Question

Please help!!!! Which tangent line gives a better approximation x=1? f(x)=-x^(3)-2x^(2)+x+1 and g(x)=ln(x+1)+1?

Answer 1

The equation for the first one is 3x^(2)-4x+1 and the second is (1/x+1).

Answer 2

The tangent line equation for the first one is y=x+1 and it is the same for the second one y=x+1.

Answer 3

How do I see which tangent line gives a better approximation at x=1? Thanks!

Answer 4

f(x)=-x^(3)-2x^(2)+x+1 and g(x)=ln(x+1)+1

f'(x)=-3x^(2)-4x+1 and g'(x)=1/(x+1)

f'(1) = -6 g'(1) = 1/2

im not sure how you can to the conclusion that they have the same tangent lines since the clearly have different slopes

Answer 5

The question part b said "Show that g(x) has the same tangent line as f(x) at x=0".

Answer 6

ah, at x=0 ...

Answer 7

Yeah, both at x=0.

Answer 8

f(0)=1 and g(0)=1

and the derivatives have the same slope at x=0

tanf = tang = x+1

Answer 9

at x=1 we have a linear approximation of 2 right?

Answer 10

Yeah!

Answer 11

now we just compare this to the actual values to see which is the better approximation

Answer 12

what is f(1) and g(1)

Answer 13

f(1)= -1

Answer 14

so at some error:

tanf(1) + error = f(1)

2 + error = -1

error = -3, of as a distance, we are off by 3

compare tang(1) + error = g(1) and define the error for it

Answer 15

Does that mean that it's not a good approx.?

Answer 16

dunno, we are seeing which function it approximates better, so the one with the smallest error wins

Answer 17

I thought somewhere we had to use f(a)+f'(a)(x-a) for linear approximations?

Answer 18

we did

Answer 19

thats how we made the tangent lines

Answer 20

tanf = f(0)+f'(0)(x-0)

tang = g(0)+g'(0)(x-0)

Answer 21

Oh okay. So at x=1 it is supposed to be close to 2?

Answer 22

no, at x=1, the approximation of the tangent line is 2

the functions will be whatever they are going to be. linear approximations are only so good for a limited interval ... at least as far as an error rate is concerned

Answer 23

all we are doing in this is seeing how which function has the better approximation is all

Answer 24

I'm sorry. I was really confused!

Answer 25

does it makes sense yet?

Answer 26

at x=0, everything is great ....

as we move away from 0, the curves bend away from the line. one curves further away than the other.

Answer 27

could I graph this on a calculator to see?

Answer 28

http://www.wolframalpha.com/input/?i=y%3D-x%5E3-2x%5E2%2Bx%2B1%2C+y%3Dln%28x%2B1%29%2B1%2C+y%3Dx%2B1

of course

Answer 29

So the ln(x+1)+1 function better approximates at x=1?

Answer 30

the tangent line better approximates ln(x+1)+1 at x=1, yes it has a smaller error

Answer 31

Okay! I get it now. Thank you. I think I will just graph it from now on to understand!

Answer 32

if that helps :) good luck