Question

Consider the function f(x)=cosh(3x). For the following questions, express your answers in terms of exponential functions.
a) Find the local linearization of f(x) near x=1
b) Does the approximation overestimate or underestimate f(2)?

I really have no idea where to start with this.

Answer 1

How I would start this and Im not 100% sure but i would plug 1 into x

Answer 2

It should look like cos(3*1) then plug that into your calculator for part A
For part B take and do the same thing except with the number 2

Answer 3

this is how u start:
linear approximation
\[\Delta y= f'(x_0) \Delta x\]

Answer 4

So, would I need to differentiate cosh(x) first?

Answer 5

yep

Answer 6

So that would come to \[\frac{ e^x-e^-x }{ 2 }\] ?

Answer 7

i'm sorry xD i dont remember derivates of cosh or anything about hyperbolic functions,
but i know what the question is asking for, i can help you with the steps

Answer 8

okay i know I got my derivative wrong, it is supposed to be a + instead of -, but what would I do from here?

Answer 9

you evaluate f '(x) at x_0,
here it says linearize near x=1, so x_0=1
substitute, get a number for f '(x_0)

Answer 10

@Revircs
following?

Answer 11

yeah, im just writing everything down for notes, sorry

Answer 12

no prob :)

Answer 13

so after evaluating at x=1, I would get to \[\frac{ e^1+e^-1 }{ 2 }\]

Answer 14

or \[\frac{ e+1/e }{ 2 }\]

Answer 15

ok, you can leave it that way for now, but later use a calculator, and actually find a number

ok so now were done with f ' ( )
the question asks you to find f (2) using the linear approx, \(\Delta x\) represents change in x, so how much has x changed when we move from f(1) to f(2) ?

Answer 16

oh, okay. my professors usually have us leave it in exact form. So from there I would evaluate at f(x)=2 and find the difference?

Answer 17

noo...
you should realize that \(\Delta x=1\)
substitute, f '(1) and \(\Delta x =1\) to find \(\Delta y \)

Answer 18

oh, so since the change in x is just 1, the change in y would be the same as the derivative in this case?

Answer 19

yes

Answer 20

i hope you are catching on to what linear approximation means,

essentially, we are assuming that the graph of f(x) between the points x=1 and x=2 is a straight line with the slope f ' (1)

Answer 21

* sorry the two points marked on the x axis are 1 and 2