Easy math problem, just want to ask a few q's?

Question

The question is this:

Find the linearization (linear approximation) of f(x) = ln(x) at a =1, then use it to approximate ln (0.98) and ln (1.05). Draw the linearization at a=1, are your approximations over or under approximations? Check on your calculator, how close are you?


I did the first part, I got y=x as the linear approximation. I just don't know how to use this to approximate ln (0.98) and ln (1.05). Would it be 0.98 and 1.05? I'm so confused. Please help! All work should be shown since I need to understand this completely and not just know the answer. Thanks in advance!

Also, I forgot the other question. It is as follows:

2. What function would you use to create a linear approximation for (8.06) ^ (2/3)? Approximate the value using linear approximation or differentials.

Now, I'm thinking that we know the value for 8^ (2/3) and it would just be (8 + x) ^ (2/3) but I'm not sure if this is right and what to do from here

Answer 1

The linearization formula is

y - f(a) = f'(a)(x-a)

Here f(x) = ln(x) and a=1, so the linear approximation is

y - ln(1) = (1/1)(x-1) or

y = x-1

So ln(0.98) ≈ 0.98 - 1 = -0.02 and ln(1.05) ≈ 1.05 - 1 = 0.05

My calculator gives -0.020202707 for ln(0.98) and 0.48790164 for ln(1.05). So the relative error in ln(0.98) is about 1%, and the relative error in ln(1.05) is about 2½%
(about what we expect, since 1.05 is 2½ times as far from 1 as 0.98 is). In both cases the approximations are overapproximations. If you graph y = ln(x) and draw the approximating line y = x-1 (which is tangent to the graph of y = ln(x) at (1,0)) and compare the two graphs at x=0.98 and x=1.05 you will see why.

For the second problem, y = (8+x)^(2/3) would work, but I think they want a simpler function, like y = x^(2/3) with a=8

Answer 2

y = x-2/x² + 5 u = 2/x² du/dx = [2]' * x² - 2[x²] ------------------ = -2/x^4 '' '' [2]' potential the derivant of two (x²)² dy/dx = a million+2/x^4 dy/dx = 0 a million+2/x^4 =0 2/x^4 = -a million -x^4 = 2 x^4 = -2 No solutions, because of the fact x² > 0 and so is x^4>0 and so forth (x^2n>0, with N being any organic type) Assuming you meant: Y=x !>+

Answer 3

im pretty sure you subtract 8 from both side and end up with x^-22/3