0)= -1

now, substitute to get , y = - 1/2 x^2

Answer 2

I should have mentioned that the question suggests that you solve the problem algebraically using quadratic approximation formulas for basic functions, for instance I used Q(e^x)=1+x+(1/2)x^2 and Q(cosx)=1-(1/2)x^2. edit: I know Q(y)=-(1/2)x^2-(1/2)(ln(cosx))^2 is the wrong answer, which is evident by using the actual quadratic approximation formula, but I don't see why I cannot also solve the problem algebraically using Q(e^x) and Q(cosx).

Answer 3

Okay. So remember that ln (x) = ln [1+(x-1)]

Using quadratic approximation,
ln [1+(x-1)]= (x-1) - (x-1)^2 /2 = 2x - x^2 /2 -3/2-----(1)

so, ln (cos x) = ln [ 1+ ( cos x -1)]

hence, ln (cos x) = 2cos x -1/2 cos^2 x -3/2 [using equation 1]

= 2 (1 - x^2/2)- 1/2 ( 1- x^2 /2) ^2 - 3/2 [using approximation for cos x]
= 2 - x^2 - 1/2 ( 1- x^2)- 3/2 [ we don't need to write terms involving higher powers of x than x^2]
= 2- x^2 - 1/2- x^2 /2-3/2
= -1/2 x^2

In this, only the quadratic approximation formula for ln (1+x) and cos x are used.

Answer 4

The problem with your "Q(y)" approximation is that the term "(1/2)(ln(cosx))^2" is either higher than degree 2 or equal to 0 (depend on whether you choose to approximate ln(cosx) linearly or quadraticly). So, it will be negligible as you already learned from the lecture. Also, the fact that you involve "(1/2)(ln(cosx))^2" as part of your answer for the approximation of ln(cosx) is illogical because if Q(y) is correct, then it's like saying in order to approximate ln(cosx) you had to already know the value of (ln(cosx))^2.