## anonymous 3 years ago Find the linear approximation to f(x) at x = x o. Graph the function and its linear approximation. f(x) = sin 3x, x 0, = 0

1. anonymous

2. anonymous

this is the same as "find the equation of the line tangent to the graph of $$y=\sin(3x)$$ at $$(0,0)$$"

3. anonymous

take the derivative, replace $$x$$ by 0 for the slope (you will get 3) and then use the point slope formula

4. anonymous

since the point is $$(0,0)$$ the equation of the line will be $$y=3x$$

5. anonymous

ok so find the derivative of sin 3x right?

6. anonymous

yes

7. anonymous

Ok so i got 3cos(3x)

8. anonymous

yes now evaluate at $$x=0$$ to find your slope

9. anonymous

ok i got 3

10. anonymous

yes

11. anonymous

that is the slope of your line

12. anonymous

okay so now i have to use the point slop formula?

13. anonymous

yes

14. anonymous

but the point is $$(0,0)$$ so it is just $$y=3x$$

15. anonymous

okay so what do i do next?

16. anonymous

you are done

17. anonymous

okay

18. anonymous

linear approximation of f(x) at x=x0: $f(x)=f(x_0)+(x-x_0)f'(x_0)$

19. anonymous

you can use this way too. (notice the resemblance to equation of line)