4.5-Linear Approximation

Question 1

If we zoom in on a point with a tangent line to it

Answer 1

We see that the curve of the function looks more and more like the tangent line

Question 2

So let’s say there is a point a,f(a). The slope of the tangent line is f’(a) so then the point slope formula for this line is

Answer 2

y-f(a)= f’(a)(x-a)
y=f(a)+f’(a)(x-a)

This is the Linear Approximation

Question 3

How can you rewrite the Linear Approximation Formula to find changes in y. Let’s say a problem asked
Approximate the change in y where y= f(x)= x^9-2x+1 when x changes from 1.00 to 1.05

Answer 3

L(x)=f(x)=f(a)+f'(a)(x-a)
rewrite
f(x)-f(a)=f'(a)(x-a)
^^^this is the change in y. (x-a) is change in x soooo
delta=>
>y=f'(a)>x
delta y equals f prime times delta x

Question 4

To approimxate f near x=a use

to approximate the change in y in the dependent variable when x changes from a to a +>x use

Answer 4

L(x)

delta y=

Question 5

Prove the differential formula

Answer 5

page 235

Question 6

Definition of differential

Answer 6

a small change in x is denoted by the DIFFERENTIAL dx. The corresponding change in f is approximated by the differential
dy= f’(x)dx, that is

delta y= f(x+dx)-f(x)=dy=f’(x)dx.