Question

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

Answer 1

@satellite73 can u help please?

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

ok so find the derivative of sin 3x right?

Answer 6

yes

Answer 7

Ok so i got 3cos(3x)

Answer 8

yes
now evaluate at \(x=0\) to find your slope

Answer 9

ok i got 3

Answer 10

yes

Answer 11

that is the slope of your line

Answer 12

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

Answer 13

yes

Answer 14

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

Answer 15

okay so what do i do next?

Answer 16

you are done

Answer 17

okay

Answer 18

linear approximation of f(x) at x=x0:
\[f(x)=f(x_0)+(x-x_0)f'(x_0)\]

Answer 19

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