help_needed 3 years ago Find the linear approximation of the function f(x) = sqrt of (1 − x) at a = 0. Use L(x) to approximate the numbers sqrt of 0.9 and sqrt of 0.99. .

1. help_needed

I already found L(X) = -1/2X + 1. Idk what do do after that.

2. myininaya

$\sqrt{1-x} \approx \frac{-1}{2}x+1$ $\sqrt{0.9}=\sqrt{1-0.1} \approx \frac{-1}{2}(0.1)+1$

3. help_needed

Srry but where did u get sqrt of 1-0.1 from if x = 0.9

4. myininaya

.9=1-0.1

5. myininaya

I was trying to find out what x was

6. myininaya

x is 0.1 since we want to approximate sqrt(0.9)

7. help_needed

Ok so for sqrt of 0.99 it wud be $\sqrt{0.99} = \sqrt{1- 0.01} ~ -1/2 (0.01) + 1$

8. myininaya

You forgot the approximation symbol btw sqrt(1-0.01) and -1/2*(.01)+1 Also you should simplify -1/2*.01+1

9. help_needed

so the approximation is 0.995 right?

10. help_needed

So what is the general formula to write approxiamtions?

11. myininaya

$\frac{-1}{2} \cdot \frac{1}{100}+1 =\frac{-1}{2} \cdot \frac{1}{100}+\frac{200}{200}=\frac{200-1}{200}=\frac{199}{200} \text{ or } =.995$ So yes for sqrt(.99), that is a good approximation. To write linear approximations: Say we have a curve y=f(x) and we want to find the linear approximation at x=a for the numbers m. So we have: $\text{ lines have this form: } y=mx+b$ Well we can find the general slope (aka derivative) for this curve. $y'=f'(x)$ We actually wanted to know the slope at x=a That slope would in fact by y' evaluated at x=a So the slope of this tangent line is: $y=f'(a)x+b$ We still need to find the y-intercept, b. We do know a point on this line (a, f(a)) We can input this point in to find b. So we have: $f(a)=f'(a) \cdot a +b$ $f(a)-f'(a) \cdot a =b$ So the tangent line to y=f(x) at x=a Or the equation we will use for linear approximations is: $y=f'(a)x+f(a)-f'(a) \cdot a$ or you might prefer to write it as: $y=f'(a) \cdot (x-a) +f(a)$ So anyways this is the linear approximation to y=f(x) for values near x=a. The approximations will get nastier the further your x is away from a. So we say: $f(x) \approx f'(a)(x-a)+f(a) \text{ for values of x near x }$ This is the general form for linear approximations.

12. help_needed

Are u typing a reply? or is this a glitch?

13. myininaya

That slope m and that number m we want to use linear approximations on aren't the same. lol.

14. help_needed

WOW THANK U SOO MUCH FOR SUCH A thorugh explanation :D

15. myininaya

Let's say we want to approximate n instead of m. So we have that $n=f( * ) \approx f'(a)(*-a)+f(a)$ What we input to approximate n depends on what the function is.

16. help_needed

the star * symbol/ what does that represent in tha equation above

17. help_needed

is it x?

18. myininaya

* is the number I chose to show you that we are not inputting n but we are inputting in a value so that the function is the same as n for some value of x. Here I just chose that value of x to be *.

19. help_needed

Oh I SEE. This makes alot more sense now. Thank u soo much for putting so much time into heloing me with this question . I appreciate it.

20. myininaya

Np. I like linear approximations.

21. help_needed

Thank u :) I might need help with more just b/c sum I find hard to differentiate

22. myininaya

k. Post em separately just in case I'm away. I have chores to do :(.

23. help_needed

NO PROBLEM :)

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