Question

help with linearization

Answer 1

FIrst, why is this linearization in the first place?

Answer 2

Linearization is to calculate an approximate value of a function when an adjacent (exact) value is known. The method is to approximate the curve by a tangent line (linear), that is where the name comes from. Differentiation is used to find the slope of the tangent at the known point.
Examples of application are finding the cube roots of numbers near a known cube root, such as 8.1, or 126, etc. It could also be used to find (possibly mentally) the sine of 31 degrees.
For more descriptions and examples, follow the following link, note especially the diagram showing a function and the tangent line. This helps you visualize the concept of linearization.
https://en.wikipedia.org/wiki/Linearization

Example:
find square root of 82.5.
Let
y(x)=sqrt(x), y'(x)=(1/2)(1/sqrt(x))
We know that 9^2=81, which is near 82.5, so use x0=81
y'(x0)=(1/2)(1/sqrt(81)=1/18
By differentials,
y(x0+1.5)=y(0)+1.5*y'(x0)=9+1.5/18=9+1/12=9.0833
Exact value = 9.0829.
So linearization estimates sqrt(82.5) with an error of (9.0833-9.0829)/9.0829=0.000042, or 0.0042% overestimated.

Answer 3

I need help with the specific question

Answer 4

@usser123
Please proceed according to above instructions to as far as you can go, then we get to know where your difficulty/error is.

Answer 5

you will use a linear approximation

Answer 6

find the slope at 4

Answer 7

y=mx+b, then use your line equation and find what y=m(x+0.02) +b is